CRAN Package Check Results for Package DPQ

Last updated on 2021-12-01 10:53:19 CET.

Flavor Version Tinstall Tcheck Ttotal Status Flags
r-devel-linux-x86_64-debian-clang 0.5-0 13.21 242.82 256.03 ERROR
r-devel-linux-x86_64-debian-gcc 0.5-0 12.20 195.34 207.54 ERROR
r-devel-linux-x86_64-fedora-clang 0.5-0 343.79 ERROR
r-devel-linux-x86_64-fedora-gcc 0.5-0 332.59 ERROR
r-devel-windows-x86_64-new-UL 0.5-0 38.00 345.00 383.00 OK
r-devel-windows-x86_64-new-TK 0.5-0 ERROR
r-devel-windows-x86_64-old 0.5-0 25.00 267.00 292.00 ERROR
r-patched-linux-x86_64 0.5-0 17.78 268.06 285.84 OK
r-patched-solaris-x86 0.5-0 507.70 OK
r-release-linux-x86_64 0.5-0 12.59 269.17 281.76 OK
r-release-macos-arm64 0.5-0 ERROR
r-release-macos-x86_64 0.5-0 OK
r-release-windows-ix86+x86_64 0.5-0 33.00 827.00 860.00 OK
r-oldrel-macos-x86_64 0.5-0 ERROR
r-oldrel-windows-ix86+x86_64 0.5-0 29.00 657.00 686.00 WARN

Additional issues

M1mac rchk

Check Details

Version: 0.5-0
Check: examples
Result: ERROR
    Running examples in 'DPQ-Ex.R' failed
    The error most likely occurred in:
    
    > base::assign(".ptime", proc.time(), pos = "CheckExEnv")
    > ### Name: ppoisson
    > ### Title: Direct Computation of 'ppois()' Poisson Distribution
    > ### Probabilities
    > ### Aliases: ppoisErr ppoisD
    > ### Keywords: distribution
    >
    > ### ** Examples
    >
    > (lams <- outer(c(1,2,5), 10^(0:3)))# 10^4 is already slow!
     [,1] [,2] [,3] [,4]
    [1,] 1 10 100 1000
    [2,] 2 20 200 2000
    [3,] 5 50 500 5000
    > system.time(e1 <- sapply(lams, ppoisErr))
     user system elapsed
     0.010 0.000 0.011
    > e1 / .Machine$double.eps
     [1] 0.0 0.5 -1.0 1.0 5.5 1.5 -4.0 -3.0 1.0 -1.0 2.0 2.0
    >
    > ## Try another 'ppFUN' :---------------------------------
    > ## this relies on the fact that it's *only* used on an 'x' of the form 0:M :
    > ppD0 <- function(x, lambda, all.from.0=TRUE)
    + cumsum(dpois(if(all.from.0) 0:x else x, lambda=lambda))
    > ## and test it:
    > p0 <- ppD0 ( 1000, lambda=10)
    > p1 <- ppois(0:1000, lambda=10)
    > stopifnot(all.equal(p0,p1, tol=8*.Machine$double.eps))
    >
    > system.time(p0.slow <- ppoisD(0:1000, lambda=10, all.from.0=FALSE))# not very slow, here
     user system elapsed
     0.005 0.000 0.005
    > p0.1 <- ppoisD(1000, lambda=10)
    > if(requireNamespace("Rmpfr")) {
    + ppoisMpfr <- function(x, lambda) cumsum(Rmpfr::dpois(x, lambda=lambda))
    + p0.best <- ppoisMpfr(0:1000, lambda = Rmpfr::mpfr(10, precBits = 256))
    + AllEq. <- Rmpfr::all.equal
    + AllEq <- function(target, current, ...)
    + AllEq.(target, current, ...,
    + formatFUN = function(x, ...) Rmpfr::format(x, digits = 9))
    + print(AllEq(p0.best, p0, tol = 0)) # 2.06e-18
    + print(AllEq(p0.best, p0.slow, tol = 0)) # the "worst" (4.44e-17)
    + print(AllEq(p0.best, p0.1, tol = 0)) # 1.08e-18
    + }
    Error in target == current : comparison of these types is not implemented
    Calls: print ... AllEq -> AllEq. -> AllEq. -> .local -> all.equal.numeric
    Execution halted
Flavor: r-devel-linux-x86_64-debian-clang

Version: 0.5-0
Check: tests
Result: ERROR
     Running 'chisq-nonc-ex.R' [35s/39s]
     Running 'dnbinom-tst.R' [19s/22s]
     Running 'dnchisq-tst.R' [0s/1s]
     Running 'hyper-dist-ex.R' [27s/31s]
     Running 'pnbeta-tst.R' [0s/1s]
     Running 'pnt-prec.R' [27s/32s]
     Running 'ppois-ex.R' [2s/2s]
     Running 'qPoisBinom-ex.R' [0s/1s]
     Running 'qbeta-dist.R' [12s/13s]
     Running 'qbeta-tst.R' [0s/1s]
     Running 'qgamma-ex.R' [11s/13s]
     Running 'stirlerr-tst.R' [2s/3s]
     Running 't-nonc-tst.R' [6s/6s]
     Running 'wienergerm-pchisq-tst.R' [0s/1s]
     Running 'wienergerm_nchisq.R' [7s/9s]
    Running the tests in 'tests/dnbinom-tst.R' failed.
    Complete output:
     > #### Testing 1) dbinom_raw(), dnbinomR() and dnbinom.mu()
     > #### 2) log1pmx(), logcf() etc
     > require(DPQ)
     Loading required package: DPQ
     > source(system.file(package="Matrix", "test-tools-1.R", mustWork=TRUE))
     Loading required package: tools
     > ## -> showProc.time(), assertError()
     >
     > (doExtras <- DPQ:::doExtras() && !grepl("valgrind", R.home()))
     [1] FALSE
     >
     > if(!dev.interactive(orNone=TRUE)) pdf("wienergerm-accuracy.pdf")
     >
     >
     > ### 1. Testing dbinom_raw(), dnbinomR() and dnbinom.mu() >>> ../R/dbinom-nbinom.R <<<
     > ### ---------- ../man/dbinom_raw.Rd & ../man/dnbinomR.Rd
     >
     > ## "FIXME:" use sfsmisc :: relErrV() already here
     >
     > ### dbinom() vs dbinom.raw() :
     >
     > for(n in 1:20) {
     + cat("n=",n," ")
     + for(x in 0:n)
     + cat(".")
     + for(p in c(0, .1, .5, .8, 1)) {
     + stopifnot(all.equal(dbinom_raw(x, n, p, q=1-p, log=FALSE),
     + dbinom (x, n, p, log=FALSE)),
     + all.equal(dbinom_raw(x, n, p, q=1-p, log =TRUE),
     + dbinom (x, n, p, log =TRUE)))
     + }
     + cat("\n")
     + }
     n= 1 ..
     n= 2 ...
     n= 3 ....
     n= 4 .....
     n= 5 ......
     n= 6 .......
     n= 7 ........
     n= 8 .........
     n= 9 ..........
     n= 10 ...........
     n= 11 ............
     n= 12 .............
     n= 13 ..............
     n= 14 ...............
     n= 15 ................
     n= 16 .................
     n= 17 ..................
     n= 18 ...................
     n= 19 ....................
     n= 20 .....................
     > showProc.time()
     Time (user system elapsed): 1.795 0.059 2.257
     >
     > ### dnbinom*() :
     > stopifnot(exprs = {
     + dnbinomR(0, 1, 1) == 1
     + })
     >
     > ### exploring 'eps' == "true" tests must be done with Rmpfr !!
     >
     > ### 2. Testing log1pmx(), logcf() etc
     > ### ----------
     >
     > ### 2a: logcf()
     > ## == =======
     > x <- c((-20:3)/4, (25:31)/32) # close (but not too close) to upper bound 1
     >
     > (lC <- logcf (x, i=2, d=3, eps=1e-9))
     [1] 0.2225364 0.2273165 0.2323964 0.2378089 0.2435921 0.2497910 0.2564582
     [8] 0.2636564 0.2714610 0.2799634 0.2892758 0.2995378 0.3109259 0.3236668
     [15] 0.3380584 0.3545014 0.3735507 0.3960035 0.4230578 0.4566237 0.5000000
     [22] 0.5595850 0.6503112 0.8221318 0.8574109 0.8989257 0.9490572 1.0118259
     [29] 1.0948318 1.2152882 1.4286636
     > lCt <- logcf (x, i=2, d=3, eps=1e-9, trace=TRUE) ; stopifnot(identical(lCt, lC))
     it= 0: ==> |b2|=162720
     it= 1: ==> |b2|=1.68458e+08
     it= 2: ==> |b2|=3.02689e+11
     it= 3: ==> |b2|=8.40216e+14
     it= 4: ==> |b2|=3.33607e+18
     it= 5: ==> |b2|=1.79478e+22
     it= 6: ==> |b2|=1.25703e+26
     it= 7: ==> |b2|=1.11146e+30
     it= 8: ==> |b2|=1.21086e+34
     it= 9: ==> |b2|=1.5936e+38
     it=10: ==> |b2|=2.49268e+42
     logcf(*) used 11 iterations.
     it= 0: ==> |b2|=151400
     it= 1: ==> |b2|=1.519e+08
     it= 2: ==> |b2|=2.64707e+11
     it= 3: ==> |b2|=7.12814e+14
     it= 4: ==> |b2|=2.74588e+18
     it= 5: ==> |b2|=1.4333e+22
     it= 6: ==> |b2|=9.73998e+25
     it= 7: ==> |b2|=8.35605e+29
     it= 8: ==> |b2|=8.83286e+33
     it= 9: ==> |b2|=1.12795e+38
     it=10: ==> |b2|=1.71192e+42
     logcf(*) used 11 iterations.
     it= 0: ==> |b2|=140480
     it= 1: ==> |b2|=1.36437e+08
     it= 2: ==> |b2|=2.30332e+11
     it= 3: ==> |b2|=6.0102e+14
     it= 4: ==> |b2|=2.24367e+18
     it= 5: ==> |b2|=1.135e+22
     it= 6: ==> |b2|=7.47503e+25
     it= 7: ==> |b2|=6.21522e+29
     it= 8: ==> |b2|=6.3674e+33
     it= 9: ==> |b2|=7.88061e+37
     it=10: ==> |b2|=1.15921e+42
     logcf(*) used 11 iterations.
     it= 0: ==> |b2|=129960
     it= 1: ==> |b2|=1.22034e+08
     it= 2: ==> |b2|=1.99336e+11
     it= 3: ==> |b2|=5.03394e+14
     it= 4: ==> |b2|=1.81889e+18
     it= 5: ==> |b2|=8.90621e+21
     it= 6: ==> |b2|=5.67763e+25
     it= 7: ==> |b2|=4.56957e+29
     it= 8: ==> |b2|=4.53158e+33
     it= 9: ==> |b2|=5.429e+37
     logcf(*) used 10 iterations.
     it= 0: ==> |b2|=119840
     it= 1: ==> |b2|=1.08655e+08
     it= 2: ==> |b2|=1.71497e+11
     it= 3: ==> |b2|=4.18587e+14
     it= 4: ==> |b2|=1.46194e+18
     it= 5: ==> |b2|=6.91963e+21
     it= 6: ==> |b2|=4.26415e+25
     it= 7: ==> |b2|=3.31759e+29
     it= 8: ==> |b2|=3.18042e+33
     it= 9: ==> |b2|=3.68336e+37
     logcf(*) used 10 iterations.
     it= 0: ==> |b2|=110120
     it= 1: ==> |b2|=9.62638e+07
     it= 2: ==> |b2|=1.46601e+11
     it= 3: ==> |b2|=3.45334e+14
     it= 4: ==> |b2|=1.16411e+18
     it= 5: ==> |b2|=5.31835e+21
     it= 6: ==> |b2|=3.16349e+25
     it= 7: ==> |b2|=2.37577e+29
     it= 8: ==> |b2|=2.19845e+33
     it= 9: ==> |b2|=2.45771e+37
     logcf(*) used 10 iterations.
     it= 0: ==> |b2|=100800
     it= 1: ==> |b2|=8.48232e+07
     it= 2: ==> |b2|=1.24442e+11
     it= 3: ==> |b2|=2.82452e+14
     it= 4: ==> |b2|=9.17519e+17
     it= 5: ==> |b2|=4.03952e+21
     it= 6: ==> |b2|=2.3156e+25
     it= 7: ==> |b2|=1.67591e+29
     it= 8: ==> |b2|=1.49457e+33
     logcf(*) used 9 iterations.
     it= 0: ==> |b2|=91880
     it= 1: ==> |b2|=7.42974e+07
     it= 2: ==> |b2|=1.04819e+11
     it= 3: ==> |b2|=2.28837e+14
     it= 4: ==> |b2|=7.15064e+17
     it= 5: ==> |b2|=3.02848e+21
     it= 6: ==> |b2|=1.67007e+25
     it= 7: ==> |b2|=1.1628e+29
     it= 8: ==> |b2|=9.97611e+32
     logcf(*) used 9 iterations.
     it= 0: ==> |b2|=83360
     it= 1: ==> |b2|=6.46501e+07
     it= 2: ==> |b2|=8.75389e+10
     it= 3: ==> |b2|=1.83464e+14
     it= 4: ==> |b2|=5.50387e+17
     it= 5: ==> |b2|=2.23803e+21
     it= 6: ==> |b2|=1.18496e+25
     it= 7: ==> |b2|=7.92152e+28
     it= 8: ==> |b2|=6.52535e+32
     logcf(*) used 9 iterations.
     it= 0: ==> |b2|=75240
     it= 1: ==> |b2|=5.58449e+07
     it= 2: ==> |b2|=7.24171e+10
     it= 3: ==> |b2|=1.45381e+14
     it= 4: ==> |b2|=4.17809e+17
     it= 5: ==> |b2|=1.6276e+21
     it= 6: ==> |b2|=8.25594e+24
     it= 7: ==> |b2|=5.28764e+28
     logcf(*) used 8 iterations.
     it= 0: ==> |b2|=67520
     it= 1: ==> |b2|=4.78456e+07
     it= 2: ==> |b2|=5.92745e+10
     it= 3: ==> |b2|=1.13708e+14
     it= 4: ==> |b2|=3.12287e+17
     it= 5: ==> |b2|=1.16261e+21
     it= 6: ==> |b2|=5.6361e+24
     it= 7: ==> |b2|=3.44989e+28
     logcf(*) used 8 iterations.
     it= 0: ==> |b2|=60200
     it= 1: ==> |b2|=4.06158e+07
     it= 2: ==> |b2|=4.79397e+10
     it= 3: ==> |b2|=8.76351e+13
     it= 4: ==> |b2|=2.2937e+17
     it= 5: ==> |b2|=8.13827e+20
     it= 6: ==> |b2|=3.76013e+24
     it= 7: ==> |b2|=2.19363e+28
     logcf(*) used 8 iterations.
     it= 0: ==> |b2|=53280
     it= 1: ==> |b2|=3.41194e+07
     it= 2: ==> |b2|=3.82483e+10
     it= 3: ==> |b2|=6.64186e+13
     it= 4: ==> |b2|=1.6515e+17
     it= 5: ==> |b2|=5.56707e+20
     it= 6: ==> |b2|=2.44378e+24
     logcf(*) used 7 iterations.
     it= 0: ==> |b2|=46760
     it= 1: ==> |b2|=2.83198e+07
     it= 2: ==> |b2|=3.0043e+10
     it= 3: ==> |b2|=4.93794e+13
     it= 4: ==> |b2|=1.16224e+17
     it= 5: ==> |b2|=3.70875e+20
     it= 6: ==> |b2|=1.54119e+24
     logcf(*) used 7 iterations.
     it= 0: ==> |b2|=40640
     it= 1: ==> |b2|=2.3181e+07
     it= 2: ==> |b2|=2.31738e+10
     it= 3: ==> |b2|=3.59e+13
     it= 4: ==> |b2|=7.96488e+16
     it= 5: ==> |b2|=2.39588e+20
     logcf(*) used 6 iterations.
     it= 0: ==> |b2|=34920
     it= 1: ==> |b2|=1.86664e+07
     it= 2: ==> |b2|=1.74976e+10
     it= 3: ==> |b2|=2.5422e+13
     it= 4: ==> |b2|=5.29017e+16
     it= 5: ==> |b2|=1.49263e+20
     logcf(*) used 6 iterations.
     it= 0: ==> |b2|=29600
     it= 1: ==> |b2|=1.474e+07
     it= 2: ==> |b2|=1.28785e+10
     it= 3: ==> |b2|=1.74436e+13
     it= 4: ==> |b2|=3.38438e+16
     logcf(*) used 5 iterations.
     it= 0: ==> |b2|=24680
     it= 1: ==> |b2|=1.13653e+07
     it= 2: ==> |b2|=9.18785e+09
     it= 3: ==> |b2|=1.1517e+13
     it= 4: ==> |b2|=2.06815e+16
     logcf(*) used 5 iterations.
     it= 0: ==> |b2|=20160
     it= 1: ==> |b2|=8.50608e+06
     it= 2: ==> |b2|=6.30386e+09
     it= 3: ==> |b2|=7.24564e+12
     logcf(*) used 4 iterations.
     it= 0: ==> |b2|=16040
     it= 1: ==> |b2|=6.12601e+06
     it= 2: ==> |b2|=4.11202e+09
     logcf(*) used 3 iterations.
     logcf(*) used 0 iterations.
     it= 0: ==> |b2|=9000
     it= 1: ==> |b2|=2.65815e+06
     it= 2: ==> |b2|=1.38218e+09
     logcf(*) used 3 iterations.
     it= 0: ==> |b2|=6080
     it= 1: ==> |b2|=1.49776e+06
     it= 2: ==> |b2|=6.50656e+08
     it= 3: ==> |b2|=4.39124e+11
     it= 4: ==> |b2|=4.24985e+14
     logcf(*) used 5 iterations.
     it= 0: ==> |b2|=3560
     it= 1: ==> |b2|=671330
     it= 2: ==> |b2|=2.24237e+08
     it= 3: ==> |b2|=1.16565e+11
     it= 4: ==> |b2|=8.69636e+13
     it= 5: ==> |b2|=8.80714e+16
     it= 6: ==> |b2|=1.16246e+20
     it= 7: ==> |b2|=1.93847e+23
     it= 8: ==> |b2|=3.98491e+26
     logcf(*) used 9 iterations.
     it= 0: ==> |b2|=3273.12
     it= 1: ==> |b2|=589700
     it= 2: ==> |b2|=1.88377e+08
     it= 3: ==> |b2|=9.36959e+10
     it= 4: ==> |b2|=6.68994e+13
     it= 5: ==> |b2|=6.48488e+16
     it= 6: ==> |b2|=8.19327e+19
     it= 7: ==> |b2|=1.30789e+23
     it= 8: ==> |b2|=2.57381e+26
     it= 9: ==> |b2|=6.12129e+29
     logcf(*) used 10 iterations.
     it= 0: ==> |b2|=2992.5
     it= 1: ==> |b2|=512650
     it= 2: ==> |b2|=1.55894e+08
     it= 3: ==> |b2|=7.3859e+10
     it= 4: ==> |b2|=5.02475e+13
     it= 5: ==> |b2|=4.64164e+16
     it= 6: ==> |b2|=5.58911e+19
     it= 7: ==> |b2|=8.50347e+22
     it= 8: ==> |b2|=1.595e+26
     it= 9: ==> |b2|=3.61574e+29
     it=10: ==> |b2|=9.74479e+32
     logcf(*) used 11 iterations.
     it= 0: ==> |b2|=2718.12
     it= 1: ==> |b2|=440109
     it= 2: ==> |b2|=1.26644e+08
     it= 3: ==> |b2|=5.68225e+10
     it= 4: ==> |b2|=3.66244e+13
     it= 5: ==> |b2|=3.20598e+16
     it= 6: ==> |b2|=3.65864e+19
     it= 7: ==> |b2|=5.27587e+22
     it= 8: ==> |b2|=9.37997e+25
     it= 9: ==> |b2|=2.01557e+29
     it=10: ==> |b2|=5.14924e+32
     it=11: ==> |b2|=1.54257e+36
     logcf(*) used 12 iterations.
     it= 0: ==> |b2|=2450
     it= 1: ==> |b2|=372006
     it= 2: ==> |b2|=1.00485e+08
     it= 3: ==> |b2|=4.23633e+10
     it= 4: ==> |b2|=2.56713e+13
     it= 5: ==> |b2|=2.11343e+16
     it= 6: ==> |b2|=2.26869e+19
     it= 7: ==> |b2|=3.07772e+22
     it= 8: ==> |b2|=5.14811e+25
     it= 9: ==> |b2|=1.04082e+29
     it=10: ==> |b2|=2.50192e+32
     it=11: ==> |b2|=7.05238e+35
     it=12: ==> |b2|=2.30384e+39
     logcf(*) used 13 iterations.
     it= 0: ==> |b2|=2188.12
     it= 1: ==> |b2|=308271
     it= 2: ==> |b2|=7.72745e+07
     it= 3: ==> |b2|=3.02658e+10
     it= 4: ==> |b2|=1.70531e+13
     it= 5: ==> |b2|=1.30605e+16
     it= 6: ==> |b2|=1.30466e+19
     it= 7: ==> |b2|=1.64734e+22
     it= 8: ==> |b2|=2.56499e+25
     it= 9: ==> |b2|=4.82765e+28
     it=10: ==> |b2|=1.08039e+32
     it=11: ==> |b2|=2.83535e+35
     it=12: ==> |b2|=8.62389e+38
     it=13: ==> |b2|=3.00926e+42
     it=14: ==> |b2|=1.19409e+46
     it=15: ==> |b2|=5.34632e+49
     logcf(*) used 16 iterations.
     it= 0: ==> |b2|=1932.5
     it= 1: ==> |b2|=248832
     it= 2: ==> |b2|=5.68734e+07
     it= 3: ==> |b2|=2.03226e+10
     it= 4: ==> |b2|=1.04577e+13
     it= 5: ==> |b2|=7.32086e+15
     it= 6: ==> |b2|=6.68834e+18
     it= 7: ==> |b2|=7.72653e+21
     it= 8: ==> |b2|=1.10096e+25
     it= 9: ==> |b2|=1.89662e+28
     it=10: ==> |b2|=3.88536e+31
     it=11: ==> |b2|=9.33474e+34
     it=12: ==> |b2|=2.59938e+38
     it=13: ==> |b2|=8.30457e+41
     it=14: ==> |b2|=3.01718e+45
     it=15: ==> |b2|=1.23692e+49
     it=16: ==> |b2|=5.68258e+52
     it=17: ==> |b2|=2.90768e+56
     it=18: ==> |b2|=1.64796e+60
     logcf(*) used 19 iterations.
     it= 0: ==> |b2|=1683.12
     it= 1: ==> |b2|=193619
     it= 2: ==> |b2|=3.91439e+07
     it= 3: ==> |b2|=1.23338e+10
     it= 4: ==> |b2|=5.59551e+12
     it= 5: ==> |b2|=3.4562e+15
     it= 6: ==> |b2|=2.78868e+18
     it= 7: ==> |b2|=2.84748e+21
     it= 8: ==> |b2|=3.58854e+24
     it= 9: ==> |b2|=5.4701e+27
     it=10: ==> |b2|=9.91885e+30
     it=11: ==> |b2|=2.10987e+34
     it=12: ==> |b2|=5.20269e+37
     it=13: ==> |b2|=1.47211e+41
     it=14: ==> |b2|=4.73732e+44
     it=15: ==> |b2|=1.72036e+48
     it=16: ==> |b2|=7.00164e+51
     it=17: ==> |b2|=3.17394e+55
     it=18: ==> |b2|=1.59374e+59
     it=19: ==> |b2|=8.8205e+62
     it=20: ==> |b2|=5.35623e+66
     it=21: ==> |b2|=3.5541e+70
     it=22: ==> |b2|=2.56724e+74
     it=23: ==> |b2|=2.01172e+78 Lrg |b2|
     it=24: ==> |b2|=147221
     it=25: ==> |b2|=1.34508e+09
     it=26: ==> |b2|=1.32142e+13
     it=27: ==> |b2|=1.39232e+17
     logcf(*) used 28 iterations.
     > (lR <- logcfR(x, i=2, d=3, eps=1e-9))
     [1] 0.2225364 0.2273165 0.2323964 0.2378089 0.2435921 0.2497910 0.2564582
     [8] 0.2636564 0.2714610 0.2799634 0.2892758 0.2995378 0.3109259 0.3236668
     [15] 0.3380584 0.3545014 0.3735507 0.3960035 0.4230578 0.4566237 0.5000000
     [22] 0.5595850 0.6503112 0.8221318 0.8574109 0.8989257 0.9490572 1.0118259
     [29] 1.0948318 1.2152882 1.4286636
     > all.equal(lC, lR, tol = 0) # to see if ..
     [1] TRUE
     > stopifnot(all.equal(lC, lR, tol = 4e-16))
     > lRt <- logcfR(x, i=2, d=3, eps=1e-9, trace=TRUE) ; stopifnot(identical(lRt, lR))
     logcf(x[ 1]= -5, i=2, d=3, eps=1e-09) logcf(*) end: after 11 iterations.
     logcf(x[ 2]= -4.75, i=2, d=3, eps=1e-09) logcf(*) end: after 11 iterations.
     logcf(x[ 3]= -4.5, i=2, d=3, eps=1e-09) logcf(*) end: after 11 iterations.
     logcf(x[ 4]= -4.25, i=2, d=3, eps=1e-09) logcf(*) end: after 10 iterations.
     logcf(x[ 5]= -4, i=2, d=3, eps=1e-09) logcf(*) end: after 10 iterations.
     logcf(x[ 6]= -3.75, i=2, d=3, eps=1e-09) logcf(*) end: after 10 iterations.
     logcf(x[ 7]= -3.5, i=2, d=3, eps=1e-09) logcf(*) end: after 9 iterations.
     logcf(x[ 8]= -3.25, i=2, d=3, eps=1e-09) logcf(*) end: after 9 iterations.
     logcf(x[ 9]= -3, i=2, d=3, eps=1e-09) logcf(*) end: after 9 iterations.
     logcf(x[10]= -2.75, i=2, d=3, eps=1e-09) logcf(*) end: after 8 iterations.
     logcf(x[11]= -2.5, i=2, d=3, eps=1e-09) logcf(*) end: after 8 iterations.
     logcf(x[12]= -2.25, i=2, d=3, eps=1e-09) logcf(*) end: after 8 iterations.
     logcf(x[13]= -2, i=2, d=3, eps=1e-09) logcf(*) end: after 7 iterations.
     logcf(x[14]= -1.75, i=2, d=3, eps=1e-09) logcf(*) end: after 7 iterations.
     logcf(x[15]= -1.5, i=2, d=3, eps=1e-09) logcf(*) end: after 6 iterations.
     logcf(x[16]= -1.25, i=2, d=3, eps=1e-09) logcf(*) end: after 6 iterations.
     logcf(x[17]= -1, i=2, d=3, eps=1e-09) logcf(*) end: after 5 iterations.
     logcf(x[18]= -0.75, i=2, d=3, eps=1e-09) logcf(*) end: after 5 iterations.
     logcf(x[19]= -0.5, i=2, d=3, eps=1e-09) logcf(*) end: after 4 iterations.
     logcf(x[20]= -0.25, i=2, d=3, eps=1e-09) logcf(*) end: after 3 iterations.
     logcf(x[21]= 0, i=2, d=3, eps=1e-09) logcf(*) end: after 0 iterations.
     logcf(x[22]= 0.25, i=2, d=3, eps=1e-09) logcf(*) end: after 3 iterations.
     logcf(x[23]= 0.5, i=2, d=3, eps=1e-09) logcf(*) end: after 5 iterations.
     logcf(x[24]= 0.75, i=2, d=3, eps=1e-09) logcf(*) end: after 9 iterations.
     logcf(x[25]= 0.78125, i=2, d=3, eps=1e-09) logcf(*) end: after 10 iterations.
     logcf(x[26]= 0.8125, i=2, d=3, eps=1e-09) logcf(*) end: after 11 iterations.
     logcf(x[27]= 0.84375, i=2, d=3, eps=1e-09) logcf(*) end: after 12 iterations.
     logcf(x[28]= 0.875, i=2, d=3, eps=1e-09) logcf(*) end: after 13 iterations.
     logcf(x[29]= 0.90625, i=2, d=3, eps=1e-09) logcf(*) end: after 16 iterations.
     logcf(x[30]= 0.9375, i=2, d=3, eps=1e-09) logcf(*) end: after 19 iterations.
     logcf(x[31]= 0.96875, i=2, d=3, eps=1e-09) logcf(*) end: after 28 iterations.
     > lRt2 <- logcfR(x, i=2, d=3, eps=1e-9, trace= 2) ; stopifnot(identical(lRt2,lR))
     logcf(x[ 1]= -5, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 162720 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.0168627
     it= 2: ==> B2= 1.68458e+08 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00303811
     it= 3: ==> B2= 3.02689e+11 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000541327
     it= 4: ==> B2= 8.40216e+14 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 9.60626e-05
     it= 5: ==> B2= 3.33607e+18 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.70167e-05
     it= 6: ==> B2= 1.79478e+22 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.01154e-06
     it= 7: ==> B2= 1.25703e+26 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 5.32664e-07
     it= 8: ==> B2= 1.11146e+30 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 9.4179e-08
     it= 9: ==> B2= 1.21086e+34 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.66472e-08
     it=10: ==> B2= 1.5936e+38 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.942e-09
     it=11: ==> B2= 2.49268e+42 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 5.19854e-10
     logcf(*) end: after 11 iterations.
     logcf(x[ 2]= -4.75, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 151400 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.0157061
     it= 2: ==> B2= 1.519e+08 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00271234
     it= 3: ==> B2= 2.64707e+11 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000463242
     it= 4: ==> B2= 7.12814e+14 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 7.88006e-05
     it= 5: ==> B2= 2.74588e+18 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.33808e-05
     it= 6: ==> B2= 1.4333e+22 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.26999e-06
     it= 7: ==> B2= 9.73998e+25 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.84872e-07
     it= 8: ==> B2= 8.35605e+29 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 6.52292e-08
     it= 9: ==> B2= 8.83286e+33 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.10523e-08
     it=10: ==> B2= 1.12795e+38 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.8723e-09
     it=11: ==> B2= 1.71192e+42 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.1713e-10
     logcf(*) end: after 11 iterations.
     logcf(x[ 3]= -4.5, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 140480 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.014539
     it= 2: ==> B2= 1.36437e+08 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00239872
     it= 3: ==> B2= 2.30332e+11 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000391409
     it= 4: ==> B2= 6.0102e+14 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 6.36152e-05
     it= 5: ==> B2= 2.24367e+18 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.03211e-05
     it= 6: ==> B2= 1.135e+22 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.67293e-06
     it= 7: ==> B2= 7.47503e+25 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.71005e-07
     it= 8: ==> B2= 6.21522e+29 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 4.38843e-08
     it= 9: ==> B2= 6.3674e+33 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 7.10431e-09
     it=10: ==> B2= 7.88061e+37 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.14987e-09
     it=11: ==> B2= 1.15921e+42 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.86085e-10
     logcf(*) end: after 11 iterations.
     logcf(x[ 4]= -4.25, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 129960 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.0133641
     it= 2: ==> B2= 1.22034e+08 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00209863
     it= 3: ==> B2= 1.99336e+11 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000325959
     it= 4: ==> B2= 5.03394e+14 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 5.04302e-05
     it= 5: ==> B2= 1.81889e+18 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 7.78852e-06
     it= 6: ==> B2= 8.90621e+21 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.20172e-06
     it= 7: ==> B2= 5.67763e+25 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.85309e-07
     it= 8: ==> B2= 4.56957e+29 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.85641e-08
     it= 9: ==> B2= 4.53158e+33 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 4.40173e-09
     it=10: ==> B2= 5.429e+37 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 6.78171e-10
     logcf(*) end: after 10 iterations.
     logcf(x[ 5]= -4, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 119840 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.0121853
     it= 2: ==> B2= 1.08655e+08 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00181353
     it= 3: ==> B2= 1.71497e+11 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000266983
     it= 4: ==> B2= 4.18587e+14 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.91528e-05
     it= 5: ==> B2= 1.46194e+18 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 5.73167e-06
     it= 6: ==> B2= 6.91963e+21 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 8.38266e-07
     it= 7: ==> B2= 4.26415e+25 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.22525e-07
     it= 8: ==> B2= 3.31759e+29 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.79016e-08
     it= 9: ==> B2= 3.18042e+33 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.6148e-09
     it=10: ==> B2= 3.68336e+37 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.81854e-10
     logcf(*) end: after 10 iterations.
     logcf(x[ 6]= -3.75, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 110120 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.0110067
     it= 2: ==> B2= 9.62638e+07 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00154491
     it= 3: ==> B2= 1.46601e+11 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00021452
     it= 4: ==> B2= 3.45334e+14 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.96738e-05
     it= 5: ==> B2= 1.16411e+18 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 4.0975e-06
     it= 6: ==> B2= 5.31835e+21 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 5.65252e-07
     it= 7: ==> B2= 3.16349e+25 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 7.79297e-08
     it= 8: ==> B2= 2.37577e+29 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.07396e-08
     it= 9: ==> B2= 2.19845e+33 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.47962e-09
     it=10: ==> B2= 2.45771e+37 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.03808e-10
     logcf(*) end: after 10 iterations.
     logcf(x[ 7]= -3.5, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 100800 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00983372
     it= 2: ==> B2= 8.48232e+07 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00129431
     it= 3: ==> B2= 1.24442e+11 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000168553
     it= 4: ==> B2= 2.82452e+14 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.18673e-05
     it= 5: ==> B2= 9.17519e+17 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.83198e-06
     it= 6: ==> B2= 4.03952e+21 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.66405e-07
     it= 7: ==> B2= 2.3156e+25 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 4.73767e-08
     it= 8: ==> B2= 1.67591e+29 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 6.12337e-09
     it= 9: ==> B2= 1.49457e+33 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 7.91207e-10
     logcf(*) end: after 9 iterations.
     logcf(x[ 8]= -3.25, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 91880 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00867282
     it= 2: ==> B2= 7.42974e+07 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00106327
     it= 3: ==> B2= 1.04819e+11 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000128994
     it= 4: ==> B2= 2.28837e+14 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.55909e-05
     it= 5: ==> B2= 7.15064e+17 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.88109e-06
     it= 6: ==> B2= 3.02848e+21 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.26734e-07
     it= 7: ==> B2= 1.67007e+25 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.7312e-08
     it= 8: ==> B2= 1.1628e+29 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.28859e-09
     it= 9: ==> B2= 9.97611e+32 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.95855e-10
     logcf(*) end: after 9 iterations.
     logcf(x[ 9]= -3, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 83360 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00753175
     it= 2: ==> B2= 6.46501e+07 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000853254
     it= 3: ==> B2= 8.75389e+10 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 9.5671e-05
     it= 4: ==> B2= 1.83464e+14 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.06874e-05
     it= 5: ==> B2= 5.50387e+17 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.19178e-06
     it= 6: ==> B2= 2.23803e+21 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.32765e-07
     it= 7: ==> B2= 1.18496e+25 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.47808e-08
     it= 8: ==> B2= 7.92152e+28 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.64484e-09
     it= 9: ==> B2= 6.52535e+32 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.82987e-10
     logcf(*) end: after 9 iterations.
     logcf(x[10]= -2.75, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 75240 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00641978
     it= 2: ==> B2= 5.58449e+07 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000665641
     it= 3: ==> B2= 7.24171e+10 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 6.83222e-05
     it= 4: ==> B2= 1.45381e+14 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 6.98689e-06
     it= 5: ==> B2= 4.17809e+17 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 7.13234e-07
     it= 6: ==> B2= 1.6276e+21 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 7.27338e-08
     it= 7: ==> B2= 8.25594e+24 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 7.41242e-09
     it= 8: ==> B2= 5.28764e+28 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 7.55083e-10
     logcf(*) end: after 8 iterations.
     logcf(x[11]= -2.5, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 67520 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00534792
     it= 2: ==> B2= 4.78456e+07 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.0005016
     it= 3: ==> B2= 5.92745e+10 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 4.65818e-05
     it= 4: ==> B2= 1.13708e+14 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 4.31004e-06
     it= 5: ==> B2= 3.12287e+17 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.98074e-07
     it= 6: ==> B2= 1.16261e+21 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.67278e-08
     it= 7: ==> B2= 5.6361e+24 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.38642e-09
     it= 8: ==> B2= 3.44989e+28 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.12099e-10
     logcf(*) end: after 8 iterations.
     logcf(x[12]= -2.25, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 60200 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00432911
     it= 2: ==> B2= 4.06158e+07 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00036199
     it= 3: ==> B2= 4.79397e+10 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.99753e-05
     it= 4: ==> B2= 8.76351e+13 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.47309e-06
     it= 5: ==> B2= 2.2937e+17 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.03667e-07
     it= 6: ==> B2= 8.13827e+20 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.67549e-08
     it= 7: ==> B2= 3.76013e+24 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.37743e-09
     it= 8: ==> B2= 2.19363e+28 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.13188e-10
     logcf(*) end: after 8 iterations.
     logcf(x[13]= -2, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 53280 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00337838
     it= 2: ==> B2= 3.41194e+07 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00024722
     it= 3: ==> B2= 3.82483e+10 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.79187e-05
     it= 4: ==> B2= 6.64186e+13 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.29399e-06
     it= 5: ==> B2= 1.6515e+17 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 9.32713e-08
     it= 6: ==> B2= 5.56707e+20 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 6.71575e-09
     it= 7: ==> B2= 2.44378e+24 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 4.83216e-10
     logcf(*) end: after 7 iterations.
     logcf(x[14]= -1.75, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 46760 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00251277
     it= 2: ==> B2= 2.83198e+07 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000157067
     it= 3: ==> B2= 3.0043e+10 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 9.72602e-06
     it= 4: ==> B2= 4.93794e+13 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 6.00029e-07
     it= 5: ==> B2= 1.16224e+17 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.69475e-08
     it= 6: ==> B2= 3.70875e+20 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.27255e-09
     it= 7: ==> B2= 1.54119e+24 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.39681e-10
     logcf(*) end: after 7 iterations.
     logcf(x[15]= -1.5, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 40640 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.001751
     it= 2: ==> B2= 2.3181e+07 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 9.04805e-05
     it= 3: ==> B2= 2.31738e+10 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 4.63221e-06
     it= 4: ==> B2= 3.59e+13 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.36258e-07
     it= 5: ==> B2= 7.96488e+16 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.20265e-08
     it= 6: ==> B2= 2.39588e+20 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 6.11497e-10
     logcf(*) end: after 6 iterations.
     logcf(x[16]= -1.25, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 34920 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00111245
     it= 2: ==> B2= 1.86664e+07 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 4.53708e-05
     it= 3: ==> B2= 1.74976e+10 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.8334e-06
     it= 4: ==> B2= 2.5422e+13 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 7.38016e-08
     it= 5: ==> B2= 5.29017e+16 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.96486e-09
     it= 6: ==> B2= 1.49263e+20 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.18968e-10
     logcf(*) end: after 6 iterations.
     logcf(x[17]= -1, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 29600 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000614941
     it= 2: ==> B2= 1.474e+07 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.84647e-05
     it= 3: ==> B2= 1.28785e+10 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 5.49306e-07
     it= 4: ==> B2= 1.74436e+13 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.62767e-08
     it= 5: ==> B2= 3.38438e+16 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 4.81303e-10
     logcf(*) end: after 5 iterations.
     logcf(x[18]= -0.75, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 24680 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000270385
     it= 2: ==> B2= 1.13653e+07 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 5.33194e-06
     it= 3: ==> B2= 9.18785e+09 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.04153e-07
     it= 4: ==> B2= 1.1517e+13 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.02617e-09
     it= 5: ==> B2= 2.06815e+16 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.93312e-11
     logcf(*) end: after 5 iterations.
     logcf(x[19]= -0.5, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 20160 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 7.69704e-05
     it= 2: ==> B2= 8.50608e+06 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 8.02466e-07
     it= 3: ==> B2= 6.30386e+09 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 8.28445e-09
     it= 4: ==> B2= 7.24564e+12 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 8.51583e-11
     logcf(*) end: after 4 iterations.
     logcf(x[20]= -0.25, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 16040 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 7.26392e-06
     it= 2: ==> B2= 6.12601e+06 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.30773e-08
     it= 3: ==> B2= 4.11202e+09 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 7.25571e-11
     logcf(*) end: after 3 iterations.
     logcf(x[21]= 0, i=2, d=3, eps=1e-09) iterations:
     logcf(*) end: after 0 iterations.
     logcf(x[22]= 0.25, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 9000 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.04918e-05
     it= 2: ==> B2= 2.65815e+06 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.08623e-07
     it= 3: ==> B2= 1.38218e+09 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 5.68393e-10
     logcf(*) end: after 3 iterations.
     logcf(x[23]= 0.5, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 6080 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000659523
     it= 2: ==> B2= 1.49776e+06 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.00942e-05
     it= 3: ==> B2= 6.50656e+08 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 6.02264e-07
     it= 4: ==> B2= 4.39124e+11 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.79273e-08
     it= 5: ==> B2= 4.24985e+14 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 5.3174e-10
     logcf(*) end: after 5 iterations.
     logcf(x[24]= 0.75, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 3560 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00856402
     it= 2: ==> B2= 671330 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00100335
     it= 3: ==> B2= 2.24237e+08 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000114482
     it= 4: ==> B2= 1.16565e+11 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.2922e-05
     it= 5: ==> B2= 8.69636e+13 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.45089e-06
     it= 6: ==> B2= 8.80714e+16 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.62421e-07
     it= 7: ==> B2= 1.16246e+20 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.81486e-08
     it= 8: ==> B2= 1.93847e+23 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.02539e-09
     it= 9: ==> B2= 3.98491e+26 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.25837e-10
     logcf(*) end: after 9 iterations.
     logcf(x[25]= 0.78125, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 3273.12 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.0116683
     it= 2: ==> B2= 589700 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00162549
     it= 3: ==> B2= 1.88377e+08 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000220072
     it= 4: ==> B2= 9.36959e+10 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.94426e-05
     it= 5: ==> B2= 6.68994e+13 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.9163e-06
     it= 6: ==> B2= 6.48488e+16 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 5.19244e-07
     it= 7: ==> B2= 8.19327e+19 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 6.87066e-08
     it= 8: ==> B2= 1.30789e+23 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 9.07922e-09
     it= 9: ==> B2= 2.57381e+26 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.19866e-09
     it=10: ==> B2= 6.12129e+29 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.58143e-10
     logcf(*) end: after 10 iterations.
     logcf(x[26]= 0.8125, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 2992.5 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.0159401
     it= 2: ==> B2= 512650 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00265674
     it= 3: ==> B2= 1.55894e+08 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000429426
     it= 4: ==> B2= 7.3859e+10 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 6.84941e-05
     it= 5: ==> B2= 5.02475e+13 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.08546e-05
     it= 6: ==> B2= 4.64164e+16 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.71404e-06
     it= 7: ==> B2= 5.58911e+19 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.70071e-07
     it= 8: ==> B2= 8.50347e+22 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 4.2492e-08
     it= 9: ==> B2= 1.595e+26 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 6.6788e-09
     it=10: ==> B2= 3.61574e+29 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.04899e-09
     it=11: ==> B2= 9.74479e+32 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.64668e-10
     logcf(*) end: after 11 iterations.
     logcf(x[27]= 0.84375, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 2718.12 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.0218736
     it= 2: ==> B2= 440109 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00440022
     it= 3: ==> B2= 1.26644e+08 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000856838
     it= 4: ==> B2= 5.68225e+10 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000164362
     it= 5: ==> B2= 3.66244e+13 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.12964e-05
     it= 6: ==> B2= 3.20598e+16 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 5.93505e-06
     it= 7: ==> B2= 3.65864e+19 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.12276e-06
     it= 8: ==> B2= 5.27587e+22 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.12056e-07
     it= 9: ==> B2= 9.37997e+25 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 4.00067e-08
     it=10: ==> B2= 2.01557e+29 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 7.54162e-09
     it=11: ==> B2= 5.14924e+32 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.42081e-09
     it=12: ==> B2= 1.54257e+36 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.67552e-10
     logcf(*) end: after 12 iterations.
     logcf(x[28]= 0.875, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 2450 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.0302147
     it= 2: ==> B2= 372006 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00742718
     it= 3: ==> B2= 1.00485e+08 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00176572
     it= 4: ==> B2= 4.23633e+10 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000412688
     it= 5: ==> B2= 2.56713e+13 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 9.56191e-05
     it= 6: ==> B2= 2.11343e+16 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.20491e-05
     it= 7: ==> B2= 2.26869e+19 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 5.0698e-06
     it= 8: ==> B2= 3.07772e+22 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.16356e-06
     it= 9: ==> B2= 5.14811e+25 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.66706e-07
     it=10: ==> B2= 1.04082e+29 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 6.10778e-08
     it=11: ==> B2= 2.50192e+32 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.39778e-08
     it=12: ==> B2= 7.05238e+35 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.19721e-09
     it=13: ==> B2= 2.30384e+39 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 7.31016e-10
     logcf(*) end: after 13 iterations.
     logcf(x[29]= 0.90625, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 2188.12 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.0421192
     it= 2: ==> B2= 308271 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.0128742
     it= 3: ==> B2= 7.72745e+07 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00381265
     it= 4: ==> B2= 3.02658e+10 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00110791
     it= 5: ==> B2= 1.70531e+13 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000318587
     it= 6: ==> B2= 1.30605e+16 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 9.10705e-05
     it= 7: ==> B2= 1.30466e+19 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.5941e-05
     it= 8: ==> B2= 1.64734e+22 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 7.37247e-06
     it= 9: ==> B2= 2.56499e+25 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.09206e-06
     it=10: ==> B2= 4.82765e+28 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 5.93012e-07
     it=11: ==> B2= 1.08039e+32 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.6796e-07
     it=12: ==> B2= 2.83535e+35 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 4.75428e-08
     it=13: ==> B2= 8.62389e+38 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.34511e-08
     it=14: ==> B2= 3.00926e+42 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.80425e-09
     it=15: ==> B2= 1.19409e+46 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.07559e-09
     it=16: ==> B2= 5.34632e+49 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.04032e-10
     logcf(*) end: after 16 iterations.
     logcf(x[30]= 0.9375, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 1932.5 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.0594391
     it= 2: ==> B2= 248832 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.02317
     it= 3: ==> B2= 5.68734e+07 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00882488
     it= 4: ==> B2= 2.03226e+10 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00329775
     it= 5: ==> B2= 1.04577e+13 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00121712
     it= 6: ==> B2= 7.32086e+15 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000445765
     it= 7: ==> B2= 6.68834e+18 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00016248
     it= 8: ==> B2= 7.72653e+21 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 5.90414e-05
     it= 9: ==> B2= 1.10096e+25 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.14101e-05
     it=10: ==> B2= 1.89662e+28 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 7.7527e-06
     it=11: ==> B2= 3.88536e+31 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.80437e-06
     it=12: ==> B2= 9.33474e+34 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.01363e-06
     it=13: ==> B2= 2.59938e+38 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.6615e-07
     it=14: ==> B2= 8.30457e+41 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.32201e-07
     it=15: ==> B2= 3.01718e+45 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 4.77137e-08
     it=16: ==> B2= 1.23692e+49 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.72154e-08
     it=17: ==> B2= 5.68258e+52 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 6.20986e-09
     it=18: ==> B2= 2.90768e+56 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.23951e-09
     it=19: ==> B2= 1.64796e+60 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 8.07503e-10
     logcf(*) end: after 19 iterations.
     logcf(x[31]= 0.96875, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 1683.12 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.0852619
     it= 2: ==> B2= 193619 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.0440308
     it= 3: ==> B2= 3.91439e+07 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.0227933
     it= 4: ==> B2= 1.23338e+10 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.0116823
     it= 5: ==> B2= 5.59551e+12 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00592272
     it= 6: ==> B2= 3.4562e+15 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00297607
     it= 7: ==> B2= 2.78868e+18 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00148555
     it= 8: ==> B2= 2.84748e+21 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00073801
     it= 9: ==> B2= 3.58854e+24 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000365396
     it=10: ==> B2= 5.4701e+27 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000180472
     it=11: ==> B2= 9.91885e+30 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 8.89791e-05
     it=12: ==> B2= 2.10987e+34 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 4.38122e-05
     it=13: ==> B2= 5.20269e+37 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.15512e-05
     it=14: ==> B2= 1.47211e+41 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.05929e-05
     it=15: ==> B2= 4.73732e+44 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 5.20351e-06
     it=16: ==> B2= 1.72036e+48 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.55486e-06
     it=17: ==> B2= 7.00164e+51 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.25391e-06
     it=18: ==> B2= 3.17394e+55 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 6.15209e-07
     it=19: ==> B2= 1.59374e+59 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.0176e-07
     it=20: ==> B2= 8.8205e+62 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.47979e-07
     it=21: ==> B2= 5.35623e+66 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 7.25526e-08
     it=22: ==> B2= 3.5541e+70 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.55657e-08
     it=23: ==> B2= 2.56724e+74 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.7432e-08
     it=24: ==> B2= 2.01172e+78 Lrg m.B2
     --> crit. |A2*b1 - a1*B2|/|b1*B2| = 8.54292e-09
     it=25: ==> B2= 147221 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 4.18617e-09
     it=26: ==> B2= 1.34508e+09 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.05108e-09
     it=27: ==> B2= 1.32142e+13 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.00487e-09
     it=28: ==> B2= 1.39232e+17 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 4.92273e-10
     logcf(*) end: after 28 iterations.
     >
     > lR. <- logcfR.(x, i=2, d=3, eps=1e-9)
     > lR.t <- logcfR.(x, i=2, d=3, eps=1e-9, trace=TRUE) ; stopifnot(identical(lR.t, lR.))
     logcf(x[], i=2, d=3, eps=1e-09) iterations:
     it= 1: needIt: 30 TRUE, and 1 F.; length(x[<todo>])=30, m.B2= 23307.2
     it= 2: needIt: 30 TRUE; length(x[<todo>])=30, m.B2= 1.01159e+07
     it= 3: needIt: 30 TRUE; length(x[<todo>])=30, m.B2= 7.70605e+09
     it= 4: needIt: 28 TRUE, and 2 F.; length(x[<todo>])=28, m.B2= 1.00852e+13
     it= 5: needIt: 27 TRUE, and 1 F.; length(x[<todo>])=27, m.B2= 1.76419e+16
     it= 6: needIt: 24 TRUE, and 3 F.; length(x[<todo>])=24, m.B2= 4.75316e+19
     it= 7: needIt: 22 TRUE, and 2 F.; length(x[<todo>])=22, m.B2= 1.2798e+23
     it= 8: needIt: 20 TRUE, and 2 F.; length(x[<todo>])=20, m.B2= 3.63581e+26
     it= 9: needIt: 17 TRUE, and 3 F.; length(x[<todo>])=17, m.B2= 6.8674e+29
     it=10: needIt: 13 TRUE, and 4 F.; length(x[<todo>])=13, m.B2= 1.03776e+33
     it=11: needIt: 9 TRUE, and 4 F.; length(x[<todo>])= 9, m.B2= 3.09233e+35
     it=12: needIt: 5 TRUE, and 4 F.; length(x[<todo>])= 5, m.B2= 2.27357e+35
     it=13: needIt: 4 TRUE, and 1 F.; length(x[<todo>])= 4, m.B2= 4.04868e+38
     it=14: needIt: 3 TRUE, and 1 F.; length(x[<todo>])= 3, m.B2= 7.16537e+41
     it=15: needIt: 3 TRUE; length(x[<todo>])= 3, m.B2= 2.57468e+45
     it=16: needIt: 3 TRUE; length(x[<todo>])= 3, m.B2= 1.04393e+49
     it=17: needIt: 2 TRUE, and 1 F.; length(x[<todo>])= 2, m.B2= 1.99468e+52
     it=18: needIt: 2 TRUE; length(x[<todo>])= 2, m.B2= 9.60666e+55
     it=19: needIt: 2 TRUE; length(x[<todo>])= 2, m.B2= 5.12487e+59
     it=20: needIt: 1 TRUE, and 1 F.; length(x[<todo>])= 1, m.B2= 8.8205e+62
     it=21: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 5.35623e+66
     it=22: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 3.5541e+70
     it=23: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 2.56724e+74
     it=24: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 2.01172e+78 Lrg m.B2
     it=25: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 147221
     it=26: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 1.34508e+09
     it=27: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 1.32142e+13
     it=28: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 1.39232e+17
     logcf(*) end: after 28 iterations.
     >
     > all.equal(lC, lR., tol = 0) # TRUE !! (every where ?)
     [1] TRUE
     > all.equal(lR, lR., tol = 0) # TRUE !! " "
     [1] TRUE
     > stopifnot(all.equal(lC, lR., tol = 1e-15))
     > ## (even though they used eps=1e-9 .. i.e., are not *so* accurate)
     > showProc.time()
     Time (user system elapsed): 0.028 0.02 0.048
     >
     > ##--- now with improved logcfR.() {<< will become the new logcfR() at least for MPFR !}:
     >
     > ##require(Rmpfr) may be not, see if NS loading (via "::") is sufficient:
     > requireNamespace("Rmpfr") || quit("no")
     Loading required namespace: Rmpfr
     [1] TRUE
     > ## ----- ----------
     > xM <- Rmpfr::mpfr(x, 512)
     > (ct.14 <- system.time(lR.14 <- logcfR.(xM, i=2, d=3, eps=1e-20, trace=TRUE))) # 0.55 sec
     logcf(x[], i=2, d=3, eps=1e-20) iterations:
     it= 1: needIt: 30 TRUE, and 1 F.; length(x[<todo>])=30, m.B2= 23307.2
     it= 2: needIt: 30 TRUE; length(x[<todo>])=30, m.B2= 1.01159e+07
     it= 3: needIt: 30 TRUE; length(x[<todo>])=30, m.B2= 7.70605e+09
     it= 4: needIt: 30 TRUE; length(x[<todo>])=30, m.B2= 9.10781e+12
     it= 5: needIt: 30 TRUE; length(x[<todo>])=30, m.B2= 1.54287e+16
     it= 6: needIt: 30 TRUE; length(x[<todo>])=30, m.B2= 3.54543e+19
     it= 7: needIt: 30 TRUE; length(x[<todo>])=30, m.B2= 1.06137e+23
     it= 8: needIt: 29 TRUE, and 1 F.; length(x[<todo>])=29, m.B2= 4.19177e+26
     it= 9: needIt: 28 TRUE, and 1 F.; length(x[<todo>])=28, m.B2= 2.26761e+30
     it=10: needIt: 27 TRUE, and 1 F.; length(x[<todo>])=27, m.B2= 1.33011e+34
     it=11: needIt: 27 TRUE; length(x[<todo>])=27, m.B2= 9.0823e+37
     it=12: needIt: 26 TRUE, and 1 F.; length(x[<todo>])=26, m.B2= 7.15387e+41
     it=13: needIt: 25 TRUE, and 1 F.; length(x[<todo>])=25, m.B2= 6.21918e+45
     it=14: needIt: 24 TRUE, and 1 F.; length(x[<todo>])=24, m.B2= 9.51187e+49
     it=15: needIt: 23 TRUE, and 1 F.; length(x[<todo>])=23, m.B2= 1.04428e+54
     it=16: needIt: 22 TRUE, and 1 F.; length(x[<todo>])=22, m.B2= 1.19866e+58
     it=17: needIt: 21 TRUE, and 1 F.; length(x[<todo>])=21, m.B2= 1.40641e+62
     it=18: needIt: 20 TRUE, and 1 F.; length(x[<todo>])=20, m.B2= 1.64566e+66
     it=19: needIt: 19 TRUE, and 1 F.; length(x[<todo>])=19, m.B2= 1.86787e+70
     it=20: needIt: 17 TRUE, and 2 F.; length(x[<todo>])=17, m.B2= 9.5095e+73
     it=21: needIt: 15 TRUE, and 2 F.; length(x[<todo>])=15, m.B2= 2.07684e+78 Lrg m.B2
     it=22: needIt: 14 TRUE, and 1 F.; length(x[<todo>])=14, m.B2= 122830
     it=23: needIt: 11 TRUE, and 3 F.; length(x[<todo>])=11, m.B2= 3.76273e+08
     it=24: needIt: 10 TRUE, and 1 F.; length(x[<todo>])=10, m.B2= 7.77428e+11
     it=25: needIt: 7 TRUE, and 3 F.; length(x[<todo>])= 7, m.B2= 4.17254e+13
     it=26: needIt: 6 TRUE, and 1 F.; length(x[<todo>])= 6, m.B2= 1.55243e+15
     it=27: needIt: 5 TRUE, and 1 F.; length(x[<todo>])= 5, m.B2= 2.47748e+15
     it=28: needIt: 4 TRUE, and 1 F.; length(x[<todo>])= 4, m.B2= 1.06982e+19
     it=29: needIt: 4 TRUE; length(x[<todo>])= 4, m.B2= 1.40477e+23
     it=30: needIt: 4 TRUE; length(x[<todo>])= 4, m.B2= 1.9693e+27
     it=31: needIt: 3 TRUE, and 1 F.; length(x[<todo>])= 3, m.B2= 7.6538e+30
     it=32: needIt: 3 TRUE; length(x[<todo>])= 3, m.B2= 1.16488e+35
     it=33: needIt: 3 TRUE; length(x[<todo>])= 3, m.B2= 1.88175e+39
     it=34: needIt: 3 TRUE; length(x[<todo>])= 3, m.B2= 3.22081e+43
     it=35: needIt: 3 TRUE; length(x[<todo>])= 3, m.B2= 5.83159e+47
     it=36: needIt: 3 TRUE; length(x[<todo>])= 3, m.B2= 1.11521e+52
     it=37: needIt: 2 TRUE, and 1 F.; length(x[<todo>])= 2, m.B2= 3.51533e+55
     it=38: needIt: 2 TRUE; length(x[<todo>])= 2, m.B2= 7.10714e+59
     it=39: needIt: 2 TRUE; length(x[<todo>])= 2, m.B2= 1.51138e+64
     it=40: needIt: 2 TRUE; length(x[<todo>])= 2, m.B2= 3.37644e+68
     it=41: needIt: 2 TRUE; length(x[<todo>])= 2, m.B2= 7.91477e+72
     it=42: needIt: 2 TRUE; length(x[<todo>])= 2, m.B2= 1.94455e+77 Lrg m.B2
     it=43: needIt: 2 TRUE; length(x[<todo>])= 2, m.B2= 43197.4
     it=44: needIt: 2 TRUE; length(x[<todo>])= 2, m.B2= 1.16214e+09
     it=45: needIt: 1 TRUE, and 1 F.; length(x[<todo>])= 1, m.B2= 2.11103e+12
     it=46: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 5.83147e+16
     it=47: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 1.68004e+21
     it=48: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 5.04365e+25
     it=49: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 1.57649e+30
     it=50: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 5.12638e+34
     it=51: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 1.7329e+39
     it=52: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 6.08495e+43
     it=53: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 2.21796e+48
     it=54: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 8.38622e+52
     it=55: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 3.28706e+57
     it=56: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 1.33476e+62
     it=57: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 5.61156e+66
     it=58: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 2.44114e+71
     it=59: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 1.0982e+76
     it=60: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 5.10636e+80 Lrg m.B2
     it=61: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 2.1182e+08
     it=62: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 1.05047e+13
     it=63: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 5.37602e+17
     logcf(*) end: after 63 iterations.
     user system elapsed
     1.897 0.007 2.597
     > (ct14 <- system.time(lR14 <- logcfR (xM, i=2, d=3, eps=1e-20, trace=TRUE))) # 4 sec
     logcf(x[ 1]= -5, i=2, d=3, eps=1e-20) logcf(*) end: after 26 iterations.
     logcf(x[ 2]= -4.75, i=2, d=3, eps=1e-20) logcf(*) end: after 25 iterations.
     logcf(x[ 3]= -4.5, i=2, d=3, eps=1e-20) logcf(*) end: after 24 iterations.
     logcf(x[ 4]= -4.25, i=2, d=3, eps=1e-20) logcf(*) end: after 24 iterations.
     logcf(x[ 5]= -4, i=2, d=3, eps=1e-20) logcf(*) end: after 23 iterations.
     logcf(x[ 6]= -3.75, i=2, d=3, eps=1e-20) logcf(*) end: after 22 iterations.
     logcf(x[ 7]= -3.5, i=2, d=3, eps=1e-20) logcf(*) end: after 22 iterations.
     logcf(x[ 8]= -3.25, i=2, d=3, eps=1e-20) logcf(*) end: after 21 iterations.
     logcf(x[ 9]= -3, i=2, d=3, eps=1e-20) logcf(*) end: after 20 iterations.
     logcf(x[10]= -2.75, i=2, d=3, eps=1e-20) logcf(*) end: after 19 iterations.
     logcf(x[11]= -2.5, i=2, d=3, eps=1e-20) logcf(*) end: after 19 iterations.
     logcf(x[12]= -2.25, i=2, d=3, eps=1e-20) logcf(*) end: after 18 iterations.
     logcf(x[13]= -2, i=2, d=3, eps=1e-20) logcf(*) end: after 17 iterations.
     logcf(x[14]= -1.75, i=2, d=3, eps=1e-20) logcf(*) end: after 16 iterations.
     logcf(x[15]= -1.5, i=2, d=3, eps=1e-20) logcf(*) end: after 15 iterations.
     logcf(x[16]= -1.25, i=2, d=3, eps=1e-20) logcf(*) end: after 14 iterations.
     logcf(x[17]= -1, i=2, d=3, eps=1e-20) logcf(*) end: after 12 iterations.
     logcf(x[18]= -0.75, i=2, d=3, eps=1e-20) logcf(*) end: after 11 iterations.
     logcf(x[19]= -0.5, i=2, d=3, eps=1e-20) logcf(*) end: after 9 iterations.
     logcf(x[20]= -0.25, i=2, d=3, eps=1e-20) logcf(*) end: after 7 iterations.
     logcf(x[21]= 0, i=2, d=3, eps=1e-20) logcf(*) end: after 0 iterations.
     logcf(x[22]= 0.25, i=2, d=3, eps=1e-20) logcf(*) end: after 8 iterations.
     logcf(x[23]= 0.5, i=2, d=3, eps=1e-20) logcf(*) end: after 13 iterations.
     logcf(x[24]= 0.75, i=2, d=3, eps=1e-20) logcf(*) end: after 20 iterations.
     logcf(x[25]= 0.78125, i=2, d=3, eps=1e-20) logcf(*) end: after 22 iterations.
     logcf(x[26]= 0.8125, i=2, d=3, eps=1e-20) logcf(*) end: after 24 iterations.
     logcf(x[27]= 0.84375, i=2, d=3, eps=1e-20) logcf(*) end: after 27 iterations.
     logcf(x[28]= 0.875, i=2, d=3, eps=1e-20) logcf(*) end: after 30 iterations.
     logcf(x[29]= 0.90625, i=2, d=3, eps=1e-20) logcf(*) end: after 36 iterations.
     logcf(x[30]= 0.9375, i=2, d=3, eps=1e-20) logcf(*) end: after 44 iterations.
     logcf(x[31]= 0.96875, i=2, d=3, eps=1e-20) logcf(*) end: after 63 iterations.
     user system elapsed
     12.420 0.003 13.627
     >
     > all.equal(lR.14, lR14, tol=0) # TRUE
     [1] TRUE
     > identical(lR.14, lR14) # TRUE !! (not sure if on all platforms!)
     [1] TRUE
     >
     > SS <- function(ch, digits=7)
     + sub(paste0("([0-9]{1,",digits,"})[0-9]*e"), "\\1e", ch)
     > ## double prec <--> MPFR: vvvv (same eps)
     > lR.9 <- logcfR.(xM, 2,3, eps=1e-9)
     > ## show:
     > SS(Rmpfr::all.equal(Rmpfr::roundMpfr(lR.9, 64), lR, tol=0))# .. 5.1138e-16
     Error in target == current : comparison of these types is not implemented
     Calls: SS ... <Anonymous> -> <Anonymous> -> .local -> all.equal.numeric
     Execution halted
    Running the tests in 'tests/qgamma-ex.R' failed.
    Complete output:
     > library(DPQ)
     >
     > ###---> Automatically find places where qgamma() is not so precise (PR#2214) :
     > ### For PR#2214, had '1e-8' below and found quite a bit
     > ## see /u/maechler/R/MM/NUMERICS/dpq-functions/beta-gamma-etc/qgamma-ex.R ..
     >
     > ## FIXME: Timing ! --- partly these matplot() partly get quite slow ~?
     > source(system.file(package="Matrix", "test-tools-1.R", mustWork=TRUE))
     Loading required package: tools
     > ##--> showProc.time(), assertError(), relErrV(), ...
     > showProc.time()
     Time (user system elapsed): 1.639 0.056 1.8
     >
     > (doExtras <- DPQ:::doExtras())
     [1] FALSE
     > (sdir <- system.file("safe", package="DPQ")) ## save directory (to read from)
     [1] "/home/hornik/tmp/R.check/r-devel-clang/Work/build/Packages/DPQ/safe"
     >
     > ### Nowadays finds cases in a special region for really small p and cutoff 1e-11 :
     > set.seed(47)
     > n <- if(doExtras) 100 else 32
     > res <- cbind(p=1,df=1,rE=1)[-1,]
     > for(M in 1:(if(doExtras) 20 else 10))
     + for(p in runif(n)) for(df in rlnorm(n)) {
     + r <- 1- pchisq(qchisq(p, df),df)/p
     + if(abs(r) > 1e-11) res <- rbind(res, c(p,df,r))
     + }
     >
     > ### use df in U[0,1]: finds two cases with bound 1e-11
     > for(p in runif(n)/2) for(df in runif(n)) {
     + qq <- qchisq(p, df)
     + if(qq > 0 && p > 0) {
     + r <- 1- pchisq(qq, df) / p
     + if(abs(r) > 1e-11) res <- rbind(res, c(p,df,r))
     + }
     + }
     >
     > ### now df very close to 0 : ==> finds more cases
     > for(p in sort(c(runif(64)/2, exp(-(1+rlnorm(256))))))
     + for(df in 2^-rlnorm(256, mean=2, sdlog=1.5)) {
     + qq <- qchisq(p, df)
     + if(qq > 0 && p > 0) {
     + r <- 1- pchisq(qq, df) / p
     + if(abs(r) > 1e-11) res <- rbind(res, c(p,df,r))
     + }
     + }
     > showProc.time()
     Time (user system elapsed): 0.911 0.028 0.954
     >
     > require(graphics)
     > if(!dev.interactive(orNone=TRUE)) pdf("qgamma-appr.pdf")
     > eaxis <- sfsmisc::eaxis
     >
     > showProc.time()
     Time (user system elapsed): 0.068 0 0.068
     > ## if(nrow(res) > 0) {
     > cat("Found inaccurate examples where pchisq(qchisq(p, df),df) != p\n")
     Found inaccurate examples where pchisq(qchisq(p, df),df) != p
     > ## sort in p, then df:
     > res <- res[order(res[,"p"], res[,"df"]), ]
     > rE <- res[,"rE"]
     > if(nrow(res) > 20) { hist(rE, breaks = 30); rug(rE) }
     > plot(res[,1:2])##--> quite interesting : all along one curve
     > ## p <= 1/2 and df <= 1 (about) !!
     > res <- cbind(res, nDig = round(-log10(abs(rE)), 1))
     > print(res, digits=12)
     p df rE nDig
     [1,] 0.000194375438651 0.02334079639198 -4.05718514340e-08 7.4
     [2,] 0.000605300028912 0.02041606754775 -1.99857908001e-11 10.7
     [3,] 0.001012316063255 0.01855615147677 -2.59145555106e-04 3.6
     [4,] 0.001248285290785 0.01838201076117 -2.84196000067e-10 9.5
     [5,] 0.001682388899865 0.01720736646288 5.53088974600e-04 3.3
     [6,] 0.001746787400790 0.01731189518997 -5.86897839217e-08 7.2
     [7,] 0.002664451237518 0.01599317398629 1.48421342013e-04 3.8
     [8,] 0.002664451237518 0.01618024201222 -3.82806282229e-08 7.4
     [9,] 0.003159421860255 0.01557612780310 -7.92117005632e-06 5.1
     [10,] 0.003159421860255 0.01568183691729 -4.52237520765e-08 7.3
     [11,] 0.004055462418244 0.01493858731306 4.15166391654e-06 5.4
     [12,] 0.004400694140827 0.01459101672970 9.07907026434e-04 3.0
     [13,] 0.004458811277768 0.01457506850867 9.03139988533e-05 4.0
     [14,] 0.004481882165743 0.01468883074316 -3.23309491179e-07 6.5
     [15,] 0.004939609905705 0.01440168350452 -2.81810098879e-06 5.6
     [16,] 0.008824465120182 0.01276352706510 1.21107345756e-04 3.9
     [17,] 0.009040265960535 0.01273711629661 1.38964402733e-05 4.9
     [18,] 0.010839089634828 0.01242499920422 2.63413624246e-10 9.6
     [19,] 0.011642124851282 0.01201471267173 1.44956234150e-04 3.8
     [20,] 0.014753716559535 0.01155624353203 1.52962087441e-10 9.8
     [21,] 0.015499213434879 0.01125420134457 -9.69695930770e-05 4.0
     [22,] 0.015499213434879 0.01135920381800 -9.55739012376e-08 7.0
     [23,] 0.018603016576955 0.01071716109330 1.63971046474e-03 2.8
     [24,] 0.018603016576955 0.01073655493589 2.14388784340e-04 3.7
     [25,] 0.022624242394389 0.01033379525113 -3.37865757594e-09 8.5
     [26,] 0.022624242394389 0.01034206121729 -2.76332994265e-08 7.6
     [27,] 0.023730217356634 0.01016252135853 -1.07732682708e-06 6.0
     [28,] 0.032427027472295 0.00942923095016 5.11205522358e-11 10.3
     [29,] 0.044753525441333 0.00839626444749 1.22224173549e-05 4.9
     [30,] 0.081818424963746 0.00686007746204 8.92777740624e-10 9.0
     [31,] 0.081818424963746 0.00689856335721 2.28502772259e-11 10.6
     [32,] 0.082800309102258 0.00681234719059 4.17997558788e-09 8.4
     [33,] 0.083507718914457 0.00680676700443 9.77167236016e-11 10.0
     [34,] 0.090821658072474 0.00655269761981 -7.16033632386e-09 8.1
     [35,] 0.102294760453517 0.00623563107239 3.69438657444e-09 8.4
     [36,] 0.110869751789691 0.00603268830251 -3.44006823028e-10 9.5
     [37,] 0.123950804624116 0.00571305309327 2.84683721041e-10 9.5
     [38,] 0.127405857731893 0.00562369059572 6.60541454867e-09 8.2
     [39,] 0.135229634154169 0.00540073357520 -2.34762594200e-05 4.6
     [40,] 0.137732279982451 0.00533092076413 2.99285844990e-04 3.5
     [41,] 0.138112917548194 0.00535138710974 -2.05335777981e-06 5.7
     [42,] 0.141100635980184 0.00527305771429 4.31593832968e-05 4.4
     [43,] 0.141100635980184 0.00537073537183 -3.00640179418e-10 9.5
     [44,] 0.142905299416015 0.00523680041306 3.48180824883e-04 3.5
     [45,] 0.145624557210331 0.00526923971034 -1.94501770245e-09 8.7
     [46,] 0.154606872884529 0.00506806894407 -4.59924667240e-07 6.3
     [47,] 0.154606872884529 0.00507366168703 2.72301046933e-07 6.6
     [48,] 0.163535630067488 0.00497650928578 3.39664962823e-11 10.5
     [49,] 0.169741036539408 0.00484181845356 5.31400978776e-09 8.3
     [50,] 0.177327576288650 0.00465956102839 5.53404362603e-05 4.3
     [51,] 0.178169157856761 0.00471949961255 4.79807527043e-10 9.3
     [52,] 0.190094017358772 0.00450373552308 -1.29698447116e-06 5.9
     [53,] 0.190147641510530 0.00453468705710 5.66235636157e-09 8.2
     [54,] 0.200112534472267 0.00442273120514 7.20473680715e-11 10.1
     [55,] 0.201518808589718 0.00439936964342 1.58748569845e-11 10.8
     [56,] 0.201518808589718 0.00439976887947 -9.97182336704e-11 10.0
     [57,] 0.210803673024037 0.00427351441034 -1.70232938856e-10 9.8
     [58,] 0.213058614771766 0.00426179831847 1.10152997834e-11 11.0
     [59,] 0.214780951412088 0.00419869272965 9.79194836326e-09 8.0
     [60,] 0.232805106603566 0.00395399315002 -9.17581020055e-08 7.0
     [61,] 0.249102914025652 0.00380019404026 -1.15818465929e-10 9.9
     [62,] 0.249102914025652 0.00382493512126 -1.39670497390e-11 10.9
     [63,] 0.252076511947811 0.00374903834738 -8.83337205604e-08 7.1
     [64,] 0.253082914021191 0.00375259362798 3.65436092498e-09 8.4
     [65,] 0.253922058700076 0.00371237348323 3.28994798726e-06 5.5
     [66,] 0.254289278570932 0.00374343873151 -1.05664899053e-09 9.0
     [67,] 0.260017499519858 0.00366179605930 2.34859742765e-07 6.6
     [68,] 0.270323906831467 0.00351999192121 -1.56164756277e-04 3.8
     [69,] 0.271699356057456 0.00355068132680 5.13092990317e-09 8.3
     [70,] 0.275516196070002 0.00346804047756 -4.35171547588e-04 3.4
     [71,] 0.280722231049885 0.00348224101220 5.48759926389e-10 9.3
     [72,] 0.284601233201101 0.00344936339590 1.57145851887e-10 9.8
     [73,] 0.290188543054775 0.00336613521112 -5.64443074502e-08 7.2
     [74,] 0.290579022038283 0.00334423496113 1.02667567892e-07 7.0
     [75,] 0.290579022038283 0.00336764858994 2.26061565023e-08 7.6
     [76,] 0.291850198713803 0.00333552811650 -1.27338760580e-06 5.9
     [77,] 0.296521136452775 0.00330308865102 2.25309977453e-07 6.6
     [78,] 0.298034174946132 0.00330462333485 8.42470393447e-09 8.1
     [79,] 0.300556783277253 0.00323922530004 4.66003314391e-05 4.3
     [80,] 0.303182283998467 0.00328704590597 -1.46205270113e-11 10.8
     [81,] 0.322319846303892 0.00306134512927 -1.15130830540e-05 4.9
     [82,] 0.322319846303892 0.00310689001755 8.57751647487e-11 10.1
     [83,] 0.325071272052651 0.00302343293053 -2.47088704493e-04 3.6
     [84,] 0.325071272052651 0.00304146419577 3.18761056051e-06 5.5
     [85,] 0.331888412404218 0.00300837121343 -4.96098895297e-09 8.3
     [86,] 0.362278153188527 0.00278204202032 4.53939330569e-10 9.3
     [87,] 0.385389476781711 0.00260981704384 7.37274796769e-10 9.1
     [88,] 0.425333956955001 0.00232995789362 1.82823025607e-08 7.7
     [89,] 0.439503709203564 0.00222452690840 -4.53585193982e-06 5.3
     [90,] 0.439503709203564 0.00224964327069 -3.02331937263e-10 9.5
     [91,] 0.450804624124430 0.00216770324934 -4.59455036239e-08 7.3
     >
     > if(requireNamespace("scatterplot3d")) {
     + scatterplot3d::scatterplot3d(res[,1:3], type ='h') ## quite interesting:
     + ## the inaccurate (p,df) points are on nice monotone curve !!!
     + ## this is *less* revealing
     + scatterplot3d::scatterplot3d(res[,c("p","df","nDig")], type ='h')
     + }
     Loading required namespace: scatterplot3d
     > rL <- res[abs(res[,'rE']) > 1e-9,]
     > rL <- rL[order(rL[,1],rL[,2]),]
     > rL
     p df rE nDig
     [1,] 0.0001943754 0.023340796 -4.057185e-08 7.4
     [2,] 0.0010123161 0.018556151 -2.591456e-04 3.6
     [3,] 0.0016823889 0.017207366 5.530890e-04 3.3
     [4,] 0.0017467874 0.017311895 -5.868978e-08 7.2
     [5,] 0.0026644512 0.015993174 1.484213e-04 3.8
     [6,] 0.0026644512 0.016180242 -3.828063e-08 7.4
     [7,] 0.0031594219 0.015576128 -7.921170e-06 5.1
     [8,] 0.0031594219 0.015681837 -4.522375e-08 7.3
     [9,] 0.0040554624 0.014938587 4.151664e-06 5.4
     [10,] 0.0044006941 0.014591017 9.079070e-04 3.0
     [11,] 0.0044588113 0.014575069 9.031400e-05 4.0
     [12,] 0.0044818822 0.014688831 -3.233095e-07 6.5
     [13,] 0.0049396099 0.014401684 -2.818101e-06 5.6
     [14,] 0.0088244651 0.012763527 1.211073e-04 3.9
     [15,] 0.0090402660 0.012737116 1.389644e-05 4.9
     [16,] 0.0116421249 0.012014713 1.449562e-04 3.8
     [17,] 0.0154992134 0.011254201 -9.696959e-05 4.0
     [18,] 0.0154992134 0.011359204 -9.557390e-08 7.0
     [19,] 0.0186030166 0.010717161 1.639710e-03 2.8
     [20,] 0.0186030166 0.010736555 2.143888e-04 3.7
     [21,] 0.0226242424 0.010333795 -3.378658e-09 8.5
     [22,] 0.0226242424 0.010342061 -2.763330e-08 7.6
     [23,] 0.0237302174 0.010162521 -1.077327e-06 6.0
     [24,] 0.0447535254 0.008396264 1.222242e-05 4.9
     [25,] 0.0828003091 0.006812347 4.179976e-09 8.4
     [26,] 0.0908216581 0.006552698 -7.160336e-09 8.1
     [27,] 0.1022947605 0.006235631 3.694387e-09 8.4
     [28,] 0.1274058577 0.005623691 6.605415e-09 8.2
     [29,] 0.1352296342 0.005400734 -2.347626e-05 4.6
     [30,] 0.1377322800 0.005330921 2.992858e-04 3.5
     [31,] 0.1381129175 0.005351387 -2.053358e-06 5.7
     [32,] 0.1411006360 0.005273058 4.315938e-05 4.4
     [33,] 0.1429052994 0.005236800 3.481808e-04 3.5
     [34,] 0.1456245572 0.005269240 -1.945018e-09 8.7
     [35,] 0.1546068729 0.005068069 -4.599247e-07 6.3
     [36,] 0.1546068729 0.005073662 2.723010e-07 6.6
     [37,] 0.1697410365 0.004841818 5.314010e-09 8.3
     [38,] 0.1773275763 0.004659561 5.534044e-05 4.3
     [39,] 0.1900940174 0.004503736 -1.296984e-06 5.9
     [40,] 0.1901476415 0.004534687 5.662356e-09 8.2
     [41,] 0.2147809514 0.004198693 9.791948e-09 8.0
     [42,] 0.2328051066 0.003953993 -9.175810e-08 7.0
     [43,] 0.2520765119 0.003749038 -8.833372e-08 7.1
     [44,] 0.2530829140 0.003752594 3.654361e-09 8.4
     [45,] 0.2539220587 0.003712373 3.289948e-06 5.5
     [46,] 0.2542892786 0.003743439 -1.056649e-09 9.0
     [47,] 0.2600174995 0.003661796 2.348597e-07 6.6
     [48,] 0.2703239068 0.003519992 -1.561648e-04 3.8
     [49,] 0.2716993561 0.003550681 5.130930e-09 8.3
     [50,] 0.2755161961 0.003468040 -4.351715e-04 3.4
     [51,] 0.2901885431 0.003366135 -5.644431e-08 7.2
     [52,] 0.2905790220 0.003344235 1.026676e-07 7.0
     [53,] 0.2905790220 0.003367649 2.260616e-08 7.6
     [54,] 0.2918501987 0.003335528 -1.273388e-06 5.9
     [55,] 0.2965211365 0.003303089 2.253100e-07 6.6
     [56,] 0.2980341749 0.003304623 8.424704e-09 8.1
     [57,] 0.3005567833 0.003239225 4.660033e-05 4.3
     [58,] 0.3223198463 0.003061345 -1.151308e-05 4.9
     [59,] 0.3250712721 0.003023433 -2.470887e-04 3.6
     [60,] 0.3250712721 0.003041464 3.187611e-06 5.5
     [61,] 0.3318884124 0.003008371 -4.960989e-09 8.3
     [62,] 0.4253339570 0.002329958 1.828230e-08 7.7
     [63,] 0.4395037092 0.002224527 -4.535852e-06 5.3
     [64,] 0.4508046241 0.002167703 -4.594550e-08 7.3
     > plot(rL[,1:2], type = "b", main = "inaccurate pchisq/qchisq pairs")
     >
     > plot(rL[,1:2], type = "b", log = "x", ylim = range(0, rL[,"df"]),
     + xaxt = "n",
     + main = "inaccurate pchisq/qchisq pairs"); abline(h = 0, lty=2)
     > ## aha -- a perfect line !!
     > lines(res[,1:2], col = adjustcolor(1, 0.5))
     > eaxis(1); axis(1, at = 1/2)
     >
     > d <- as.data.frame(res)
     > plot (df ~ log(p), data = d, type = "b", cex=1/4, col="gray")
     > points(df ~ log(p), data = as.data.frame(rL), col=2, cex = 1/2)
     >
     > summary(fm <- lm (df ~ log(p), data = d, weights = -log(abs(rE))))
    
     Call:
     lm(formula = df ~ log(p), data = d, weights = -log(abs(rE)))
    
     Weighted Residuals:
     Min 1Q Median 3Q Max
     -6.924e-04 -1.443e-04 -2.096e-05 7.786e-05 1.079e-03
    
     Coefficients:
     Estimate Std. Error t value Pr(>|t|)
     (Intercept) 5.168e-06 1.149e-05 0.45 0.654
     log(p) -2.725e-03 3.683e-06 -739.99 <2e-16 ***
     ---
     Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
    
     Residual standard error: 0.0002575 on 89 degrees of freedom
     Multiple R-squared: 0.9998, Adjusted R-squared: 0.9998
     F-statistic: 5.476e+05 on 1 and 89 DF, p-value: < 2.2e-16
    
     > ## R^2 = 0.9998
     >
     > p0 <- 2^seq(-50,-1, by=1/8)
     > dN <- data.frame(p = p0,
     + df = predict(fm, newdata = data.frame(p = p0)))
     > rE <- with(dN, 1- pchisq(qchisq(p, df),df)/p)
     > dN <- cbind(dN, rE = rE, nDig = round(-log10(abs(rE)), 1))
     > print(dN, digits=10)
     p df rE nDig
     1 8.881784197e-16 0.094454797738 -6.103206185e-07 6.2
     2 9.685654347e-16 0.094218673664 -2.417772682e-07 6.6
     3 1.056228096e-15 0.093982549590 1.482101845e-07 6.8
     4 1.151824906e-15 0.093746425517 5.596452101e-07 6.3
     5 1.256073967e-15 0.093510301443 9.925312783e-07 6.0
     6 1.369758374e-15 0.093274177369 -5.456117484e-07 6.3
     7 1.493732098e-15 0.093038053295 -6.476825187e-08 7.2
     8 1.628926404e-15 0.092801929221 4.375367337e-07 6.4
     9 1.776356839e-15 0.092565805147 9.613066765e-07 6.0
     10 1.937130869e-15 0.092329681073 -4.656782389e-07 6.3
     11 2.112456192e-15 0.092093557000 1.060769096e-07 7.0
     12 2.303649813e-15 0.091857432926 6.993074607e-07 6.2
     13 2.512147934e-15 0.091621308852 -6.429918680e-07 6.2
     14 2.739516748e-15 0.091385184778 -1.755328105e-09 8.8
     15 2.987464197e-15 0.091149060704 6.609670441e-07 6.2
     16 3.257852808e-15 0.090912936630 -5.966163394e-07 6.2
     17 3.552713679e-15 0.090676812556 1.141328021e-07 6.9
     18 3.874261739e-15 0.090440688483 8.463782111e-07 6.1
     19 4.224912384e-15 0.090204564409 -3.264588970e-07 6.5
     20 4.607299625e-15 0.089968440335 4.538340435e-07 6.3
     21 5.024295868e-15 0.089732316261 -6.607811160e-07 6.2
     22 5.479033495e-15 0.089496192187 1.675731415e-07 6.8
     23 5.974928394e-15 0.089260068113 -8.888069507e-07 6.1
     24 6.515705616e-15 0.089023944039 -1.237758629e-08 7.9
     25 7.105427358e-15 0.088787819965 8.855723791e-07 6.1
     26 7.748523477e-15 0.088551695892 -8.599123835e-08 7.1
     27 8.449824769e-15 0.088315571818 8.600545690e-07 6.1
     28 9.214599250e-15 0.088079447744 -5.324092944e-08 7.3
     29 1.004859174e-14 0.087843323670 -9.348371277e-07 6.0
     30 1.095806699e-14 0.087607199596 8.590019329e-08 7.1
     31 1.194985679e-14 0.087371075522 -7.374085143e-07 6.1
     32 1.303141123e-14 0.087134951448 3.314589819e-07 6.5
     33 1.421085472e-14 0.086898827375 -4.335491499e-07 6.4
     34 1.549704695e-14 0.086662703301 6.834622716e-07 6.2
     35 1.689964954e-14 0.086426579227 -2.323222548e-08 7.6
     36 1.842919850e-14 0.086190455153 -6.982056668e-07 6.2
     37 2.009718347e-14 0.085954331079 4.935690683e-07 6.3
     38 2.191613398e-14 0.085718207005 -1.230714122e-07 6.9
     39 2.389971358e-14 0.085482082931 -7.079816475e-07 6.1
     40 2.606282246e-14 0.085245958858 5.585870368e-07 6.3
     41 2.842170943e-14 0.085009834784 3.202904830e-08 7.5
     42 3.099409391e-14 0.084773710710 -4.627895094e-07 6.3
     43 3.379929908e-14 0.084537586636 8.786037373e-07 6.1
     44 3.685839700e-14 0.084301462562 4.421566921e-07 6.4
     45 4.019436694e-14 0.084065338488 3.745824395e-08 7.4
     46 4.383226796e-14 0.083829214414 -3.354886757e-07 6.5
     47 4.779942715e-14 0.083593090340 -6.766811509e-07 6.2
     48 5.212564492e-14 0.083356966267 7.928489877e-07 6.1
     49 5.684341886e-14 0.083120842193 5.100597127e-07 6.3
     50 6.198818782e-14 0.082884718119 2.590340281e-07 6.6
     51 6.759859815e-14 0.082648594045 3.977485741e-08 7.4
     52 7.371679400e-14 0.082412469971 -1.477148934e-07 6.8
     53 8.038873388e-14 0.082176345897 -3.034323055e-07 6.5
     54 8.766453592e-14 0.081940221823 -4.273744663e-07 6.4
     55 9.559885430e-14 0.081704097750 -5.195384647e-07 6.3
     56 1.042512898e-13 0.081467973676 -5.799213925e-07 6.2
     57 1.136868377e-13 0.081231849602 -6.085203379e-07 6.2
     58 1.239763756e-13 0.080995725528 -6.053324058e-07 6.2
     59 1.351971963e-13 0.080759601454 -5.703546833e-07 6.2
     60 1.474335880e-13 0.080523477380 -5.035842794e-07 6.3
     61 1.607774678e-13 0.080287353306 -4.050182891e-07 6.4
     62 1.753290718e-13 0.080051229233 -2.746538159e-07 6.6
     63 1.911977086e-13 0.079815105159 -1.124879638e-07 6.9
     64 2.085025797e-13 0.079578981085 8.148216124e-08 7.1
     65 2.273736754e-13 0.079342857011 3.072594555e-07 6.5
     66 2.479527513e-13 0.079106732937 5.648467992e-07 6.2
     67 2.703943926e-13 0.078870608863 -8.276082903e-07 6.1
     68 2.948671760e-13 0.078634484789 -5.012849573e-07 6.3
     69 3.215549355e-13 0.078398360715 -1.431432968e-07 6.8
     70 3.506581437e-13 0.078162236642 2.468195778e-07 6.6
     71 3.823954172e-13 0.077926112568 6.686065366e-07 6.2
     72 4.170051594e-13 0.077689988494 -5.340348010e-07 6.3
     73 4.547473509e-13 0.077453864420 -4.348594862e-08 7.4
     74 4.959055026e-13 0.077217740346 4.788952396e-07 6.3
     75 5.407887852e-13 0.076981616272 -6.077622940e-07 6.2
     76 5.897343520e-13 0.076745492198 -1.660412186e-08 7.8
     77 6.431098711e-13 0.076509368125 6.063946030e-07 6.2
     78 7.013162874e-13 0.076273244051 -3.642605502e-07 6.4
     79 7.647908344e-13 0.076037119977 3.275302165e-07 6.5
     80 8.340103188e-13 0.075800995903 -5.640570402e-07 6.2
     81 9.094947018e-13 0.075564871829 1.965354515e-07 6.7
     82 9.918110051e-13 0.075328747755 -6.159766142e-07 6.2
     83 1.081577570e-12 0.075092623681 2.134272701e-07 6.7
     84 1.179468704e-12 0.074856499608 -5.200023394e-07 6.3
     85 1.286219742e-12 0.074620375534 3.782225700e-07 6.4
     86 1.402632575e-12 0.074384251460 -2.761173419e-07 6.6
     87 1.529581669e-12 0.074148127386 6.909382196e-07 6.2
     88 1.668020638e-12 0.073912003312 1.156952232e-07 6.9
     89 1.818989404e-12 0.073675879238 -4.174157540e-07 6.4
     90 1.983622010e-12 0.073439755164 6.554521572e-07 6.2
     91 2.163155141e-12 0.073203631090 2.014481738e-07 6.7
     92 2.358937408e-12 0.072967507017 -2.104198829e-07 6.7
     93 2.572439484e-12 0.072731382943 -5.801509786e-07 6.2
     94 2.805265149e-12 0.072495258869 6.355293285e-07 6.2
     95 3.059163338e-12 0.072259134795 3.449181070e-07 6.5
     96 3.336041275e-12 0.072023010721 9.644772814e-08 7.0
     97 3.637978807e-12 0.071786886647 -1.098808029e-07 7.0
     98 3.967244020e-12 0.071550762573 -2.740664691e-07 6.6
     99 4.326310282e-12 0.071314638500 -3.961082631e-07 6.4
     100 4.717874816e-12 0.071078514426 -4.760051799e-07 6.3
     101 5.144878969e-12 0.070842390352 -5.137562191e-07 6.3
     102 5.610530299e-12 0.070606266278 -5.093603785e-07 6.3
     103 6.118326675e-12 0.070370142204 -4.628166750e-07 6.3
     104 6.672082550e-12 0.070134018130 -3.741241081e-07 6.4
     105 7.275957614e-12 0.069897894056 -2.432817023e-07 6.6
     106 7.934488041e-12 0.069661769983 -7.028846949e-08 7.2
     107 8.652620563e-12 0.069425645909 1.448565687e-07 6.8
     108 9.435749632e-12 0.069189521835 4.021543881e-07 6.4
     109 1.028975794e-11 0.068953397761 7.016059603e-07 6.2
     110 1.122106060e-11 0.068717273687 -4.176447372e-07 6.4
     111 1.223665335e-11 0.068481149613 -2.873865945e-08 7.5
     112 1.334416510e-11 0.068245025539 4.023232466e-07 6.4
     113 1.455191523e-11 0.068008901466 -5.698258618e-07 6.2
     114 1.586897608e-11 0.067772777392 -4.930799502e-08 7.3
     115 1.730524113e-11 0.067536653318 5.133677458e-07 6.3
     116 1.887149926e-11 0.067300529244 -3.116849310e-07 6.5
     117 2.057951587e-11 0.067064405170 3.404481462e-07 6.5
     118 2.244212120e-11 0.066828281096 -3.848111123e-07 6.4
     119 2.447330670e-11 0.066592157022 3.567795747e-07 6.4
     120 2.668833020e-11 0.066356032948 -2.686901401e-07 6.6
     121 2.910383046e-11 0.066119908875 5.623583754e-07 6.2
     122 3.173795216e-11 0.065883784801 3.667429294e-08 7.4
     123 3.461048225e-11 0.065647660727 -4.365232460e-07 6.4
     124 3.774299853e-11 0.065411536653 5.312784427e-07 6.3
     125 4.115903175e-11 0.065175412579 1.578594558e-07 6.8
     126 4.488424239e-11 0.064939288505 -1.630783120e-07 6.8
     127 4.894661340e-11 0.064703164431 -4.315370583e-07 6.4
     128 5.337666040e-11 0.064467040358 -6.475189818e-07 6.2
     129 5.820766091e-11 0.064230916284 5.516171563e-07 6.3
     130 6.347590433e-11 0.063994792210 4.354002830e-07 6.4
     131 6.922096451e-11 0.063758668136 3.716548337e-07 6.4
     132 7.548599706e-11 0.063522544062 3.603785849e-07 6.4
     133 8.231806350e-11 0.063286419988 4.015693077e-07 6.4
     134 8.976848478e-11 0.063050295914 4.952247737e-07 6.3
     135 9.789322680e-11 0.062814171841 6.413427337e-07 6.2
     136 1.067533208e-10 0.062578047767 -4.864750427e-07 6.3
     137 1.164153218e-10 0.062341923693 -2.302656055e-07 6.6
     138 1.269518087e-10 0.062105799619 7.839833394e-08 7.1
     139 1.384419290e-10 0.061869675545 4.395145157e-07 6.4
     140 1.509719941e-10 0.061633551471 -4.525763320e-07 6.3
     141 1.646361270e-10 0.061397427397 1.860554089e-08 7.7
     142 1.795369696e-10 0.061161303323 5.422315954e-07 6.3
     143 1.957864536e-10 0.060925179250 -1.718042137e-07 6.8
     144 2.135066416e-10 0.060689055176 4.618673999e-07 6.3
     145 2.328306437e-10 0.060452931102 -1.317493019e-07 6.9
     146 2.539036173e-10 0.060216807028 6.119534922e-07 6.2
     147 2.768838580e-10 0.059980682954 1.387356391e-07 6.9
     148 3.019439882e-10 0.059744558880 -2.716851728e-07 6.6
     149 3.292722540e-10 0.059508434806 -6.193156503e-07 6.2
     150 3.590739391e-10 0.059272310733 3.495618646e-07 6.5
     151 3.915729072e-10 0.059036186659 1.222864269e-07 6.9
     152 4.270132832e-10 0.058800062585 -4.221716110e-08 7.4
     153 4.656612873e-10 0.058563938511 -1.439556299e-07 6.8
     154 5.078072346e-10 0.058327814437 -1.829357232e-07 6.7
     155 5.537677160e-10 0.058091690363 -1.591641836e-07 6.8
     156 6.038879765e-10 0.057855566289 -7.264775914e-08 7.1
     157 6.585445080e-10 0.057619442216 7.660678825e-08 7.1
     158 7.181478783e-10 0.057383318142 2.885926981e-07 6.5
     159 7.831458144e-10 0.057147194068 5.633032045e-07 6.2
     160 8.540265665e-10 0.056911069994 -3.010137886e-07 6.5
     161 9.313225746e-10 0.056674945920 1.043142313e-07 7.0
     162 1.015614469e-09 0.056438821846 5.723448235e-07 6.2
     163 1.107535432e-09 0.056202697772 -8.306453525e-08 7.1
     164 1.207775953e-09 0.055966573698 5.155342088e-07 6.3
     165 1.317089016e-09 0.055730449625 1.091064128e-09 9.0
     166 1.436295757e-09 0.055494325551 -4.402707552e-07 6.4
     167 1.566291629e-09 0.055258201477 3.567050869e-07 6.4
     168 1.708053133e-09 0.055022077403 5.624478649e-08 7.2
     169 1.862645149e-09 0.054785953329 -1.711696178e-07 6.8
     170 2.031228938e-09 0.054549829255 -3.255506589e-07 6.5
     171 2.215070864e-09 0.054313705181 -4.069108679e-07 6.4
     172 2.415551906e-09 0.054077581108 -4.152627815e-07 6.4
     173 2.634178032e-09 0.053841457034 -3.506189497e-07 6.5
     174 2.872591513e-09 0.053605332960 -2.129919214e-07 6.7
     175 3.132583258e-09 0.053369208886 -2.394249909e-09 8.6
     176 3.416106266e-09 0.053133084812 2.811614974e-07 6.6
     177 3.725290298e-09 0.052896960738 -4.754652609e-07 6.3
     178 4.062457877e-09 0.052660836664 -4.082860428e-08 7.4
     179 4.430141728e-09 0.052424712591 4.667262948e-07 6.3
     180 4.831103812e-09 0.052188588517 -5.029516048e-08 7.3
     181 5.268356064e-09 0.051952464443 -4.839870040e-07 6.3
     182 5.745183026e-09 0.051716340369 2.526344739e-07 6.6
     183 6.265166516e-09 0.051480216295 -1.969243901e-08 7.7
     184 6.832212532e-09 0.051244092221 -2.087459541e-07 6.7
     185 7.450580597e-09 0.051007968147 -3.145456657e-07 6.5
     186 8.124915754e-09 0.050771844073 -3.371111803e-07 6.5
     187 8.860283457e-09 0.050535720000 -2.764621003e-07 6.6
     188 9.662207623e-09 0.050299595926 -1.326180317e-07 6.9
     189 1.053671213e-08 0.050063471852 9.440140747e-08 7.0
     190 1.149036605e-08 0.049827347778 4.045766026e-07 6.4
     191 1.253033303e-08 0.049591223704 -2.420871066e-07 6.6
     192 1.366442506e-08 0.049355099630 2.395533774e-07 6.6
     193 1.490116119e-08 0.049118975556 -2.252220010e-07 6.6
     194 1.624983151e-08 0.048882851483 4.277948780e-07 6.4
     195 1.772056691e-08 0.048646727409 1.448078844e-07 6.8
     196 1.932441525e-08 0.048410603335 -4.470068493e-08 7.3
     197 2.107342426e-08 0.048174479261 -1.407587547e-07 6.9
     198 2.298073210e-08 0.047938355187 -1.433942325e-07 6.8
     199 2.506066606e-08 0.047702231113 -5.263503899e-08 7.3
     200 2.732885013e-08 0.047466107039 1.314909133e-07 6.9
     201 2.980232239e-08 0.047229982966 4.089557111e-07 6.4
     202 3.249966302e-08 0.046993858892 -2.026429911e-07 6.7
     203 3.544113383e-08 0.046757734818 2.666381839e-07 6.6
     204 3.864883049e-08 0.046521610744 -1.427213570e-07 6.8
     205 4.214684851e-08 0.046285486670 -4.483846767e-07 6.3
     206 4.596146421e-08 0.046049362596 3.109955485e-07 6.5
     207 5.012133212e-08 0.045813238522 2.073633435e-07 6.7
     208 5.465770025e-08 0.045577114448 2.073183537e-07 6.7
     209 5.960464478e-08 0.045340990375 3.108231207e-07 6.5
     210 6.499932603e-08 0.045104866301 -4.225693764e-07 6.4
     211 7.088226765e-08 0.044868742227 -1.068421438e-07 7.0
     212 7.729766099e-08 0.044632618153 3.123190967e-07 6.5
     213 8.429369702e-08 0.044396494079 -8.979926758e-08 7.0
     214 9.192292842e-08 0.044160370005 -3.780712474e-07 6.4
     215 1.002426642e-07 0.043924245931 3.616102325e-07 6.4
     216 1.093154005e-07 0.043688121858 2.956315699e-07 6.5
     217 1.192092896e-07 0.043451997784 3.433583751e-07 6.5
     218 1.299986521e-07 0.043215873710 -3.936604087e-07 6.4
     219 1.417645353e-07 0.042979749636 -1.134241445e-07 6.9
     220 1.545953220e-07 0.042743625562 2.803691416e-07 6.6
     221 1.685873940e-07 0.042507501488 -9.497759912e-08 7.0
     222 1.838458568e-07 0.042271377414 -3.463650020e-07 6.5
     223 2.004853285e-07 0.042035253341 3.982654885e-07 6.4
     224 2.186308010e-07 0.041799129267 3.893573700e-07 6.4
     225 2.384185791e-07 0.041563005193 -3.573736593e-07 6.4
     226 2.599973041e-07 0.041326881119 -1.135260472e-07 6.9
     227 2.835290706e-07 0.041090757045 2.539798655e-07 6.6
     228 3.091906439e-07 0.040854632971 -1.007498764e-07 7.0
     229 3.371747881e-07 0.040618508897 -3.214330988e-07 6.5
     230 3.676917137e-07 0.040382384823 -4.081432583e-07 6.4
     231 4.009706570e-07 0.040146260750 -3.609537691e-07 6.4
     232 4.372616020e-07 0.039910136676 -1.799379994e-07 6.7
     233 4.768371582e-07 0.039674012602 1.348307247e-07 6.9
     234 5.199946082e-07 0.039437888528 -2.309643412e-07 6.6
     235 5.670581412e-07 0.039201764454 3.563334678e-07 6.4
     236 6.183812879e-07 0.038965640380 2.734302492e-07 6.6
     237 6.743495762e-07 0.038729516306 3.344816584e-07 6.5
     238 7.353834273e-07 0.038493392233 -2.537711326e-07 6.6
     239 8.019413140e-07 0.038257268159 1.001817561e-07 7.0
     240 8.745232040e-07 0.038021144085 -1.848126916e-07 6.7
     241 9.536743164e-07 0.037785020011 -3.156868371e-07 6.5
     242 1.039989216e-06 0.037548895937 -2.925440488e-07 6.5
     243 1.134116282e-06 0.037312771863 -1.154876221e-07 6.9
     244 1.236762576e-06 0.037076647789 2.153792392e-07 6.7
     245 1.348699152e-06 0.036840523716 -5.632338751e-08 7.2
     246 1.470766855e-06 0.036604399642 -1.638943248e-07 6.8
     247 1.603882628e-06 0.036368275568 -1.074536702e-07 7.0
     248 1.749046408e-06 0.036132151494 1.128785959e-07 6.9
     249 1.907348633e-06 0.035896027420 -2.381848818e-07 6.6
     250 2.079978433e-06 0.035659903346 3.148260044e-07 6.5
     251 2.268232565e-06 0.035423779272 3.067288504e-07 6.5
     252 2.473525152e-06 0.035187655198 -2.467805615e-07 6.6
     253 2.697398305e-06 0.034951531125 9.809279378e-08 7.0
     254 2.941533709e-06 0.034715407051 -9.217531383e-08 7.0
     255 3.207765256e-06 0.034479282977 -9.840820714e-08 7.0
     256 3.498092816e-06 0.034243158903 7.923717715e-08 7.1
     257 3.814697266e-06 0.034007034829 -2.523280838e-07 6.6
     258 4.159956866e-06 0.033770910755 2.978638209e-07 6.5
     259 4.536465130e-06 0.033534786681 -3.332980778e-07 6.5
     260 4.947050303e-06 0.033298662608 -8.305711496e-08 7.1
     261 5.394796609e-06 0.033062538534 -3.110326594e-07 6.5
     262 5.883067419e-06 0.032826414460 3.314533775e-07 6.5
     263 6.415530512e-06 0.032590290386 -1.553354063e-07 6.8
     264 6.996185632e-06 0.032354166312 2.279407690e-07 6.6
     265 7.629394531e-06 0.032118042238 1.638951825e-07 6.8
     266 8.319913732e-06 0.031881918164 3.135417774e-07 6.5
     267 9.072930260e-06 0.031645794091 3.656237890e-08 7.4
     268 9.894100606e-06 0.031409670017 -1.661173443e-08 7.8
     269 1.078959322e-05 0.031173545943 1.537749293e-07 6.8
     270 1.176613484e-05 0.030937421869 -7.675076596e-08 7.1
     271 1.283106102e-05 0.030701297795 -7.365044086e-08 7.1
     272 1.399237126e-05 0.030465173721 1.628068312e-07 6.8
     273 1.525878906e-05 0.030229049647 2.398761112e-08 7.6
     274 1.663982746e-05 0.029992925573 1.285369383e-07 6.9
     275 1.814586052e-05 0.029756801500 -1.216179157e-07 6.9
     276 1.978820121e-05 0.029520677426 -1.184344594e-07 6.9
     277 2.157918644e-05 0.029284553352 1.377655910e-07 6.9
     278 2.353226967e-05 0.029048429278 6.475006253e-08 7.2
     279 2.566212205e-05 0.028812305204 2.546562226e-07 6.6
     280 2.798474253e-05 0.028576181130 1.358057637e-07 6.9
     281 3.051757812e-05 0.028340057056 -2.762893581e-07 6.6
     282 3.327965493e-05 0.028103932983 1.553088241e-07 6.8
     283 3.629172104e-05 0.027867808909 -2.519743056e-07 6.6
     284 3.957640242e-05 0.027631684835 1.836182478e-07 6.7
     285 4.315837288e-05 0.027395560761 -1.887623498e-07 6.7
     286 4.706453935e-05 0.027159436687 -2.586992773e-07 6.6
     287 5.132424410e-05 0.026923312613 -2.666510612e-08 7.6
     288 5.596948506e-05 0.026687188539 -2.210357075e-08 7.7
     289 6.103515625e-05 0.026451064466 -2.296375117e-07 6.6
     290 6.655930986e-05 0.026214940392 -1.155410669e-07 6.9
     291 7.258344208e-05 0.025978816318 -1.934344240e-07 6.7
     292 7.915280485e-05 0.025742692244 5.977598905e-08 7.2
     293 8.631674575e-05 0.025506568170 1.410127927e-07 6.9
     294 9.412907870e-05 0.025270444096 6.554035215e-08 7.2
     295 1.026484882e-04 0.025034320022 -1.513968153e-07 6.8
     296 1.119389701e-04 0.024798195948 -7.984736650e-09 8.1
     297 1.220703125e-04 0.024562071875 1.377293413e-08 7.9
     298 1.331186197e-04 0.024325947801 -7.096807053e-08 7.1
     299 1.451668842e-04 0.024089823727 2.236347612e-07 6.7
     300 1.583056097e-04 0.023853699653 -3.406038118e-08 7.5
     301 1.726334915e-04 0.023617575579 1.071465702e-07 7.0
     302 1.882581574e-04 0.023381451505 1.915567694e-07 6.7
     303 2.052969764e-04 0.023145327431 -2.153880145e-07 6.7
     304 2.238779402e-04 0.022909203358 -1.943811789e-07 6.7
     305 2.441406250e-04 0.022673079284 -1.853585263e-07 6.7
     306 2.662372394e-04 0.022436955210 -1.734718758e-07 6.8
     307 2.903337683e-04 0.022200831136 -1.439218900e-07 6.8
     308 3.166112194e-04 0.021964707062 -8.196069534e-08 7.1
     309 3.452669830e-04 0.021728582988 2.710548919e-08 7.6
     310 3.765163148e-04 0.021492458914 1.979137831e-07 6.7
     311 4.105939528e-04 0.021256334841 3.784053582e-08 7.4
     312 4.477558805e-04 0.021020210767 -2.082512274e-08 7.7
     313 4.882812500e-04 0.020784086693 3.635452028e-08 7.4
     314 5.324744788e-04 0.020547962619 -1.675743422e-07 6.8
     315 5.806675366e-04 0.020311838545 1.696075650e-07 6.8
     316 6.332224388e-04 0.020075714471 -9.599203343e-08 7.0
     317 6.905339660e-04 0.019839590397 -1.676104155e-07 6.8
     318 7.530326296e-04 0.019603466323 -3.122730075e-08 7.5
     319 8.211879055e-04 0.019367342250 -3.779823499e-08 7.4
     320 8.955117609e-04 0.019131218176 -1.576477535e-07 6.8
     321 9.765625000e-04 0.018895094102 -6.902106886e-09 8.2
     322 1.064948958e-03 0.018658970028 7.899459131e-08 7.1
     323 1.161335073e-03 0.018422845954 1.293463390e-07 6.9
     324 1.266444878e-03 0.018186721880 -1.651599366e-07 6.8
     325 1.381067932e-03 0.017950597806 -9.328012252e-08 7.0
     326 1.506065259e-03 0.017714473733 3.008480842e-08 7.5
     327 1.642375811e-03 0.017478349659 -8.903183435e-08 7.1
     328 1.791023522e-03 0.017242225585 -8.893825099e-08 7.1
     329 1.953125000e-03 0.017006101511 5.866337671e-08 7.2
     330 2.129897915e-03 0.016769977437 7.502565913e-08 7.1
     331 2.322670146e-03 0.016533853363 3.812254845e-09 8.4
     332 2.532889755e-03 0.016297729289 -1.115639656e-07 7.0
     333 2.762135864e-03 0.016061605216 6.307704159e-08 7.2
     334 3.012130518e-03 0.015825481142 -1.675984129e-08 7.8
     335 3.284751622e-03 0.015589357068 -1.223428781e-08 7.9
     336 3.582047044e-03 0.015353232994 1.188536293e-07 6.9
     337 3.906250000e-03 0.015117108920 -1.217006735e-07 6.9
     338 4.259795831e-03 0.014880984846 -1.314145703e-07 6.9
     339 4.645340293e-03 0.014644860772 -1.288061258e-07 6.9
     340 5.065779510e-03 0.014408736698 -5.759581589e-08 7.2
     341 5.524271728e-03 0.014172612625 -1.110605841e-07 7.0
     342 6.024261037e-03 0.013936488551 2.552848555e-08 7.6
     343 6.569503244e-03 0.013700364477 -7.038977068e-08 7.2
     344 7.164094088e-03 0.013464240403 -8.014783059e-08 7.1
     345 7.812500000e-03 0.013228116329 6.514376105e-08 7.2
     346 8.519591661e-03 0.012991992255 -1.226700341e-08 7.9
     347 9.290680586e-03 0.012755868181 3.829306428e-09 8.4
     348 1.013155902e-02 0.012519744108 -1.720011800e-08 7.8
     349 1.104854346e-02 0.012283620034 2.105846508e-08 7.7
     350 1.204852207e-02 0.012047495960 1.170782316e-08 7.9
     351 1.313900649e-02 0.011811371886 6.422403376e-08 7.2
     352 1.432818818e-02 0.011575247812 9.486125396e-08 7.0
     353 1.562500000e-02 0.011339123738 3.894984368e-08 7.4
     354 1.703918332e-02 0.011102999664 3.208461985e-08 7.5
     355 1.858136117e-02 0.010866875591 3.155170891e-08 7.5
     356 2.026311804e-02 0.010630751517 1.297326291e-08 7.9
     357 2.209708691e-02 0.010394627443 -3.001926907e-08 7.5
     358 2.409704415e-02 0.010158503369 7.470777241e-08 7.1
     359 2.627801298e-02 0.009922379295 2.881776751e-08 7.5
     360 2.865637635e-02 0.009686255221 4.355808669e-08 7.4
     361 3.125000000e-02 0.009450131147 3.486256517e-08 7.5
     362 3.407836665e-02 0.009214007073 -4.540193843e-08 7.3
     363 3.716272234e-02 0.008977883000 6.057274970e-08 7.2
     364 4.052623608e-02 0.008741758926 -4.748039517e-08 7.3
     365 4.419417382e-02 0.008505634852 -4.947995325e-08 7.3
     366 4.819408829e-02 0.008269510778 5.303493644e-09 8.3
     367 5.255602595e-02 0.008033386704 2.750297434e-09 8.6
     368 5.731275270e-02 0.007797262630 7.649789580e-09 8.1
     369 6.250000000e-02 0.007561138556 2.575244529e-08 7.6
     370 6.815673329e-02 0.007325014483 2.588700820e-08 7.6
     371 7.432544469e-02 0.007088890409 -3.873515997e-08 7.4
     372 8.105247217e-02 0.006852766335 -2.868914595e-08 7.5
     373 8.838834765e-02 0.006616642261 8.820984831e-09 8.1
     374 9.638817659e-02 0.006380518187 -1.249980452e-08 7.9
     375 1.051120519e-01 0.006144394113 -2.542298283e-08 7.6
     376 1.146255054e-01 0.005908270039 -3.411814475e-08 7.5
     377 1.250000000e-01 0.005672145966 1.097271574e-08 8.0
     378 1.363134666e-01 0.005436021892 -5.883856513e-09 8.2
     379 1.486508894e-01 0.005199897818 -1.778869496e-08 7.7
     380 1.621049443e-01 0.004963773744 -2.477514438e-08 7.6
     381 1.767766953e-01 0.004727649670 -2.316977810e-08 7.6
     382 1.927763532e-01 0.004491525596 -1.391213433e-08 7.9
     383 2.102241038e-01 0.004255401522 4.032074008e-09 8.4
     384 2.292510108e-01 0.004019277448 1.322844057e-10 9.9
     385 2.500000000e-01 0.003783153375 1.667496141e-09 8.8
     386 2.726269332e-01 0.003547029301 -6.843543954e-09 8.2
     387 2.973017788e-01 0.003310905227 -5.830278704e-09 8.2
     388 3.242098887e-01 0.003074781153 -4.568749379e-09 8.3
     389 3.535533906e-01 0.002838657079 -6.372713468e-09 8.2
     390 3.855527064e-01 0.002602533005 4.293532641e-09 8.4
     391 4.204482076e-01 0.002366408931 4.466793713e-09 8.4
     392 4.585020216e-01 0.002130284858 -3.886739153e-09 8.4
     393 5.000000000e-01 0.001894160784 1.495896629e-09 8.8
     >
     > ## } ## only when we find inaccurate regions
     > showProc.time()
     Time (user system elapsed): 0.101 0.016 0.129
     >
     >
     > ## Oops: another qgamma() / qchisq() problem: mostly NaN's == all solved now
     > curve(qgamma(x, 20), 1e-16, 1e-10, log='x')
     > curve(qgamma(x, 20), 1e-300, .99 , log='xy') # and add the critical region from above:
     > abline(v=c(1e-16, 1e-10), col="light blue")
     > curve(qgamma(x, 20), 1e-26, 1e-07, log='x')
     > ##-> now using log=TRUE in same region:
     > curve(qgamma(x, 20, log=TRUE), -38, -16)## no problem!!
     > curve(qgamma(exp(x), 20), add=TRUE, col="green3", n=2001)
     > ## had problem here, but no longer !
     >
     > ##--> Further fix for qgamma: when 'x' is very small: use "log=TRUE of log(x)"!
     >
     > ## had bug (gave NaN), but no longer:
     > (q_12 <- qgamma(1e-12, 20))
     [1] 2.330042
     > all.equal(1e-12, pgamma(q_12, 20), tol=0)# show rel.err (Lnx 64-bit: 4.04e-16)
     [1] "Mean relative difference: 4.038968e-16"
     > stopifnot(
     + all.equal(1e-12, pgamma(q_12, 20), tolerance = 1e-14)
     + )
     >
     >
     > ## --- Nice graphic : --- but amazingly *S..L..O..W*
     >
     > p.qgammaSml <- function(from= 1e-110, to = 1e-5, ylim = c(0.4, 1000),
     + n = 201, k.lab = 3,
     + a1 = c(10, seq(10.1,20, by=.2), 21:105),
     + a2 = seq(110,330, by=10),
     + a3 = seq(350,1600, by=50))
     + {
     + ## Purpose: nice qgamma() lines ``for small x'' aka p
     + ## ----------------------------------------------------------------------
     + ## Author: Martin Maechler, Date: 22 Mar 2004, 14:23
     + x <- exp(seq(log(from), log(to), length = n))
     +
     + op <- par(las=1, lab = c(10,10, 7), xaxs = "i", mex = 0.8)
     + on.exit(par(op))
     + plot(x, qgamma(x, a1[1]), log="xy", ylim=ylim, type='l', xaxt = "n",
     + main = paste("qgamma(x, a) for very small x, a in [",
     + formatC(a1[1]),", ",formatC(max(a1,a2,a3)),"] - log-log", sep=''),
     + sub = R.version.string)
     + lab.x <- pretty(log10(c(from,to)), 20)
     + axis(1, at=10^lab.x, lab = paste("10^",formatC(lab.x),sep=''))
     + if(is.nan(qgamma(1e-12, 20)))
     + text(1e-60, 20, "all NaN", cex = 2)
     + if(!is.finite(qgamma(1e-140, 155)))
     + text(1e-240, 5, "all +Inf", cex = 2)
     +
     + lines.txt <- function(a.s, col = par("col")) {
     + col <- rep(col, length=length(a.s))
     + for(i in seq(along=a.s)) {
     + qx <- qgamma(x, (a <- a.s[i]))
     + if(i %% k.lab == 0 &&
     + any(ifi <- is.finite(qx) & qx >= ylim[1])) {
     + ik <- (i%%(2*k.lab))/k.lab # = 0 or 1
     + j <- quantile(which(ifi), c(.02,(1:3)/4+ ik/10, .98))
     + ## "segments" around the labels :
     + i0 <- 1
     + for(jj in j) {
     + ii <- i0:(jj-1)
     + i2 <- jj + -1:1
     + lines(x[ii], qx[ii], col=col[i])
     + lines(x[i2], qx[i2], col=col[i], type = 'c')
     + i0 <- jj+1
     + }
     + text(x[j], qx[j], formatC(a), col= "gray40", cex = 0.8)
     + }
     + else
     + lines(x, qx, col=col[i])
     +
     + }
     + }
     + oo <- options(warn = -1)
     + lines.txt(a1[-1])
     + lines.txt(a2, col= 2)
     + lines.txt(a3, col= rainbow(length(a3), .8, .8,
     + start = (max(a3)-min(a3))/(1+max(a3))))
     + invisible(options(oo))
     + }
     >
     > showProc.time()
     Time (user system elapsed): 0.028 0 0.034
     >
     > p.qgammaSml()
     > p.qgammaSml(1e-300)
     > p.qgammaSml(1e-300,1e-50, a2= seq(100,360, by=4), a3=seq(350,1500, by=10))
     >
     > showProc.time()
     Time (user system elapsed): 1.471 0.015 1.826
     >
     > ## The "upper" problematic corner:
     > p.qgammaSml(1e-19, 1e-3, a2=NULL,a3=NULL, ylim=c(.1,20))
     > p.qgammaSml(1e-19, 1e-3, a2=seq(1,12, by=.04), ylim=c(.1,20),a3=NULL,k.lab=10)
     > ## now shows the problem (quite well):
     > ## could it be in pgamma()'s inaccuracy, leading to qgamma() bias ?
     > aa <- c(seq(9, 22, by=0.25),seq(22.3,40,by=0.4))
     > caa <- formatC(range(aa))
     > sfsmisc::mult.fig(2)
     > curve(pgamma(x, sh=aa[1]), 0.5, 20, log = 'xy', ylim = c(1e-60, .2),
     + main = sprintf("pgamma(x, a) for a in [%s,%s]", caa[1],caa[2]))
     > for(sh in aa) curve(pgamma(x, sh), add = TRUE, col=2)
     > abline(h=c(1e-15), col="light blue", lty=2)
     >
     > curve(pgamma(x, sh=aa[1]), 0.5, 20, log = 'xy', ylim = c(1e-15, .8),
     + main = sprintf("pgamma(x, a) for a in [%s,%s]", caa[1],caa[2]))
     > for(sh in aa) curve(pgamma(x, sh), add = TRUE, col=2)
     > ## the "border curve" between "Pearson" and "Continued fraction (upper tail)"
     > ## in pgamma.c :
     > curve(pgamma(max(1,x), x), add = TRUE, col=4)
     > ## ==> pgamma() is perfect here {series expansion up to eps_C accuracy}!
     >
     > aa <- c(seq(9, 22, by=0.25),seq(22.3,40.4,by=0.4))
     > p.qgammaSml(1e-24, 1e-5, a1=aa, a2=NULL,a3=NULL, ylim=c(.8,8))
     > ## -------- save the above?
     > aa1 <- c(aa,seq(40.5,90, by=0.5))
     > p.qgammaSml(1e-60, 1e-5, a1=aa1, a2=NULL,a3=NULL, ylim=c(.9, 16))
     > aa2 <- c(aa1, seq(91,150, by= 1))
     > p.qgammaSml(1e-90, 1e-5, a1=aa2, a2=NULL,a3=NULL, ylim=c(.9, 35))
     > aa3 <- c(aa2, seq(150,250, by= 2), seq(253, 400, by=5))
     > p.qgammaSml(1e-200, 1e-5, a1=aa3, a2=NULL,a3=NULL, ylim=c(.9, 100))
     > p.qgammaSml(1e-200, 1e-5, a1=aa3, a2=NULL,a3=NULL, ylim=c(.9, 200),k.lab=9e9)
     > p.qgammaSml(1e-60, 1e-5, a1=aa3, a2=NULL,a3=NULL, ylim=c(.9, 200),k.lab=9e9)
     >
     > showProc.time()
     Time (user system elapsed): 3.865 0.024 4.328
     >
     > ## lower a \> 10
     >
     > curve(qgamma(x, 19), 1e-14, 1e-9, log='x')
     > curve(qgamma(x, 18), 1e-14, 1e-9, log='x')
     > curve(qgamma(x, 15), 1e-11, 5e-9, log='x')
     > curve(qgamma(x, 13), 5e-10, 1e-8, log='x')
     > curve(qgamma(x, 11), 1e-8, 5e-8, log='x')
     > curve(qgamma(x, 10.5), 4.2e-8, 6e-8, log='x')
     > curve(qgamma(x, 10.3), 6e-8, 7e-8, log='x')
     > curve(qgamma(x, 10.2), 7.1e-8, 7.6e-8, log='x')
     > curve(qgamma(x, 10.15),7.7e-8, 7.9e-8, log='x')
     > curve(qgamma(x, 10.14),7.88e-8,7.92e-8, log='x',n=10001)
     >
     > ## no more problems for smaller a!! here:
     > curve(qgamma(x, 10.13), 1e-10, 5e-4, log='x',n=20001)
     > curve(qgamma(x, 10.12), 1e-10, 5e-4, log='x',n=20001)
     > curve(qgamma(x, 10.1), 1e-10, 5e-4, log='x',n=20001)
     >
     > showProc.time()
     Time (user system elapsed): 0.596 0.016 0.618
     >
     > ##--- the "+Inf" / premature "0" case:
     > curve(qgamma(x, 155, log=TRUE), -1500, 0, log='y', n=2001,col=2)
     > curve(qgamma(x, 1e3, log=TRUE), -1500, 0, log='y', n=2001,col=2)
     > ## now works, but slowly and with kink
     > curve(qgamma (x, 1e5, log=TRUE), -3e5, 0, log='y', n=2001,col=2,lwd=3)
     > curve(qgammaAppr(x, 1e5, log=TRUE), add = TRUE, n=2001, col="blue",lwd=.4)
     > ## --- curves are almost "identical"
     > ## ===> the kink *does* come from the initial approx... hmm
     >
     > ## still "identical"
     > curve(qgamma (x, 1e4, log=TRUE), -3e4, 0, log='y', n=2001,col=2)
     > curve(qgammaAppr(x, 1e4, log=TRUE), add = TRUE, n=2001, col="tomato3")
     >
     > ## now see some difference (approx. has kink at ~ -165)
     > curve(qgamma (x, 100, log=TRUE), -200, 0, log='y', n=2001,col=2)
     > curve(qgammaAppr(x, 100, log=TRUE), add = TRUE, n=2001, col="tomato3")
     > ##
     > (kk <- 100 * 2/1.24)# 161.29
     [1] 161.2903
     > curve(qgamma (x, 100, log=TRUE), -1.1*kk, -.95*kk, log='y', n=2001,col=2)
     > curve(qgammaAppr(x, 100, log=TRUE), add = TRUE, n=2001, col="tomato3")
     > abline(v = -kk, col='blue', lty=2)# exactly: kink is at a * 2 / 1.24 = a / .62
     > curve(qgammaAppr(x - 100/.62, 100,log=TRUE), -1e-3, +1e-3)
     >
     > showProc.time()
     Time (user system elapsed): 0.145 0.004 0.149
     >
     > p.qgammaLog <- function(alpha, xl.f = 1.5, xr.f = 0.4, n = 2001)
     + {
     + ## Purpose:
     + ## ----------------------------------------------------------------------
     + ## Arguments:
     + ## ----------------------------------------------------------------------
     + ## Author: Martin Maechler, Date: 30 Mar 2004, 18:44
     + kk <- -alpha / .62 # = (alpha * 2) / (-1.24)
     + curve(qgamma(x, alpha, log=TRUE), xl.f*kk, xr.f*kk, log='y',
     + n=n, col=2, lwd=3.6, lty = 4,
     + main= paste("qgamma(x, alpha=",formatC(alpha,digits=10),", log = TRUE)"))
     + lines(kk, qgamma(kk, alpha, log=TRUE), type = 'h', lty = 3)
     + curve(qgamma (exp(x), alpha), add = TRUE, col="orange", n=n, lwd= 2)
     + curve(qgammaAppr(x, alpha, log=TRUE), add = TRUE, col=3, n=n,lwd = .4)
     + }
     >
     > showProc.time()
     Time (user system elapsed): 0.001 0 0
     >
     > p.qgammaLog(25)
     > p.qgammaLog(16)# ~ [-25, -20]
     > p.qgammaLog(12, 1.2, 0.8)# small problem remaining
     > p.qgammaLog(11, 1.2, 0.8)# even smaller
     > p.qgammaLog(10.5, 1.1, 0.9)# even smaller
     > p.qgammaLog(10.25, 1.1, 0.9)# even smaller
     > ## 2019-08: __nothing__ visible from here on:
     > p.qgammaLog(10.18, 1.02, 0.98)# even smaller
     > p.qgammaLog(10.15, 1.02, 0.98)# even smaller
     > p.qgammaLog(10.14, 1.001, 0.999)# even smaller
     > p.qgammaLog(10.139, 1.0002, 0.9998)#
     > p.qgammaLog(10.138, 1.0002, 0.9998)#
     > p.qgammaLog(10.137, 1.00001, 0.99999)#
     > p.qgammaLog(10.13699, 1.0000001, 0.9999999)#
     > p.qgammaLog(10.1369899, 1.0000001, 0.9999999)#
     > p.qgammaLog(10.1369894, 1.0000001, 0.9999999)#
     > p.qgammaLog(10.1369893, 1.0000001, 0.9999999)# even smaller at -16.34998
     >
     > showProc.time()
     Time (user system elapsed): 0.53 0.016 0.584
     >
     > ##-- here is the boundary --- for 64-bit AMD Opteron ---
     > ## and for 32-bit AMD Athlon
     >
     > p.qgammaLog(10.1369892, 1.0000001, 0.9999999)# no more
     > p.qgammaLog(10.136989, 1.0000001, 0.9999999)#
     > p.qgammaLog(10.136988, 1.0000001, 0.9999999)#
     > p.qgammaLog(10.136985, 1.0000001, 0.9999999)#
     > p.qgammaLog(10.13698, 1.0000001, 0.9999999)#
     > p.qgammaLog(10.13697, 1.0000001, 0.9999999)#
     > p.qgammaLog(10.13695, 1.0000001, 0.9999999)#
     > p.qgammaLog(10.1368, 1.000001, 0.999999)#
     > p.qgammaLog(10.1365, 1.000001, 0.999999)#
     > p.qgammaLog(10.136, 1.000001, 0.999999)#
     > p.qgammaLog(10.125, 1.1, 0.9)# --- see it now
     > p.qgammaLog(10, 1.2, 0.8)
     > p.qgammaLog(9)
     >
     > showProc.time()
     Time (user system elapsed): 0.393 0.024 0.419
     >
     > ## For large alpha: show difference to see problem better
     > ## ---> for alpha >= 10, the x problem starts *roughly* at x = -0.8*alpha
     > ##
     >
     > sfsmisc::mult.fig(2)
     > curve(qgammaAppr(x, 5, log=TRUE), - 8.1, -8, n=2001)
     > curve(qgammaAppr(x- 5/.62, 5, log=TRUE), -1e-15, 0)
     >
     > ## is the kink from pgamma() ? : no: this looks fine,
     > curve(pgamma(x, 1e5, log=TRUE), 1, 2e5, log='x', n=2001,col=2)
     > ## and this does too:
     > curve( dgamma(x, 1e5), .5e5, 2e5); par(new=TRUE)
     > curve( dgamma(x, 1e5, log=TRUE), .5e5, 2e5, col=2, yaxt="n")
     > axis(4,col.axis=2); par(new=TRUE)
     > curve( pgamma(x, 1e5), .5e5, 2e5, n=2001, col=3); par(new=TRUE)
     > curve( pgamma(x, 1e5, log=TRUE), .5e5, 2e5, n=2001, col=4); par(new=TRUE)
     > curve(-pgamma(x, 1e5, log=TRUE,lower=FALSE), .5e5, 2e5, n=2001, col=4)
     > ## all looking nice
     >
     >
     > x <- 10^seq(2,6, length=4001)
     > qx <- qgamma(pgamma(x, 1e5, log=TRUE), 1e5, log=TRUE)
     > plot(x, qx, type ='l', col=2, asp = 1); abline(0,1, lty=3)
     >
     > showProc.time()
     Time (user system elapsed): 0.064 0.003 0.068
     > <0c>
     > ###------------- Approximations of qgamma() ------
     > ##
     >
     > ## source("/u/maechler/R/MM/NUMERICS/dpq-functions/qchisqAppr.R")
     > ##--> qchisqAppr()
     > ##--> qchisqWH [ = Wilson Hilferty ]
     > ##--> qchisqKG [ = Kennedy & Gentle's improvements "a la AS 91" ]
     > ## dyn.load("/u/maechler/R/MM/NUMERICS/dpq-functions/qchisq_appr.so")
     >
     > ## Consider the two different implementations of
     > ## lgamma1p(a) := lgamma(1+a) == log(gamma(1+a) == log(a*gamma(a)) "stable":
     >
     > if(!exists("lseq", mode="function"))
     + lseq <- if(requireNamespace("sfsmisc")) sfsmisc::lseq else
     + function(from, to, length) exp(seq(log(from), log(to), length.out = length))
     >
     > if(require("Rmpfr")) { ##---------------- MPFR numbers -------------------------
     +
     + .mpfr.all.eq <- Rmpfr::all.equal
     + AllEq <- function(target, current, ...)
     + .mpfr.all.eq(target, current, ...,
     + formatFUN = function(x, ...) Rmpfr::format(x, digits = 9))
     +
     + print(gammaE <- Const("gamma",200)); pi. <- Const("pi",200)
     + print(a0 <- (gammaE^2 + pi.^2/6)/2)
     + print(psi2.1 <- -2*zeta(mpfr(3,200)))# == psigamma(1,2) =~ -2.4041138
     + print(a1 <- (psi2.1 - gammaE*(pi.^2/2 + gammaE^2))/6)
     +
     + x <- lseq(1e-30, 0.8, length = if(doExtras) 1000 else 125)
     + x. <- mpfr(x, 200)
     + xct. <- log(x. * gamma(x.)) ## using MPFR arithmetic .. no overflow ...
     + xc2. <- log(x.) + lgamma(x.)## (ditto)
     + print(AllEq(xct., xc2., tol = 0)) # 3.15779......e-57
     + xct <- as.numeric(xct.)
     + stopifnot(exprs = {
     + AllEq(xct., xc2., tol = 1e-45)
     + AllEq(xct , xc2., tol = 1e-15)
     + ##
     + all.equal(lgamma1p(x), lgamma1p(x, tol= 1e-16), tol=0)
     + ## -> no difference; i.e., default tol = 1e-14 seems fine enough!
     + })
     + showProc.time()
     +
     + m.appr <- cbind(log(x*gamma(x)), lgamma(1+x), log(x) + lgamma(x),
     + lgamma1p.(x, k=1, cut=3e-6),
     + lgamma1p.(x, k=2, cut=1e-4),
     + lgamma1p.(x, k=3, cut=8e-4),
     + lgamma1p(x))#, tol= 1e-14), # = default
     +
     + eMat <- m.appr - xct # absolute error
     + ## Relative errors:
     + str(reMat. <- m.appr/xct. - 1)
     + str(reMat <- as(reMat., "array")) # as(., "matrix") fails in older versions
     +
     + matplot(x, eMat , log="x", type="l", lty=1) #-> problematic log(x) + lgamma(x) for "large"
     + matplot(x, abs( eMat), log="xy", type="l", lty=1) #-> but good for small; lgamma1p is much better
     + matplot(x, abs(reMat), log="xy", type="l", lty=1)
     + abline(v= 3.47548562941137e-08, col = "gray80", lwd=3)#<- the cutoff value of lgamma1p()
     + ##---> should use earlier cutoff!
     + ## zoom in:
     +
     + matplot(x, abs(reMat), log="xy", type="l", col=1:7, lty=1,
     + lwd=2, xlim=c(8e-9, 1e-3), ylim = c(1e-18, 1e-7), axes=FALSE, frame=TRUE,
     + main = expression(lgamma1p(x) == log(Gamma(x+1)) ~~~ "approximations"
     + ~~~ abs(rel.Err(.))))
     + eaxis(1); eaxis(2)
     + abline(v= 3.47548562941137e-08, col = "gray80", lwd=3)#<- the cutoff value of lgamma1p()
     + abline(h= c(1,2,4)*.Machine$double.eps, lty=3, col="skyblue")
     + legend("topright", col=1:7, lty=1,lwd=2,
     + c("log(x*gamma(x))", "lgamma(1+x)", "log(x) + lgamma(x)",
     + "lgamma1p.(x, k=1, c=3e-6)",
     + "lgamma1p.(x, k=2, c=1e-4)",
     + "lgamma1p.(x, k=3, c=8e-4)",
     + "lgamma1p(x)"), bty="n", ncol=2)
     + abline(v = c(3e-6, 1e-4, 8e-4), col=4:6, lty=2, lwd=1/2)
     +
     + ## FIXME: do the same for the lgaamma1p_series()
     +
     + ## rm(x., xct., xc2., reMat., eMat, AllEq)
     + detach("package:Rmpfr")
     + showProc.time()
     +
     + } ## if( MPFR ) ----------------------------------------------------------------
     Loading required package: Rmpfr
     Loading required package: gmp
    
     Attaching package: 'gmp'
    
     The following objects are masked from 'package:base':
    
     %*%, apply, crossprod, matrix, tcrossprod
    
     C code of R package 'Rmpfr': GMP using 64 bits per limb
    
    
     Attaching package: 'Rmpfr'
    
     The following object is masked from 'package:gmp':
    
     outer
    
     The following object is masked from 'package:DPQ':
    
     log1mexp
    
     The following objects are masked from 'package:stats':
    
     dbinom, dgamma, dnbinom, dnorm, dpois, dt, pnorm
    
     The following objects are masked from 'package:base':
    
     cbind, pmax, pmin, rbind
    
     1 'mpfr' number of precision 200 bits
     [1] 0.57721566490153286060651209008240243104215933593992359880576723
     1 'mpfr' number of precision 200 bits
     [1] 0.98905599532797255539539565150063470793918352072821409044319567
     1 'mpfr' number of precision 200 bits
     [1] -2.404113806319188570799476323022899981529972584680997763584544
     1 'mpfr' number of precision 200 bits
     [1] -0.90747907608088628901656016735627511492861144907256376094133062
     Error in target == current : comparison of these types is not implemented
     Calls: print ... .mpfr.all.eq -> .mpfr.all.eq -> .local -> all.equal.numeric
     Execution halted
    Running the tests in 'tests/stirlerr-tst.R' failed.
    Complete output:
     > #### Testing stirlerr(), bd0(), ebd0(), dpois_raw(), ...
     > #### ===============================================
     >
     > require(DPQ)
     Loading required package: DPQ
     > for(pkg in c("Rmpfr", "DPQmpfr"))
     + if(!requireNamespace(pkg)) {
     + cat("no CRAN package", sQuote(pkg), " ---> no tests here.\n")
     + q("no")
     + }
     Loading required namespace: Rmpfr
     Loading required namespace: DPQmpfr
     > require("Rmpfr")
     Loading required package: Rmpfr
     Loading required package: gmp
    
     Attaching package: 'gmp'
    
     The following objects are masked from 'package:base':
    
     %*%, apply, crossprod, matrix, tcrossprod
    
     C code of R package 'Rmpfr': GMP using 64 bits per limb
    
    
     Attaching package: 'Rmpfr'
    
     The following object is masked from 'package:gmp':
    
     outer
    
     The following object is masked from 'package:DPQ':
    
     log1mexp
    
     The following objects are masked from 'package:stats':
    
     dbinom, dgamma, dnbinom, dnorm, dpois, dt, pnorm
    
     The following objects are masked from 'package:base':
    
     cbind, pmax, pmin, rbind
    
     >
     > cutoffs <- c(15,35,80,500) # cut points, n=*, in the above "algorithm"
     > ##
     > n <- c(seq(1,15, by=1/4),seq(16, 25, by=1/2), 26:30, seq(32,50, by=2), seq(55,1000, by=5),
     + 20*c(51:99), 50*(40:80), 150*(27:48), 500*(15:20))
     > st.n <- stirlerr(n)# rather use.halves=TRUE, just here , use.halves=FALSE)
     > plot(st.n ~ n, log="xy", type="b") ## looks good now
     > nM <- mpfr(n, 2048)
     > st.nM <- stirlerr(nM, use.halves=FALSE) ## << on purpose
     > all.equal(asNumeric(st.nM), st.n)# TRUE
     [1] TRUE
     > all.equal(st.nM, as(st.n,"mpfr"))# .. difference: 1.05884..............................e-15
     Error in target == current : comparison of these types is not implemented
     Calls: all.equal -> all.equal -> .local -> all.equal.numeric
     Execution halted
Flavor: r-devel-linux-x86_64-debian-clang

Version: 0.5-0
Check: examples
Result: ERROR
    Running examples in ‘DPQ-Ex.R’ failed
    The error most likely occurred in:
    
    > base::assign(".ptime", proc.time(), pos = "CheckExEnv")
    > ### Name: ppoisson
    > ### Title: Direct Computation of 'ppois()' Poisson Distribution
    > ### Probabilities
    > ### Aliases: ppoisErr ppoisD
    > ### Keywords: distribution
    >
    > ### ** Examples
    >
    > (lams <- outer(c(1,2,5), 10^(0:3)))# 10^4 is already slow!
     [,1] [,2] [,3] [,4]
    [1,] 1 10 100 1000
    [2,] 2 20 200 2000
    [3,] 5 50 500 5000
    > system.time(e1 <- sapply(lams, ppoisErr))
     user system elapsed
     0.006 0.000 0.023
    > e1 / .Machine$double.eps
     [1] 0.0 0.5 -1.0 1.0 5.5 1.5 -4.0 -3.0 1.0 -1.0 2.0 2.0
    >
    > ## Try another 'ppFUN' :---------------------------------
    > ## this relies on the fact that it's *only* used on an 'x' of the form 0:M :
    > ppD0 <- function(x, lambda, all.from.0=TRUE)
    + cumsum(dpois(if(all.from.0) 0:x else x, lambda=lambda))
    > ## and test it:
    > p0 <- ppD0 ( 1000, lambda=10)
    > p1 <- ppois(0:1000, lambda=10)
    > stopifnot(all.equal(p0,p1, tol=8*.Machine$double.eps))
    >
    > system.time(p0.slow <- ppoisD(0:1000, lambda=10, all.from.0=FALSE))# not very slow, here
     user system elapsed
     0.002 0.000 0.019
    > p0.1 <- ppoisD(1000, lambda=10)
    > if(requireNamespace("Rmpfr")) {
    + ppoisMpfr <- function(x, lambda) cumsum(Rmpfr::dpois(x, lambda=lambda))
    + p0.best <- ppoisMpfr(0:1000, lambda = Rmpfr::mpfr(10, precBits = 256))
    + AllEq. <- Rmpfr::all.equal
    + AllEq <- function(target, current, ...)
    + AllEq.(target, current, ...,
    + formatFUN = function(x, ...) Rmpfr::format(x, digits = 9))
    + print(AllEq(p0.best, p0, tol = 0)) # 2.06e-18
    + print(AllEq(p0.best, p0.slow, tol = 0)) # the "worst" (4.44e-17)
    + print(AllEq(p0.best, p0.1, tol = 0)) # 1.08e-18
    + }
    Error in target == current : comparison of these types is not implemented
    Calls: print ... AllEq -> AllEq. -> AllEq. -> .local -> all.equal.numeric
    Execution halted
Flavor: r-devel-linux-x86_64-debian-gcc

Version: 0.5-0
Check: tests
Result: ERROR
     Running ‘chisq-nonc-ex.R’ [27s/52s]
     Running ‘dnbinom-tst.R’ [16s/29s]
     Running ‘dnchisq-tst.R’ [0s/1s]
     Running ‘hyper-dist-ex.R’ [19s/35s]
     Running ‘pnbeta-tst.R’ [0s/1s]
     Running ‘pnt-prec.R’ [21s/39s]
     Running ‘ppois-ex.R’ [1s/2s]
     Running ‘qPoisBinom-ex.R’ [0s/1s]
     Running ‘qbeta-dist.R’ [11s/18s]
     Running ‘qbeta-tst.R’ [0s/1s]
     Running ‘qgamma-ex.R’ [9s/14s]
     Running ‘stirlerr-tst.R’ [2s/3s]
     Running ‘t-nonc-tst.R’ [5s/7s]
     Running ‘wienergerm-pchisq-tst.R’ [0s/1s]
     Running ‘wienergerm_nchisq.R’ [6s/9s]
    Running the tests in ‘tests/dnbinom-tst.R’ failed.
    Complete output:
     > #### Testing 1) dbinom_raw(), dnbinomR() and dnbinom.mu()
     > #### 2) log1pmx(), logcf() etc
     > require(DPQ)
     Loading required package: DPQ
     > source(system.file(package="Matrix", "test-tools-1.R", mustWork=TRUE))
     Loading required package: tools
     > ## -> showProc.time(), assertError()
     >
     > (doExtras <- DPQ:::doExtras() && !grepl("valgrind", R.home()))
     [1] FALSE
     >
     > if(!dev.interactive(orNone=TRUE)) pdf("wienergerm-accuracy.pdf")
     >
     >
     > ### 1. Testing dbinom_raw(), dnbinomR() and dnbinom.mu() >>> ../R/dbinom-nbinom.R <<<
     > ### ---------- ../man/dbinom_raw.Rd & ../man/dnbinomR.Rd
     >
     > ## "FIXME:" use sfsmisc :: relErrV() already here
     >
     > ### dbinom() vs dbinom.raw() :
     >
     > for(n in 1:20) {
     + cat("n=",n," ")
     + for(x in 0:n)
     + cat(".")
     + for(p in c(0, .1, .5, .8, 1)) {
     + stopifnot(all.equal(dbinom_raw(x, n, p, q=1-p, log=FALSE),
     + dbinom (x, n, p, log=FALSE)),
     + all.equal(dbinom_raw(x, n, p, q=1-p, log =TRUE),
     + dbinom (x, n, p, log =TRUE)))
     + }
     + cat("\n")
     + }
     n= 1 ..
     n= 2 ...
     n= 3 ....
     n= 4 .....
     n= 5 ......
     n= 6 .......
     n= 7 ........
     n= 8 .........
     n= 9 ..........
     n= 10 ...........
     n= 11 ............
     n= 12 .............
     n= 13 ..............
     n= 14 ...............
     n= 15 ................
     n= 16 .................
     n= 17 ..................
     n= 18 ...................
     n= 19 ....................
     n= 20 .....................
     > showProc.time()
     Time (user system elapsed): 1.763 0.101 3.381
     >
     > ### dnbinom*() :
     > stopifnot(exprs = {
     + dnbinomR(0, 1, 1) == 1
     + })
     >
     > ### exploring 'eps' == "true" tests must be done with Rmpfr !!
     >
     > ### 2. Testing log1pmx(), logcf() etc
     > ### ----------
     >
     > ### 2a: logcf()
     > ## == =======
     > x <- c((-20:3)/4, (25:31)/32) # close (but not too close) to upper bound 1
     >
     > (lC <- logcf (x, i=2, d=3, eps=1e-9))
     [1] 0.2225364 0.2273165 0.2323964 0.2378089 0.2435921 0.2497910 0.2564582
     [8] 0.2636564 0.2714610 0.2799634 0.2892758 0.2995378 0.3109259 0.3236668
     [15] 0.3380584 0.3545014 0.3735507 0.3960035 0.4230578 0.4566237 0.5000000
     [22] 0.5595850 0.6503112 0.8221318 0.8574109 0.8989257 0.9490572 1.0118259
     [29] 1.0948318 1.2152882 1.4286636
     > lCt <- logcf (x, i=2, d=3, eps=1e-9, trace=TRUE) ; stopifnot(identical(lCt, lC))
     it= 0: ==> |b2|=162720
     it= 1: ==> |b2|=1.68458e+08
     it= 2: ==> |b2|=3.02689e+11
     it= 3: ==> |b2|=8.40216e+14
     it= 4: ==> |b2|=3.33607e+18
     it= 5: ==> |b2|=1.79478e+22
     it= 6: ==> |b2|=1.25703e+26
     it= 7: ==> |b2|=1.11146e+30
     it= 8: ==> |b2|=1.21086e+34
     it= 9: ==> |b2|=1.5936e+38
     it=10: ==> |b2|=2.49268e+42
     logcf(*) used 11 iterations.
     it= 0: ==> |b2|=151400
     it= 1: ==> |b2|=1.519e+08
     it= 2: ==> |b2|=2.64707e+11
     it= 3: ==> |b2|=7.12814e+14
     it= 4: ==> |b2|=2.74588e+18
     it= 5: ==> |b2|=1.4333e+22
     it= 6: ==> |b2|=9.73998e+25
     it= 7: ==> |b2|=8.35605e+29
     it= 8: ==> |b2|=8.83286e+33
     it= 9: ==> |b2|=1.12795e+38
     it=10: ==> |b2|=1.71192e+42
     logcf(*) used 11 iterations.
     it= 0: ==> |b2|=140480
     it= 1: ==> |b2|=1.36437e+08
     it= 2: ==> |b2|=2.30332e+11
     it= 3: ==> |b2|=6.0102e+14
     it= 4: ==> |b2|=2.24367e+18
     it= 5: ==> |b2|=1.135e+22
     it= 6: ==> |b2|=7.47503e+25
     it= 7: ==> |b2|=6.21522e+29
     it= 8: ==> |b2|=6.3674e+33
     it= 9: ==> |b2|=7.88061e+37
     it=10: ==> |b2|=1.15921e+42
     logcf(*) used 11 iterations.
     it= 0: ==> |b2|=129960
     it= 1: ==> |b2|=1.22034e+08
     it= 2: ==> |b2|=1.99336e+11
     it= 3: ==> |b2|=5.03394e+14
     it= 4: ==> |b2|=1.81889e+18
     it= 5: ==> |b2|=8.90621e+21
     it= 6: ==> |b2|=5.67763e+25
     it= 7: ==> |b2|=4.56957e+29
     it= 8: ==> |b2|=4.53158e+33
     it= 9: ==> |b2|=5.429e+37
     logcf(*) used 10 iterations.
     it= 0: ==> |b2|=119840
     it= 1: ==> |b2|=1.08655e+08
     it= 2: ==> |b2|=1.71497e+11
     it= 3: ==> |b2|=4.18587e+14
     it= 4: ==> |b2|=1.46194e+18
     it= 5: ==> |b2|=6.91963e+21
     it= 6: ==> |b2|=4.26415e+25
     it= 7: ==> |b2|=3.31759e+29
     it= 8: ==> |b2|=3.18042e+33
     it= 9: ==> |b2|=3.68336e+37
     logcf(*) used 10 iterations.
     it= 0: ==> |b2|=110120
     it= 1: ==> |b2|=9.62638e+07
     it= 2: ==> |b2|=1.46601e+11
     it= 3: ==> |b2|=3.45334e+14
     it= 4: ==> |b2|=1.16411e+18
     it= 5: ==> |b2|=5.31835e+21
     it= 6: ==> |b2|=3.16349e+25
     it= 7: ==> |b2|=2.37577e+29
     it= 8: ==> |b2|=2.19845e+33
     it= 9: ==> |b2|=2.45771e+37
     logcf(*) used 10 iterations.
     it= 0: ==> |b2|=100800
     it= 1: ==> |b2|=8.48232e+07
     it= 2: ==> |b2|=1.24442e+11
     it= 3: ==> |b2|=2.82452e+14
     it= 4: ==> |b2|=9.17519e+17
     it= 5: ==> |b2|=4.03952e+21
     it= 6: ==> |b2|=2.3156e+25
     it= 7: ==> |b2|=1.67591e+29
     it= 8: ==> |b2|=1.49457e+33
     logcf(*) used 9 iterations.
     it= 0: ==> |b2|=91880
     it= 1: ==> |b2|=7.42974e+07
     it= 2: ==> |b2|=1.04819e+11
     it= 3: ==> |b2|=2.28837e+14
     it= 4: ==> |b2|=7.15064e+17
     it= 5: ==> |b2|=3.02848e+21
     it= 6: ==> |b2|=1.67007e+25
     it= 7: ==> |b2|=1.1628e+29
     it= 8: ==> |b2|=9.97611e+32
     logcf(*) used 9 iterations.
     it= 0: ==> |b2|=83360
     it= 1: ==> |b2|=6.46501e+07
     it= 2: ==> |b2|=8.75389e+10
     it= 3: ==> |b2|=1.83464e+14
     it= 4: ==> |b2|=5.50387e+17
     it= 5: ==> |b2|=2.23803e+21
     it= 6: ==> |b2|=1.18496e+25
     it= 7: ==> |b2|=7.92152e+28
     it= 8: ==> |b2|=6.52535e+32
     logcf(*) used 9 iterations.
     it= 0: ==> |b2|=75240
     it= 1: ==> |b2|=5.58449e+07
     it= 2: ==> |b2|=7.24171e+10
     it= 3: ==> |b2|=1.45381e+14
     it= 4: ==> |b2|=4.17809e+17
     it= 5: ==> |b2|=1.6276e+21
     it= 6: ==> |b2|=8.25594e+24
     it= 7: ==> |b2|=5.28764e+28
     logcf(*) used 8 iterations.
     it= 0: ==> |b2|=67520
     it= 1: ==> |b2|=4.78456e+07
     it= 2: ==> |b2|=5.92745e+10
     it= 3: ==> |b2|=1.13708e+14
     it= 4: ==> |b2|=3.12287e+17
     it= 5: ==> |b2|=1.16261e+21
     it= 6: ==> |b2|=5.6361e+24
     it= 7: ==> |b2|=3.44989e+28
     logcf(*) used 8 iterations.
     it= 0: ==> |b2|=60200
     it= 1: ==> |b2|=4.06158e+07
     it= 2: ==> |b2|=4.79397e+10
     it= 3: ==> |b2|=8.76351e+13
     it= 4: ==> |b2|=2.2937e+17
     it= 5: ==> |b2|=8.13827e+20
     it= 6: ==> |b2|=3.76013e+24
     it= 7: ==> |b2|=2.19363e+28
     logcf(*) used 8 iterations.
     it= 0: ==> |b2|=53280
     it= 1: ==> |b2|=3.41194e+07
     it= 2: ==> |b2|=3.82483e+10
     it= 3: ==> |b2|=6.64186e+13
     it= 4: ==> |b2|=1.6515e+17
     it= 5: ==> |b2|=5.56707e+20
     it= 6: ==> |b2|=2.44378e+24
     logcf(*) used 7 iterations.
     it= 0: ==> |b2|=46760
     it= 1: ==> |b2|=2.83198e+07
     it= 2: ==> |b2|=3.0043e+10
     it= 3: ==> |b2|=4.93794e+13
     it= 4: ==> |b2|=1.16224e+17
     it= 5: ==> |b2|=3.70875e+20
     it= 6: ==> |b2|=1.54119e+24
     logcf(*) used 7 iterations.
     it= 0: ==> |b2|=40640
     it= 1: ==> |b2|=2.3181e+07
     it= 2: ==> |b2|=2.31738e+10
     it= 3: ==> |b2|=3.59e+13
     it= 4: ==> |b2|=7.96488e+16
     it= 5: ==> |b2|=2.39588e+20
     logcf(*) used 6 iterations.
     it= 0: ==> |b2|=34920
     it= 1: ==> |b2|=1.86664e+07
     it= 2: ==> |b2|=1.74976e+10
     it= 3: ==> |b2|=2.5422e+13
     it= 4: ==> |b2|=5.29017e+16
     it= 5: ==> |b2|=1.49263e+20
     logcf(*) used 6 iterations.
     it= 0: ==> |b2|=29600
     it= 1: ==> |b2|=1.474e+07
     it= 2: ==> |b2|=1.28785e+10
     it= 3: ==> |b2|=1.74436e+13
     it= 4: ==> |b2|=3.38438e+16
     logcf(*) used 5 iterations.
     it= 0: ==> |b2|=24680
     it= 1: ==> |b2|=1.13653e+07
     it= 2: ==> |b2|=9.18785e+09
     it= 3: ==> |b2|=1.1517e+13
     it= 4: ==> |b2|=2.06815e+16
     logcf(*) used 5 iterations.
     it= 0: ==> |b2|=20160
     it= 1: ==> |b2|=8.50608e+06
     it= 2: ==> |b2|=6.30386e+09
     it= 3: ==> |b2|=7.24564e+12
     logcf(*) used 4 iterations.
     it= 0: ==> |b2|=16040
     it= 1: ==> |b2|=6.12601e+06
     it= 2: ==> |b2|=4.11202e+09
     logcf(*) used 3 iterations.
     logcf(*) used 0 iterations.
     it= 0: ==> |b2|=9000
     it= 1: ==> |b2|=2.65815e+06
     it= 2: ==> |b2|=1.38218e+09
     logcf(*) used 3 iterations.
     it= 0: ==> |b2|=6080
     it= 1: ==> |b2|=1.49776e+06
     it= 2: ==> |b2|=6.50656e+08
     it= 3: ==> |b2|=4.39124e+11
     it= 4: ==> |b2|=4.24985e+14
     logcf(*) used 5 iterations.
     it= 0: ==> |b2|=3560
     it= 1: ==> |b2|=671330
     it= 2: ==> |b2|=2.24237e+08
     it= 3: ==> |b2|=1.16565e+11
     it= 4: ==> |b2|=8.69636e+13
     it= 5: ==> |b2|=8.80714e+16
     it= 6: ==> |b2|=1.16246e+20
     it= 7: ==> |b2|=1.93847e+23
     it= 8: ==> |b2|=3.98491e+26
     logcf(*) used 9 iterations.
     it= 0: ==> |b2|=3273.12
     it= 1: ==> |b2|=589700
     it= 2: ==> |b2|=1.88377e+08
     it= 3: ==> |b2|=9.36959e+10
     it= 4: ==> |b2|=6.68994e+13
     it= 5: ==> |b2|=6.48488e+16
     it= 6: ==> |b2|=8.19327e+19
     it= 7: ==> |b2|=1.30789e+23
     it= 8: ==> |b2|=2.57381e+26
     it= 9: ==> |b2|=6.12129e+29
     logcf(*) used 10 iterations.
     it= 0: ==> |b2|=2992.5
     it= 1: ==> |b2|=512650
     it= 2: ==> |b2|=1.55894e+08
     it= 3: ==> |b2|=7.3859e+10
     it= 4: ==> |b2|=5.02475e+13
     it= 5: ==> |b2|=4.64164e+16
     it= 6: ==> |b2|=5.58911e+19
     it= 7: ==> |b2|=8.50347e+22
     it= 8: ==> |b2|=1.595e+26
     it= 9: ==> |b2|=3.61574e+29
     it=10: ==> |b2|=9.74479e+32
     logcf(*) used 11 iterations.
     it= 0: ==> |b2|=2718.12
     it= 1: ==> |b2|=440109
     it= 2: ==> |b2|=1.26644e+08
     it= 3: ==> |b2|=5.68225e+10
     it= 4: ==> |b2|=3.66244e+13
     it= 5: ==> |b2|=3.20598e+16
     it= 6: ==> |b2|=3.65864e+19
     it= 7: ==> |b2|=5.27587e+22
     it= 8: ==> |b2|=9.37997e+25
     it= 9: ==> |b2|=2.01557e+29
     it=10: ==> |b2|=5.14924e+32
     it=11: ==> |b2|=1.54257e+36
     logcf(*) used 12 iterations.
     it= 0: ==> |b2|=2450
     it= 1: ==> |b2|=372006
     it= 2: ==> |b2|=1.00485e+08
     it= 3: ==> |b2|=4.23633e+10
     it= 4: ==> |b2|=2.56713e+13
     it= 5: ==> |b2|=2.11343e+16
     it= 6: ==> |b2|=2.26869e+19
     it= 7: ==> |b2|=3.07772e+22
     it= 8: ==> |b2|=5.14811e+25
     it= 9: ==> |b2|=1.04082e+29
     it=10: ==> |b2|=2.50192e+32
     it=11: ==> |b2|=7.05238e+35
     it=12: ==> |b2|=2.30384e+39
     logcf(*) used 13 iterations.
     it= 0: ==> |b2|=2188.12
     it= 1: ==> |b2|=308271
     it= 2: ==> |b2|=7.72745e+07
     it= 3: ==> |b2|=3.02658e+10
     it= 4: ==> |b2|=1.70531e+13
     it= 5: ==> |b2|=1.30605e+16
     it= 6: ==> |b2|=1.30466e+19
     it= 7: ==> |b2|=1.64734e+22
     it= 8: ==> |b2|=2.56499e+25
     it= 9: ==> |b2|=4.82765e+28
     it=10: ==> |b2|=1.08039e+32
     it=11: ==> |b2|=2.83535e+35
     it=12: ==> |b2|=8.62389e+38
     it=13: ==> |b2|=3.00926e+42
     it=14: ==> |b2|=1.19409e+46
     it=15: ==> |b2|=5.34632e+49
     logcf(*) used 16 iterations.
     it= 0: ==> |b2|=1932.5
     it= 1: ==> |b2|=248832
     it= 2: ==> |b2|=5.68734e+07
     it= 3: ==> |b2|=2.03226e+10
     it= 4: ==> |b2|=1.04577e+13
     it= 5: ==> |b2|=7.32086e+15
     it= 6: ==> |b2|=6.68834e+18
     it= 7: ==> |b2|=7.72653e+21
     it= 8: ==> |b2|=1.10096e+25
     it= 9: ==> |b2|=1.89662e+28
     it=10: ==> |b2|=3.88536e+31
     it=11: ==> |b2|=9.33474e+34
     it=12: ==> |b2|=2.59938e+38
     it=13: ==> |b2|=8.30457e+41
     it=14: ==> |b2|=3.01718e+45
     it=15: ==> |b2|=1.23692e+49
     it=16: ==> |b2|=5.68258e+52
     it=17: ==> |b2|=2.90768e+56
     it=18: ==> |b2|=1.64796e+60
     logcf(*) used 19 iterations.
     it= 0: ==> |b2|=1683.12
     it= 1: ==> |b2|=193619
     it= 2: ==> |b2|=3.91439e+07
     it= 3: ==> |b2|=1.23338e+10
     it= 4: ==> |b2|=5.59551e+12
     it= 5: ==> |b2|=3.4562e+15
     it= 6: ==> |b2|=2.78868e+18
     it= 7: ==> |b2|=2.84748e+21
     it= 8: ==> |b2|=3.58854e+24
     it= 9: ==> |b2|=5.4701e+27
     it=10: ==> |b2|=9.91885e+30
     it=11: ==> |b2|=2.10987e+34
     it=12: ==> |b2|=5.20269e+37
     it=13: ==> |b2|=1.47211e+41
     it=14: ==> |b2|=4.73732e+44
     it=15: ==> |b2|=1.72036e+48
     it=16: ==> |b2|=7.00164e+51
     it=17: ==> |b2|=3.17394e+55
     it=18: ==> |b2|=1.59374e+59
     it=19: ==> |b2|=8.8205e+62
     it=20: ==> |b2|=5.35623e+66
     it=21: ==> |b2|=3.5541e+70
     it=22: ==> |b2|=2.56724e+74
     it=23: ==> |b2|=2.01172e+78 Lrg |b2|
     it=24: ==> |b2|=147221
     it=25: ==> |b2|=1.34508e+09
     it=26: ==> |b2|=1.32142e+13
     it=27: ==> |b2|=1.39232e+17
     logcf(*) used 28 iterations.
     > (lR <- logcfR(x, i=2, d=3, eps=1e-9))
     [1] 0.2225364 0.2273165 0.2323964 0.2378089 0.2435921 0.2497910 0.2564582
     [8] 0.2636564 0.2714610 0.2799634 0.2892758 0.2995378 0.3109259 0.3236668
     [15] 0.3380584 0.3545014 0.3735507 0.3960035 0.4230578 0.4566237 0.5000000
     [22] 0.5595850 0.6503112 0.8221318 0.8574109 0.8989257 0.9490572 1.0118259
     [29] 1.0948318 1.2152882 1.4286636
     > all.equal(lC, lR, tol = 0) # to see if ..
     [1] TRUE
     > stopifnot(all.equal(lC, lR, tol = 4e-16))
     > lRt <- logcfR(x, i=2, d=3, eps=1e-9, trace=TRUE) ; stopifnot(identical(lRt, lR))
     logcf(x[ 1]= -5, i=2, d=3, eps=1e-09) logcf(*) end: after 11 iterations.
     logcf(x[ 2]= -4.75, i=2, d=3, eps=1e-09) logcf(*) end: after 11 iterations.
     logcf(x[ 3]= -4.5, i=2, d=3, eps=1e-09) logcf(*) end: after 11 iterations.
     logcf(x[ 4]= -4.25, i=2, d=3, eps=1e-09) logcf(*) end: after 10 iterations.
     logcf(x[ 5]= -4, i=2, d=3, eps=1e-09) logcf(*) end: after 10 iterations.
     logcf(x[ 6]= -3.75, i=2, d=3, eps=1e-09) logcf(*) end: after 10 iterations.
     logcf(x[ 7]= -3.5, i=2, d=3, eps=1e-09) logcf(*) end: after 9 iterations.
     logcf(x[ 8]= -3.25, i=2, d=3, eps=1e-09) logcf(*) end: after 9 iterations.
     logcf(x[ 9]= -3, i=2, d=3, eps=1e-09) logcf(*) end: after 9 iterations.
     logcf(x[10]= -2.75, i=2, d=3, eps=1e-09) logcf(*) end: after 8 iterations.
     logcf(x[11]= -2.5, i=2, d=3, eps=1e-09) logcf(*) end: after 8 iterations.
     logcf(x[12]= -2.25, i=2, d=3, eps=1e-09) logcf(*) end: after 8 iterations.
     logcf(x[13]= -2, i=2, d=3, eps=1e-09) logcf(*) end: after 7 iterations.
     logcf(x[14]= -1.75, i=2, d=3, eps=1e-09) logcf(*) end: after 7 iterations.
     logcf(x[15]= -1.5, i=2, d=3, eps=1e-09) logcf(*) end: after 6 iterations.
     logcf(x[16]= -1.25, i=2, d=3, eps=1e-09) logcf(*) end: after 6 iterations.
     logcf(x[17]= -1, i=2, d=3, eps=1e-09) logcf(*) end: after 5 iterations.
     logcf(x[18]= -0.75, i=2, d=3, eps=1e-09) logcf(*) end: after 5 iterations.
     logcf(x[19]= -0.5, i=2, d=3, eps=1e-09) logcf(*) end: after 4 iterations.
     logcf(x[20]= -0.25, i=2, d=3, eps=1e-09) logcf(*) end: after 3 iterations.
     logcf(x[21]= 0, i=2, d=3, eps=1e-09) logcf(*) end: after 0 iterations.
     logcf(x[22]= 0.25, i=2, d=3, eps=1e-09) logcf(*) end: after 3 iterations.
     logcf(x[23]= 0.5, i=2, d=3, eps=1e-09) logcf(*) end: after 5 iterations.
     logcf(x[24]= 0.75, i=2, d=3, eps=1e-09) logcf(*) end: after 9 iterations.
     logcf(x[25]= 0.78125, i=2, d=3, eps=1e-09) logcf(*) end: after 10 iterations.
     logcf(x[26]= 0.8125, i=2, d=3, eps=1e-09) logcf(*) end: after 11 iterations.
     logcf(x[27]= 0.84375, i=2, d=3, eps=1e-09) logcf(*) end: after 12 iterations.
     logcf(x[28]= 0.875, i=2, d=3, eps=1e-09) logcf(*) end: after 13 iterations.
     logcf(x[29]= 0.90625, i=2, d=3, eps=1e-09) logcf(*) end: after 16 iterations.
     logcf(x[30]= 0.9375, i=2, d=3, eps=1e-09) logcf(*) end: after 19 iterations.
     logcf(x[31]= 0.96875, i=2, d=3, eps=1e-09) logcf(*) end: after 28 iterations.
     > lRt2 <- logcfR(x, i=2, d=3, eps=1e-9, trace= 2) ; stopifnot(identical(lRt2,lR))
     logcf(x[ 1]= -5, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 162720 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.0168627
     it= 2: ==> B2= 1.68458e+08 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00303811
     it= 3: ==> B2= 3.02689e+11 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000541327
     it= 4: ==> B2= 8.40216e+14 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 9.60626e-05
     it= 5: ==> B2= 3.33607e+18 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.70167e-05
     it= 6: ==> B2= 1.79478e+22 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.01154e-06
     it= 7: ==> B2= 1.25703e+26 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 5.32664e-07
     it= 8: ==> B2= 1.11146e+30 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 9.4179e-08
     it= 9: ==> B2= 1.21086e+34 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.66472e-08
     it=10: ==> B2= 1.5936e+38 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.942e-09
     it=11: ==> B2= 2.49268e+42 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 5.19854e-10
     logcf(*) end: after 11 iterations.
     logcf(x[ 2]= -4.75, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 151400 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.0157061
     it= 2: ==> B2= 1.519e+08 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00271234
     it= 3: ==> B2= 2.64707e+11 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000463242
     it= 4: ==> B2= 7.12814e+14 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 7.88006e-05
     it= 5: ==> B2= 2.74588e+18 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.33808e-05
     it= 6: ==> B2= 1.4333e+22 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.26999e-06
     it= 7: ==> B2= 9.73998e+25 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.84872e-07
     it= 8: ==> B2= 8.35605e+29 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 6.52292e-08
     it= 9: ==> B2= 8.83286e+33 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.10523e-08
     it=10: ==> B2= 1.12795e+38 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.8723e-09
     it=11: ==> B2= 1.71192e+42 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.1713e-10
     logcf(*) end: after 11 iterations.
     logcf(x[ 3]= -4.5, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 140480 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.014539
     it= 2: ==> B2= 1.36437e+08 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00239872
     it= 3: ==> B2= 2.30332e+11 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000391409
     it= 4: ==> B2= 6.0102e+14 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 6.36152e-05
     it= 5: ==> B2= 2.24367e+18 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.03211e-05
     it= 6: ==> B2= 1.135e+22 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.67293e-06
     it= 7: ==> B2= 7.47503e+25 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.71005e-07
     it= 8: ==> B2= 6.21522e+29 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 4.38843e-08
     it= 9: ==> B2= 6.3674e+33 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 7.10431e-09
     it=10: ==> B2= 7.88061e+37 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.14987e-09
     it=11: ==> B2= 1.15921e+42 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.86085e-10
     logcf(*) end: after 11 iterations.
     logcf(x[ 4]= -4.25, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 129960 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.0133641
     it= 2: ==> B2= 1.22034e+08 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00209863
     it= 3: ==> B2= 1.99336e+11 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000325959
     it= 4: ==> B2= 5.03394e+14 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 5.04302e-05
     it= 5: ==> B2= 1.81889e+18 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 7.78852e-06
     it= 6: ==> B2= 8.90621e+21 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.20172e-06
     it= 7: ==> B2= 5.67763e+25 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.85309e-07
     it= 8: ==> B2= 4.56957e+29 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.85641e-08
     it= 9: ==> B2= 4.53158e+33 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 4.40173e-09
     it=10: ==> B2= 5.429e+37 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 6.78171e-10
     logcf(*) end: after 10 iterations.
     logcf(x[ 5]= -4, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 119840 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.0121853
     it= 2: ==> B2= 1.08655e+08 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00181353
     it= 3: ==> B2= 1.71497e+11 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000266983
     it= 4: ==> B2= 4.18587e+14 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.91528e-05
     it= 5: ==> B2= 1.46194e+18 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 5.73167e-06
     it= 6: ==> B2= 6.91963e+21 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 8.38266e-07
     it= 7: ==> B2= 4.26415e+25 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.22525e-07
     it= 8: ==> B2= 3.31759e+29 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.79016e-08
     it= 9: ==> B2= 3.18042e+33 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.6148e-09
     it=10: ==> B2= 3.68336e+37 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.81854e-10
     logcf(*) end: after 10 iterations.
     logcf(x[ 6]= -3.75, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 110120 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.0110067
     it= 2: ==> B2= 9.62638e+07 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00154491
     it= 3: ==> B2= 1.46601e+11 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00021452
     it= 4: ==> B2= 3.45334e+14 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.96738e-05
     it= 5: ==> B2= 1.16411e+18 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 4.0975e-06
     it= 6: ==> B2= 5.31835e+21 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 5.65252e-07
     it= 7: ==> B2= 3.16349e+25 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 7.79297e-08
     it= 8: ==> B2= 2.37577e+29 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.07396e-08
     it= 9: ==> B2= 2.19845e+33 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.47962e-09
     it=10: ==> B2= 2.45771e+37 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.03808e-10
     logcf(*) end: after 10 iterations.
     logcf(x[ 7]= -3.5, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 100800 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00983372
     it= 2: ==> B2= 8.48232e+07 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00129431
     it= 3: ==> B2= 1.24442e+11 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000168553
     it= 4: ==> B2= 2.82452e+14 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.18673e-05
     it= 5: ==> B2= 9.17519e+17 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.83198e-06
     it= 6: ==> B2= 4.03952e+21 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.66405e-07
     it= 7: ==> B2= 2.3156e+25 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 4.73767e-08
     it= 8: ==> B2= 1.67591e+29 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 6.12337e-09
     it= 9: ==> B2= 1.49457e+33 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 7.91207e-10
     logcf(*) end: after 9 iterations.
     logcf(x[ 8]= -3.25, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 91880 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00867282
     it= 2: ==> B2= 7.42974e+07 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00106327
     it= 3: ==> B2= 1.04819e+11 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000128994
     it= 4: ==> B2= 2.28837e+14 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.55909e-05
     it= 5: ==> B2= 7.15064e+17 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.88109e-06
     it= 6: ==> B2= 3.02848e+21 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.26734e-07
     it= 7: ==> B2= 1.67007e+25 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.7312e-08
     it= 8: ==> B2= 1.1628e+29 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.28859e-09
     it= 9: ==> B2= 9.97611e+32 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.95855e-10
     logcf(*) end: after 9 iterations.
     logcf(x[ 9]= -3, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 83360 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00753175
     it= 2: ==> B2= 6.46501e+07 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000853254
     it= 3: ==> B2= 8.75389e+10 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 9.5671e-05
     it= 4: ==> B2= 1.83464e+14 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.06874e-05
     it= 5: ==> B2= 5.50387e+17 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.19178e-06
     it= 6: ==> B2= 2.23803e+21 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.32765e-07
     it= 7: ==> B2= 1.18496e+25 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.47808e-08
     it= 8: ==> B2= 7.92152e+28 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.64484e-09
     it= 9: ==> B2= 6.52535e+32 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.82987e-10
     logcf(*) end: after 9 iterations.
     logcf(x[10]= -2.75, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 75240 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00641978
     it= 2: ==> B2= 5.58449e+07 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000665641
     it= 3: ==> B2= 7.24171e+10 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 6.83222e-05
     it= 4: ==> B2= 1.45381e+14 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 6.98689e-06
     it= 5: ==> B2= 4.17809e+17 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 7.13234e-07
     it= 6: ==> B2= 1.6276e+21 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 7.27338e-08
     it= 7: ==> B2= 8.25594e+24 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 7.41242e-09
     it= 8: ==> B2= 5.28764e+28 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 7.55083e-10
     logcf(*) end: after 8 iterations.
     logcf(x[11]= -2.5, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 67520 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00534792
     it= 2: ==> B2= 4.78456e+07 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.0005016
     it= 3: ==> B2= 5.92745e+10 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 4.65818e-05
     it= 4: ==> B2= 1.13708e+14 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 4.31004e-06
     it= 5: ==> B2= 3.12287e+17 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.98074e-07
     it= 6: ==> B2= 1.16261e+21 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.67278e-08
     it= 7: ==> B2= 5.6361e+24 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.38642e-09
     it= 8: ==> B2= 3.44989e+28 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.12099e-10
     logcf(*) end: after 8 iterations.
     logcf(x[12]= -2.25, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 60200 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00432911
     it= 2: ==> B2= 4.06158e+07 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00036199
     it= 3: ==> B2= 4.79397e+10 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.99753e-05
     it= 4: ==> B2= 8.76351e+13 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.47309e-06
     it= 5: ==> B2= 2.2937e+17 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.03667e-07
     it= 6: ==> B2= 8.13827e+20 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.67549e-08
     it= 7: ==> B2= 3.76013e+24 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.37743e-09
     it= 8: ==> B2= 2.19363e+28 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.13188e-10
     logcf(*) end: after 8 iterations.
     logcf(x[13]= -2, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 53280 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00337838
     it= 2: ==> B2= 3.41194e+07 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00024722
     it= 3: ==> B2= 3.82483e+10 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.79187e-05
     it= 4: ==> B2= 6.64186e+13 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.29399e-06
     it= 5: ==> B2= 1.6515e+17 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 9.32713e-08
     it= 6: ==> B2= 5.56707e+20 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 6.71575e-09
     it= 7: ==> B2= 2.44378e+24 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 4.83216e-10
     logcf(*) end: after 7 iterations.
     logcf(x[14]= -1.75, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 46760 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00251277
     it= 2: ==> B2= 2.83198e+07 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000157067
     it= 3: ==> B2= 3.0043e+10 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 9.72602e-06
     it= 4: ==> B2= 4.93794e+13 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 6.00029e-07
     it= 5: ==> B2= 1.16224e+17 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.69475e-08
     it= 6: ==> B2= 3.70875e+20 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.27255e-09
     it= 7: ==> B2= 1.54119e+24 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.39681e-10
     logcf(*) end: after 7 iterations.
     logcf(x[15]= -1.5, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 40640 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.001751
     it= 2: ==> B2= 2.3181e+07 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 9.04805e-05
     it= 3: ==> B2= 2.31738e+10 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 4.63221e-06
     it= 4: ==> B2= 3.59e+13 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.36258e-07
     it= 5: ==> B2= 7.96488e+16 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.20265e-08
     it= 6: ==> B2= 2.39588e+20 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 6.11497e-10
     logcf(*) end: after 6 iterations.
     logcf(x[16]= -1.25, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 34920 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00111245
     it= 2: ==> B2= 1.86664e+07 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 4.53708e-05
     it= 3: ==> B2= 1.74976e+10 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.8334e-06
     it= 4: ==> B2= 2.5422e+13 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 7.38016e-08
     it= 5: ==> B2= 5.29017e+16 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.96486e-09
     it= 6: ==> B2= 1.49263e+20 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.18968e-10
     logcf(*) end: after 6 iterations.
     logcf(x[17]= -1, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 29600 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000614941
     it= 2: ==> B2= 1.474e+07 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.84647e-05
     it= 3: ==> B2= 1.28785e+10 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 5.49306e-07
     it= 4: ==> B2= 1.74436e+13 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.62767e-08
     it= 5: ==> B2= 3.38438e+16 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 4.81303e-10
     logcf(*) end: after 5 iterations.
     logcf(x[18]= -0.75, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 24680 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000270385
     it= 2: ==> B2= 1.13653e+07 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 5.33194e-06
     it= 3: ==> B2= 9.18785e+09 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.04153e-07
     it= 4: ==> B2= 1.1517e+13 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.02617e-09
     it= 5: ==> B2= 2.06815e+16 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.93312e-11
     logcf(*) end: after 5 iterations.
     logcf(x[19]= -0.5, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 20160 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 7.69704e-05
     it= 2: ==> B2= 8.50608e+06 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 8.02466e-07
     it= 3: ==> B2= 6.30386e+09 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 8.28445e-09
     it= 4: ==> B2= 7.24564e+12 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 8.51583e-11
     logcf(*) end: after 4 iterations.
     logcf(x[20]= -0.25, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 16040 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 7.26392e-06
     it= 2: ==> B2= 6.12601e+06 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.30773e-08
     it= 3: ==> B2= 4.11202e+09 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 7.25571e-11
     logcf(*) end: after 3 iterations.
     logcf(x[21]= 0, i=2, d=3, eps=1e-09) iterations:
     logcf(*) end: after 0 iterations.
     logcf(x[22]= 0.25, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 9000 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.04918e-05
     it= 2: ==> B2= 2.65815e+06 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.08623e-07
     it= 3: ==> B2= 1.38218e+09 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 5.68393e-10
     logcf(*) end: after 3 iterations.
     logcf(x[23]= 0.5, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 6080 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000659523
     it= 2: ==> B2= 1.49776e+06 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.00942e-05
     it= 3: ==> B2= 6.50656e+08 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 6.02264e-07
     it= 4: ==> B2= 4.39124e+11 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.79273e-08
     it= 5: ==> B2= 4.24985e+14 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 5.3174e-10
     logcf(*) end: after 5 iterations.
     logcf(x[24]= 0.75, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 3560 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00856402
     it= 2: ==> B2= 671330 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00100335
     it= 3: ==> B2= 2.24237e+08 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000114482
     it= 4: ==> B2= 1.16565e+11 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.2922e-05
     it= 5: ==> B2= 8.69636e+13 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.45089e-06
     it= 6: ==> B2= 8.80714e+16 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.62421e-07
     it= 7: ==> B2= 1.16246e+20 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.81486e-08
     it= 8: ==> B2= 1.93847e+23 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.02539e-09
     it= 9: ==> B2= 3.98491e+26 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.25837e-10
     logcf(*) end: after 9 iterations.
     logcf(x[25]= 0.78125, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 3273.12 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.0116683
     it= 2: ==> B2= 589700 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00162549
     it= 3: ==> B2= 1.88377e+08 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000220072
     it= 4: ==> B2= 9.36959e+10 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.94426e-05
     it= 5: ==> B2= 6.68994e+13 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.9163e-06
     it= 6: ==> B2= 6.48488e+16 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 5.19244e-07
     it= 7: ==> B2= 8.19327e+19 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 6.87066e-08
     it= 8: ==> B2= 1.30789e+23 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 9.07922e-09
     it= 9: ==> B2= 2.57381e+26 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.19866e-09
     it=10: ==> B2= 6.12129e+29 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.58143e-10
     logcf(*) end: after 10 iterations.
     logcf(x[26]= 0.8125, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 2992.5 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.0159401
     it= 2: ==> B2= 512650 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00265674
     it= 3: ==> B2= 1.55894e+08 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000429426
     it= 4: ==> B2= 7.3859e+10 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 6.84941e-05
     it= 5: ==> B2= 5.02475e+13 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.08546e-05
     it= 6: ==> B2= 4.64164e+16 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.71404e-06
     it= 7: ==> B2= 5.58911e+19 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.70071e-07
     it= 8: ==> B2= 8.50347e+22 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 4.2492e-08
     it= 9: ==> B2= 1.595e+26 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 6.6788e-09
     it=10: ==> B2= 3.61574e+29 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.04899e-09
     it=11: ==> B2= 9.74479e+32 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.64668e-10
     logcf(*) end: after 11 iterations.
     logcf(x[27]= 0.84375, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 2718.12 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.0218736
     it= 2: ==> B2= 440109 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00440022
     it= 3: ==> B2= 1.26644e+08 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000856838
     it= 4: ==> B2= 5.68225e+10 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000164362
     it= 5: ==> B2= 3.66244e+13 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.12964e-05
     it= 6: ==> B2= 3.20598e+16 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 5.93505e-06
     it= 7: ==> B2= 3.65864e+19 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.12276e-06
     it= 8: ==> B2= 5.27587e+22 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.12056e-07
     it= 9: ==> B2= 9.37997e+25 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 4.00067e-08
     it=10: ==> B2= 2.01557e+29 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 7.54162e-09
     it=11: ==> B2= 5.14924e+32 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.42081e-09
     it=12: ==> B2= 1.54257e+36 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.67552e-10
     logcf(*) end: after 12 iterations.
     logcf(x[28]= 0.875, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 2450 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.0302147
     it= 2: ==> B2= 372006 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00742718
     it= 3: ==> B2= 1.00485e+08 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00176572
     it= 4: ==> B2= 4.23633e+10 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000412688
     it= 5: ==> B2= 2.56713e+13 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 9.56191e-05
     it= 6: ==> B2= 2.11343e+16 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.20491e-05
     it= 7: ==> B2= 2.26869e+19 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 5.0698e-06
     it= 8: ==> B2= 3.07772e+22 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.16356e-06
     it= 9: ==> B2= 5.14811e+25 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.66706e-07
     it=10: ==> B2= 1.04082e+29 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 6.10778e-08
     it=11: ==> B2= 2.50192e+32 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.39778e-08
     it=12: ==> B2= 7.05238e+35 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.19721e-09
     it=13: ==> B2= 2.30384e+39 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 7.31016e-10
     logcf(*) end: after 13 iterations.
     logcf(x[29]= 0.90625, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 2188.12 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.0421192
     it= 2: ==> B2= 308271 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.0128742
     it= 3: ==> B2= 7.72745e+07 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00381265
     it= 4: ==> B2= 3.02658e+10 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00110791
     it= 5: ==> B2= 1.70531e+13 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000318587
     it= 6: ==> B2= 1.30605e+16 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 9.10705e-05
     it= 7: ==> B2= 1.30466e+19 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.5941e-05
     it= 8: ==> B2= 1.64734e+22 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 7.37247e-06
     it= 9: ==> B2= 2.56499e+25 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.09206e-06
     it=10: ==> B2= 4.82765e+28 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 5.93012e-07
     it=11: ==> B2= 1.08039e+32 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.6796e-07
     it=12: ==> B2= 2.83535e+35 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 4.75428e-08
     it=13: ==> B2= 8.62389e+38 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.34511e-08
     it=14: ==> B2= 3.00926e+42 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.80425e-09
     it=15: ==> B2= 1.19409e+46 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.07559e-09
     it=16: ==> B2= 5.34632e+49 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.04032e-10
     logcf(*) end: after 16 iterations.
     logcf(x[30]= 0.9375, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 1932.5 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.0594391
     it= 2: ==> B2= 248832 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.02317
     it= 3: ==> B2= 5.68734e+07 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00882488
     it= 4: ==> B2= 2.03226e+10 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00329775
     it= 5: ==> B2= 1.04577e+13 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00121712
     it= 6: ==> B2= 7.32086e+15 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000445765
     it= 7: ==> B2= 6.68834e+18 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00016248
     it= 8: ==> B2= 7.72653e+21 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 5.90414e-05
     it= 9: ==> B2= 1.10096e+25 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.14101e-05
     it=10: ==> B2= 1.89662e+28 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 7.7527e-06
     it=11: ==> B2= 3.88536e+31 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.80437e-06
     it=12: ==> B2= 9.33474e+34 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.01363e-06
     it=13: ==> B2= 2.59938e+38 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.6615e-07
     it=14: ==> B2= 8.30457e+41 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.32201e-07
     it=15: ==> B2= 3.01718e+45 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 4.77137e-08
     it=16: ==> B2= 1.23692e+49 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.72154e-08
     it=17: ==> B2= 5.68258e+52 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 6.20986e-09
     it=18: ==> B2= 2.90768e+56 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.23951e-09
     it=19: ==> B2= 1.64796e+60 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 8.07503e-10
     logcf(*) end: after 19 iterations.
     logcf(x[31]= 0.96875, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 1683.12 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.0852619
     it= 2: ==> B2= 193619 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.0440308
     it= 3: ==> B2= 3.91439e+07 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.0227933
     it= 4: ==> B2= 1.23338e+10 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.0116823
     it= 5: ==> B2= 5.59551e+12 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00592272
     it= 6: ==> B2= 3.4562e+15 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00297607
     it= 7: ==> B2= 2.78868e+18 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00148555
     it= 8: ==> B2= 2.84748e+21 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00073801
     it= 9: ==> B2= 3.58854e+24 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000365396
     it=10: ==> B2= 5.4701e+27 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000180472
     it=11: ==> B2= 9.91885e+30 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 8.89791e-05
     it=12: ==> B2= 2.10987e+34 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 4.38122e-05
     it=13: ==> B2= 5.20269e+37 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.15512e-05
     it=14: ==> B2= 1.47211e+41 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.05929e-05
     it=15: ==> B2= 4.73732e+44 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 5.20351e-06
     it=16: ==> B2= 1.72036e+48 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.55486e-06
     it=17: ==> B2= 7.00164e+51 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.25391e-06
     it=18: ==> B2= 3.17394e+55 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 6.15209e-07
     it=19: ==> B2= 1.59374e+59 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.0176e-07
     it=20: ==> B2= 8.8205e+62 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.47979e-07
     it=21: ==> B2= 5.35623e+66 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 7.25526e-08
     it=22: ==> B2= 3.5541e+70 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.55657e-08
     it=23: ==> B2= 2.56724e+74 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.7432e-08
     it=24: ==> B2= 2.01172e+78 Lrg m.B2
     --> crit. |A2*b1 - a1*B2|/|b1*B2| = 8.54292e-09
     it=25: ==> B2= 147221 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 4.18617e-09
     it=26: ==> B2= 1.34508e+09 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.05108e-09
     it=27: ==> B2= 1.32142e+13 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.00487e-09
     it=28: ==> B2= 1.39232e+17 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 4.92273e-10
     logcf(*) end: after 28 iterations.
     >
     > lR. <- logcfR.(x, i=2, d=3, eps=1e-9)
     > lR.t <- logcfR.(x, i=2, d=3, eps=1e-9, trace=TRUE) ; stopifnot(identical(lR.t, lR.))
     logcf(x[], i=2, d=3, eps=1e-09) iterations:
     it= 1: needIt: 30 TRUE, and 1 F.; length(x[<todo>])=30, m.B2= 23307.2
     it= 2: needIt: 30 TRUE; length(x[<todo>])=30, m.B2= 1.01159e+07
     it= 3: needIt: 30 TRUE; length(x[<todo>])=30, m.B2= 7.70605e+09
     it= 4: needIt: 28 TRUE, and 2 F.; length(x[<todo>])=28, m.B2= 1.00852e+13
     it= 5: needIt: 27 TRUE, and 1 F.; length(x[<todo>])=27, m.B2= 1.76419e+16
     it= 6: needIt: 24 TRUE, and 3 F.; length(x[<todo>])=24, m.B2= 4.75316e+19
     it= 7: needIt: 22 TRUE, and 2 F.; length(x[<todo>])=22, m.B2= 1.2798e+23
     it= 8: needIt: 20 TRUE, and 2 F.; length(x[<todo>])=20, m.B2= 3.63581e+26
     it= 9: needIt: 17 TRUE, and 3 F.; length(x[<todo>])=17, m.B2= 6.8674e+29
     it=10: needIt: 13 TRUE, and 4 F.; length(x[<todo>])=13, m.B2= 1.03776e+33
     it=11: needIt: 9 TRUE, and 4 F.; length(x[<todo>])= 9, m.B2= 3.09233e+35
     it=12: needIt: 5 TRUE, and 4 F.; length(x[<todo>])= 5, m.B2= 2.27357e+35
     it=13: needIt: 4 TRUE, and 1 F.; length(x[<todo>])= 4, m.B2= 4.04868e+38
     it=14: needIt: 3 TRUE, and 1 F.; length(x[<todo>])= 3, m.B2= 7.16537e+41
     it=15: needIt: 3 TRUE; length(x[<todo>])= 3, m.B2= 2.57468e+45
     it=16: needIt: 3 TRUE; length(x[<todo>])= 3, m.B2= 1.04393e+49
     it=17: needIt: 2 TRUE, and 1 F.; length(x[<todo>])= 2, m.B2= 1.99468e+52
     it=18: needIt: 2 TRUE; length(x[<todo>])= 2, m.B2= 9.60666e+55
     it=19: needIt: 2 TRUE; length(x[<todo>])= 2, m.B2= 5.12487e+59
     it=20: needIt: 1 TRUE, and 1 F.; length(x[<todo>])= 1, m.B2= 8.8205e+62
     it=21: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 5.35623e+66
     it=22: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 3.5541e+70
     it=23: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 2.56724e+74
     it=24: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 2.01172e+78 Lrg m.B2
     it=25: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 147221
     it=26: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 1.34508e+09
     it=27: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 1.32142e+13
     it=28: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 1.39232e+17
     logcf(*) end: after 28 iterations.
     >
     > all.equal(lC, lR., tol = 0) # TRUE !! (every where ?)
     [1] TRUE
     > all.equal(lR, lR., tol = 0) # TRUE !! " "
     [1] TRUE
     > stopifnot(all.equal(lC, lR., tol = 1e-15))
     > ## (even though they used eps=1e-9 .. i.e., are not *so* accurate)
     > showProc.time()
     Time (user system elapsed): 0.023 0.016 0.079
     >
     > ##--- now with improved logcfR.() {<< will become the new logcfR() at least for MPFR !}:
     >
     > ##require(Rmpfr) may be not, see if NS loading (via "::") is sufficient:
     > requireNamespace("Rmpfr") || quit("no")
     Loading required namespace: Rmpfr
     [1] TRUE
     > ## ----- ----------
     > xM <- Rmpfr::mpfr(x, 512)
     > (ct.14 <- system.time(lR.14 <- logcfR.(xM, i=2, d=3, eps=1e-20, trace=TRUE))) # 0.55 sec
     logcf(x[], i=2, d=3, eps=1e-20) iterations:
     it= 1: needIt: 30 TRUE, and 1 F.; length(x[<todo>])=30, m.B2= 23307.2
     it= 2: needIt: 30 TRUE; length(x[<todo>])=30, m.B2= 1.01159e+07
     it= 3: needIt: 30 TRUE; length(x[<todo>])=30, m.B2= 7.70605e+09
     it= 4: needIt: 30 TRUE; length(x[<todo>])=30, m.B2= 9.10781e+12
     it= 5: needIt: 30 TRUE; length(x[<todo>])=30, m.B2= 1.54287e+16
     it= 6: needIt: 30 TRUE; length(x[<todo>])=30, m.B2= 3.54543e+19
     it= 7: needIt: 30 TRUE; length(x[<todo>])=30, m.B2= 1.06137e+23
     it= 8: needIt: 29 TRUE, and 1 F.; length(x[<todo>])=29, m.B2= 4.19177e+26
     it= 9: needIt: 28 TRUE, and 1 F.; length(x[<todo>])=28, m.B2= 2.26761e+30
     it=10: needIt: 27 TRUE, and 1 F.; length(x[<todo>])=27, m.B2= 1.33011e+34
     it=11: needIt: 27 TRUE; length(x[<todo>])=27, m.B2= 9.0823e+37
     it=12: needIt: 26 TRUE, and 1 F.; length(x[<todo>])=26, m.B2= 7.15387e+41
     it=13: needIt: 25 TRUE, and 1 F.; length(x[<todo>])=25, m.B2= 6.21918e+45
     it=14: needIt: 24 TRUE, and 1 F.; length(x[<todo>])=24, m.B2= 9.51187e+49
     it=15: needIt: 23 TRUE, and 1 F.; length(x[<todo>])=23, m.B2= 1.04428e+54
     it=16: needIt: 22 TRUE, and 1 F.; length(x[<todo>])=22, m.B2= 1.19866e+58
     it=17: needIt: 21 TRUE, and 1 F.; length(x[<todo>])=21, m.B2= 1.40641e+62
     it=18: needIt: 20 TRUE, and 1 F.; length(x[<todo>])=20, m.B2= 1.64566e+66
     it=19: needIt: 19 TRUE, and 1 F.; length(x[<todo>])=19, m.B2= 1.86787e+70
     it=20: needIt: 17 TRUE, and 2 F.; length(x[<todo>])=17, m.B2= 9.5095e+73
     it=21: needIt: 15 TRUE, and 2 F.; length(x[<todo>])=15, m.B2= 2.07684e+78 Lrg m.B2
     it=22: needIt: 14 TRUE, and 1 F.; length(x[<todo>])=14, m.B2= 122830
     it=23: needIt: 11 TRUE, and 3 F.; length(x[<todo>])=11, m.B2= 3.76273e+08
     it=24: needIt: 10 TRUE, and 1 F.; length(x[<todo>])=10, m.B2= 7.77428e+11
     it=25: needIt: 7 TRUE, and 3 F.; length(x[<todo>])= 7, m.B2= 4.17254e+13
     it=26: needIt: 6 TRUE, and 1 F.; length(x[<todo>])= 6, m.B2= 1.55243e+15
     it=27: needIt: 5 TRUE, and 1 F.; length(x[<todo>])= 5, m.B2= 2.47748e+15
     it=28: needIt: 4 TRUE, and 1 F.; length(x[<todo>])= 4, m.B2= 1.06982e+19
     it=29: needIt: 4 TRUE; length(x[<todo>])= 4, m.B2= 1.40477e+23
     it=30: needIt: 4 TRUE; length(x[<todo>])= 4, m.B2= 1.9693e+27
     it=31: needIt: 3 TRUE, and 1 F.; length(x[<todo>])= 3, m.B2= 7.6538e+30
     it=32: needIt: 3 TRUE; length(x[<todo>])= 3, m.B2= 1.16488e+35
     it=33: needIt: 3 TRUE; length(x[<todo>])= 3, m.B2= 1.88175e+39
     it=34: needIt: 3 TRUE; length(x[<todo>])= 3, m.B2= 3.22081e+43
     it=35: needIt: 3 TRUE; length(x[<todo>])= 3, m.B2= 5.83159e+47
     it=36: needIt: 3 TRUE; length(x[<todo>])= 3, m.B2= 1.11521e+52
     it=37: needIt: 2 TRUE, and 1 F.; length(x[<todo>])= 2, m.B2= 3.51533e+55
     it=38: needIt: 2 TRUE; length(x[<todo>])= 2, m.B2= 7.10714e+59
     it=39: needIt: 2 TRUE; length(x[<todo>])= 2, m.B2= 1.51138e+64
     it=40: needIt: 2 TRUE; length(x[<todo>])= 2, m.B2= 3.37644e+68
     it=41: needIt: 2 TRUE; length(x[<todo>])= 2, m.B2= 7.91477e+72
     it=42: needIt: 2 TRUE; length(x[<todo>])= 2, m.B2= 1.94455e+77 Lrg m.B2
     it=43: needIt: 2 TRUE; length(x[<todo>])= 2, m.B2= 43197.4
     it=44: needIt: 2 TRUE; length(x[<todo>])= 2, m.B2= 1.16214e+09
     it=45: needIt: 1 TRUE, and 1 F.; length(x[<todo>])= 1, m.B2= 2.11103e+12
     it=46: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 5.83147e+16
     it=47: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 1.68004e+21
     it=48: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 5.04365e+25
     it=49: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 1.57649e+30
     it=50: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 5.12638e+34
     it=51: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 1.7329e+39
     it=52: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 6.08495e+43
     it=53: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 2.21796e+48
     it=54: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 8.38622e+52
     it=55: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 3.28706e+57
     it=56: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 1.33476e+62
     it=57: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 5.61156e+66
     it=58: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 2.44114e+71
     it=59: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 1.0982e+76
     it=60: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 5.10636e+80 Lrg m.B2
     it=61: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 2.1182e+08
     it=62: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 1.05047e+13
     it=63: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 5.37602e+17
     logcf(*) end: after 63 iterations.
     user system elapsed
     1.421 0.019 1.883
     > (ct14 <- system.time(lR14 <- logcfR (xM, i=2, d=3, eps=1e-20, trace=TRUE))) # 4 sec
     logcf(x[ 1]= -5, i=2, d=3, eps=1e-20) logcf(*) end: after 26 iterations.
     logcf(x[ 2]= -4.75, i=2, d=3, eps=1e-20) logcf(*) end: after 25 iterations.
     logcf(x[ 3]= -4.5, i=2, d=3, eps=1e-20) logcf(*) end: after 24 iterations.
     logcf(x[ 4]= -4.25, i=2, d=3, eps=1e-20) logcf(*) end: after 24 iterations.
     logcf(x[ 5]= -4, i=2, d=3, eps=1e-20) logcf(*) end: after 23 iterations.
     logcf(x[ 6]= -3.75, i=2, d=3, eps=1e-20) logcf(*) end: after 22 iterations.
     logcf(x[ 7]= -3.5, i=2, d=3, eps=1e-20) logcf(*) end: after 22 iterations.
     logcf(x[ 8]= -3.25, i=2, d=3, eps=1e-20) logcf(*) end: after 21 iterations.
     logcf(x[ 9]= -3, i=2, d=3, eps=1e-20) logcf(*) end: after 20 iterations.
     logcf(x[10]= -2.75, i=2, d=3, eps=1e-20) logcf(*) end: after 19 iterations.
     logcf(x[11]= -2.5, i=2, d=3, eps=1e-20) logcf(*) end: after 19 iterations.
     logcf(x[12]= -2.25, i=2, d=3, eps=1e-20) logcf(*) end: after 18 iterations.
     logcf(x[13]= -2, i=2, d=3, eps=1e-20) logcf(*) end: after 17 iterations.
     logcf(x[14]= -1.75, i=2, d=3, eps=1e-20) logcf(*) end: after 16 iterations.
     logcf(x[15]= -1.5, i=2, d=3, eps=1e-20) logcf(*) end: after 15 iterations.
     logcf(x[16]= -1.25, i=2, d=3, eps=1e-20) logcf(*) end: after 14 iterations.
     logcf(x[17]= -1, i=2, d=3, eps=1e-20) logcf(*) end: after 12 iterations.
     logcf(x[18]= -0.75, i=2, d=3, eps=1e-20) logcf(*) end: after 11 iterations.
     logcf(x[19]= -0.5, i=2, d=3, eps=1e-20) logcf(*) end: after 9 iterations.
     logcf(x[20]= -0.25, i=2, d=3, eps=1e-20) logcf(*) end: after 7 iterations.
     logcf(x[21]= 0, i=2, d=3, eps=1e-20) logcf(*) end: after 0 iterations.
     logcf(x[22]= 0.25, i=2, d=3, eps=1e-20) logcf(*) end: after 8 iterations.
     logcf(x[23]= 0.5, i=2, d=3, eps=1e-20) logcf(*) end: after 13 iterations.
     logcf(x[24]= 0.75, i=2, d=3, eps=1e-20) logcf(*) end: after 20 iterations.
     logcf(x[25]= 0.78125, i=2, d=3, eps=1e-20) logcf(*) end: after 22 iterations.
     logcf(x[26]= 0.8125, i=2, d=3, eps=1e-20) logcf(*) end: after 24 iterations.
     logcf(x[27]= 0.84375, i=2, d=3, eps=1e-20) logcf(*) end: after 27 iterations.
     logcf(x[28]= 0.875, i=2, d=3, eps=1e-20) logcf(*) end: after 30 iterations.
     logcf(x[29]= 0.90625, i=2, d=3, eps=1e-20) logcf(*) end: after 36 iterations.
     logcf(x[30]= 0.9375, i=2, d=3, eps=1e-20) logcf(*) end: after 44 iterations.
     logcf(x[31]= 0.96875, i=2, d=3, eps=1e-20) logcf(*) end: after 63 iterations.
     user system elapsed
     10.176 0.010 20.399
     >
     > all.equal(lR.14, lR14, tol=0) # TRUE
     [1] TRUE
     > identical(lR.14, lR14) # TRUE !! (not sure if on all platforms!)
     [1] TRUE
     >
     > SS <- function(ch, digits=7)
     + sub(paste0("([0-9]{1,",digits,"})[0-9]*e"), "\\1e", ch)
     > ## double prec <--> MPFR: vvvv (same eps)
     > lR.9 <- logcfR.(xM, 2,3, eps=1e-9)
     > ## show:
     > SS(Rmpfr::all.equal(Rmpfr::roundMpfr(lR.9, 64), lR, tol=0))# .. 5.1138e-16
     Error in target == current : comparison of these types is not implemented
     Calls: SS ... <Anonymous> -> <Anonymous> -> .local -> all.equal.numeric
     Execution halted
    Running the tests in ‘tests/qgamma-ex.R’ failed.
    Complete output:
     > library(DPQ)
     >
     > ###---> Automatically find places where qgamma() is not so precise (PR#2214) :
     > ### For PR#2214, had '1e-8' below and found quite a bit
     > ## see /u/maechler/R/MM/NUMERICS/dpq-functions/beta-gamma-etc/qgamma-ex.R ..
     >
     > ## FIXME: Timing ! --- partly these matplot() partly get quite slow ~?
     > source(system.file(package="Matrix", "test-tools-1.R", mustWork=TRUE))
     Loading required package: tools
     > ##--> showProc.time(), assertError(), relErrV(), ...
     > showProc.time()
     Time (user system elapsed): 1.561 0.101 2.057
     >
     > (doExtras <- DPQ:::doExtras())
     [1] FALSE
     > (sdir <- system.file("safe", package="DPQ")) ## save directory (to read from)
     [1] "/home/hornik/tmp/R.check/r-devel-gcc/Work/build/Packages/DPQ/safe"
     >
     > ### Nowadays finds cases in a special region for really small p and cutoff 1e-11 :
     > set.seed(47)
     > n <- if(doExtras) 100 else 32
     > res <- cbind(p=1,df=1,rE=1)[-1,]
     > for(M in 1:(if(doExtras) 20 else 10))
     + for(p in runif(n)) for(df in rlnorm(n)) {
     + r <- 1- pchisq(qchisq(p, df),df)/p
     + if(abs(r) > 1e-11) res <- rbind(res, c(p,df,r))
     + }
     >
     > ### use df in U[0,1]: finds two cases with bound 1e-11
     > for(p in runif(n)/2) for(df in runif(n)) {
     + qq <- qchisq(p, df)
     + if(qq > 0 && p > 0) {
     + r <- 1- pchisq(qq, df) / p
     + if(abs(r) > 1e-11) res <- rbind(res, c(p,df,r))
     + }
     + }
     >
     > ### now df very close to 0 : ==> finds more cases
     > for(p in sort(c(runif(64)/2, exp(-(1+rlnorm(256))))))
     + for(df in 2^-rlnorm(256, mean=2, sdlog=1.5)) {
     + qq <- qchisq(p, df)
     + if(qq > 0 && p > 0) {
     + r <- 1- pchisq(qq, df) / p
     + if(abs(r) > 1e-11) res <- rbind(res, c(p,df,r))
     + }
     + }
     > showProc.time()
     Time (user system elapsed): 0.635 0.021 1.038
     >
     > require(graphics)
     > if(!dev.interactive(orNone=TRUE)) pdf("qgamma-appr.pdf")
     > eaxis <- sfsmisc::eaxis
     >
     > showProc.time()
     Time (user system elapsed): 0.057 0 0.072
     > ## if(nrow(res) > 0) {
     > cat("Found inaccurate examples where pchisq(qchisq(p, df),df) != p\n")
     Found inaccurate examples where pchisq(qchisq(p, df),df) != p
     > ## sort in p, then df:
     > res <- res[order(res[,"p"], res[,"df"]), ]
     > rE <- res[,"rE"]
     > if(nrow(res) > 20) { hist(rE, breaks = 30); rug(rE) }
     > plot(res[,1:2])##--> quite interesting : all along one curve
     > ## p <= 1/2 and df <= 1 (about) !!
     > res <- cbind(res, nDig = round(-log10(abs(rE)), 1))
     > print(res, digits=12)
     p df rE nDig
     [1,] 0.000194375438651 0.02334079639198 -4.05718514340e-08 7.4
     [2,] 0.000605300028912 0.02041606754775 -1.99857908001e-11 10.7
     [3,] 0.001012316063255 0.01855615147677 -2.59145555106e-04 3.6
     [4,] 0.001248285290785 0.01838201076117 -2.84196000067e-10 9.5
     [5,] 0.001682388899865 0.01720736646288 5.53088974600e-04 3.3
     [6,] 0.001746787400790 0.01731189518997 -5.86897839217e-08 7.2
     [7,] 0.002664451237518 0.01599317398629 1.48421342013e-04 3.8
     [8,] 0.002664451237518 0.01618024201222 -3.82806282229e-08 7.4
     [9,] 0.003159421860255 0.01557612780310 -7.92117005632e-06 5.1
     [10,] 0.003159421860255 0.01568183691729 -4.52237520765e-08 7.3
     [11,] 0.004055462418244 0.01493858731306 4.15166391654e-06 5.4
     [12,] 0.004400694140827 0.01459101672970 9.07907026434e-04 3.0
     [13,] 0.004458811277768 0.01457506850867 9.03139988533e-05 4.0
     [14,] 0.004481882165743 0.01468883074316 -3.23309491179e-07 6.5
     [15,] 0.004939609905705 0.01440168350452 -2.81810098879e-06 5.6
     [16,] 0.008824465120182 0.01276352706510 1.21107345756e-04 3.9
     [17,] 0.009040265960535 0.01273711629661 1.38964402733e-05 4.9
     [18,] 0.010839089634828 0.01242499920422 2.63413624246e-10 9.6
     [19,] 0.011642124851282 0.01201471267173 1.44956234150e-04 3.8
     [20,] 0.014753716559535 0.01155624353203 1.52962087441e-10 9.8
     [21,] 0.015499213434879 0.01125420134457 -9.69695930770e-05 4.0
     [22,] 0.015499213434879 0.01135920381800 -9.55739012376e-08 7.0
     [23,] 0.018603016576955 0.01071716109330 1.63971046474e-03 2.8
     [24,] 0.018603016576955 0.01073655493589 2.14388784340e-04 3.7
     [25,] 0.022624242394389 0.01033379525113 -3.37865757594e-09 8.5
     [26,] 0.022624242394389 0.01034206121729 -2.76332994265e-08 7.6
     [27,] 0.023730217356634 0.01016252135853 -1.07732682708e-06 6.0
     [28,] 0.032427027472295 0.00942923095016 5.11205522358e-11 10.3
     [29,] 0.044753525441333 0.00839626444749 1.22224173549e-05 4.9
     [30,] 0.081818424963746 0.00686007746204 8.92777740624e-10 9.0
     [31,] 0.081818424963746 0.00689856335721 2.28502772259e-11 10.6
     [32,] 0.082800309102258 0.00681234719059 4.17997558788e-09 8.4
     [33,] 0.083507718914457 0.00680676700443 9.77167236016e-11 10.0
     [34,] 0.090821658072474 0.00655269761981 -7.16033632386e-09 8.1
     [35,] 0.102294760453517 0.00623563107239 3.69438657444e-09 8.4
     [36,] 0.110869751789691 0.00603268830251 -3.44006823028e-10 9.5
     [37,] 0.123950804624116 0.00571305309327 2.84683721041e-10 9.5
     [38,] 0.127405857731893 0.00562369059572 6.60541454867e-09 8.2
     [39,] 0.135229634154169 0.00540073357520 -2.34762594200e-05 4.6
     [40,] 0.137732279982451 0.00533092076413 2.99285844990e-04 3.5
     [41,] 0.138112917548194 0.00535138710974 -2.05335777981e-06 5.7
     [42,] 0.141100635980184 0.00527305771429 4.31593832968e-05 4.4
     [43,] 0.141100635980184 0.00537073537183 -3.00640179418e-10 9.5
     [44,] 0.142905299416015 0.00523680041306 3.48180824883e-04 3.5
     [45,] 0.145624557210331 0.00526923971034 -1.94501770245e-09 8.7
     [46,] 0.154606872884529 0.00506806894407 -4.59924667240e-07 6.3
     [47,] 0.154606872884529 0.00507366168703 2.72301046933e-07 6.6
     [48,] 0.163535630067488 0.00497650928578 3.39664962823e-11 10.5
     [49,] 0.169741036539408 0.00484181845356 5.31400978776e-09 8.3
     [50,] 0.177327576288650 0.00465956102839 5.53404362603e-05 4.3
     [51,] 0.178169157856761 0.00471949961255 4.79807527043e-10 9.3
     [52,] 0.190094017358772 0.00450373552308 -1.29698447116e-06 5.9
     [53,] 0.190147641510530 0.00453468705710 5.66235636157e-09 8.2
     [54,] 0.200112534472267 0.00442273120514 7.20473680715e-11 10.1
     [55,] 0.201518808589718 0.00439936964342 1.58748569845e-11 10.8
     [56,] 0.201518808589718 0.00439976887947 -9.97182336704e-11 10.0
     [57,] 0.210803673024037 0.00427351441034 -1.70232938856e-10 9.8
     [58,] 0.213058614771766 0.00426179831847 1.10152997834e-11 11.0
     [59,] 0.214780951412088 0.00419869272965 9.79194836326e-09 8.0
     [60,] 0.232805106603566 0.00395399315002 -9.17581020055e-08 7.0
     [61,] 0.249102914025652 0.00380019404026 -1.15818465929e-10 9.9
     [62,] 0.249102914025652 0.00382493512126 -1.39670497390e-11 10.9
     [63,] 0.252076511947811 0.00374903834738 -8.83337205604e-08 7.1
     [64,] 0.253082914021191 0.00375259362798 3.65436092498e-09 8.4
     [65,] 0.253922058700076 0.00371237348323 3.28994798726e-06 5.5
     [66,] 0.254289278570932 0.00374343873151 -1.05664899053e-09 9.0
     [67,] 0.260017499519858 0.00366179605930 2.34859742765e-07 6.6
     [68,] 0.270323906831467 0.00351999192121 -1.56164756277e-04 3.8
     [69,] 0.271699356057456 0.00355068132680 5.13092990317e-09 8.3
     [70,] 0.275516196070002 0.00346804047756 -4.35171547588e-04 3.4
     [71,] 0.280722231049885 0.00348224101220 5.48759926389e-10 9.3
     [72,] 0.284601233201101 0.00344936339590 1.57145851887e-10 9.8
     [73,] 0.290188543054775 0.00336613521112 -5.64443074502e-08 7.2
     [74,] 0.290579022038283 0.00334423496113 1.02667567892e-07 7.0
     [75,] 0.290579022038283 0.00336764858994 2.26061565023e-08 7.6
     [76,] 0.291850198713803 0.00333552811650 -1.27338760580e-06 5.9
     [77,] 0.296521136452775 0.00330308865102 2.25309977453e-07 6.6
     [78,] 0.298034174946132 0.00330462333485 8.42470393447e-09 8.1
     [79,] 0.300556783277253 0.00323922530004 4.66003314391e-05 4.3
     [80,] 0.303182283998467 0.00328704590597 -1.46205270113e-11 10.8
     [81,] 0.322319846303892 0.00306134512927 -1.15130830540e-05 4.9
     [82,] 0.322319846303892 0.00310689001755 8.57751647487e-11 10.1
     [83,] 0.325071272052651 0.00302343293053 -2.47088704493e-04 3.6
     [84,] 0.325071272052651 0.00304146419577 3.18761056051e-06 5.5
     [85,] 0.331888412404218 0.00300837121343 -4.96098895297e-09 8.3
     [86,] 0.362278153188527 0.00278204202032 4.53939330569e-10 9.3
     [87,] 0.385389476781711 0.00260981704384 7.37274796769e-10 9.1
     [88,] 0.425333956955001 0.00232995789362 1.82823025607e-08 7.7
     [89,] 0.439503709203564 0.00222452690840 -4.53585193982e-06 5.3
     [90,] 0.439503709203564 0.00224964327069 -3.02331937263e-10 9.5
     [91,] 0.450804624124430 0.00216770324934 -4.59455036239e-08 7.3
     >
     > if(requireNamespace("scatterplot3d")) {
     + scatterplot3d::scatterplot3d(res[,1:3], type ='h') ## quite interesting:
     + ## the inaccurate (p,df) points are on nice monotone curve !!!
     + ## this is *less* revealing
     + scatterplot3d::scatterplot3d(res[,c("p","df","nDig")], type ='h')
     + }
     Loading required namespace: scatterplot3d
     > rL <- res[abs(res[,'rE']) > 1e-9,]
     > rL <- rL[order(rL[,1],rL[,2]),]
     > rL
     p df rE nDig
     [1,] 0.0001943754 0.023340796 -4.057185e-08 7.4
     [2,] 0.0010123161 0.018556151 -2.591456e-04 3.6
     [3,] 0.0016823889 0.017207366 5.530890e-04 3.3
     [4,] 0.0017467874 0.017311895 -5.868978e-08 7.2
     [5,] 0.0026644512 0.015993174 1.484213e-04 3.8
     [6,] 0.0026644512 0.016180242 -3.828063e-08 7.4
     [7,] 0.0031594219 0.015576128 -7.921170e-06 5.1
     [8,] 0.0031594219 0.015681837 -4.522375e-08 7.3
     [9,] 0.0040554624 0.014938587 4.151664e-06 5.4
     [10,] 0.0044006941 0.014591017 9.079070e-04 3.0
     [11,] 0.0044588113 0.014575069 9.031400e-05 4.0
     [12,] 0.0044818822 0.014688831 -3.233095e-07 6.5
     [13,] 0.0049396099 0.014401684 -2.818101e-06 5.6
     [14,] 0.0088244651 0.012763527 1.211073e-04 3.9
     [15,] 0.0090402660 0.012737116 1.389644e-05 4.9
     [16,] 0.0116421249 0.012014713 1.449562e-04 3.8
     [17,] 0.0154992134 0.011254201 -9.696959e-05 4.0
     [18,] 0.0154992134 0.011359204 -9.557390e-08 7.0
     [19,] 0.0186030166 0.010717161 1.639710e-03 2.8
     [20,] 0.0186030166 0.010736555 2.143888e-04 3.7
     [21,] 0.0226242424 0.010333795 -3.378658e-09 8.5
     [22,] 0.0226242424 0.010342061 -2.763330e-08 7.6
     [23,] 0.0237302174 0.010162521 -1.077327e-06 6.0
     [24,] 0.0447535254 0.008396264 1.222242e-05 4.9
     [25,] 0.0828003091 0.006812347 4.179976e-09 8.4
     [26,] 0.0908216581 0.006552698 -7.160336e-09 8.1
     [27,] 0.1022947605 0.006235631 3.694387e-09 8.4
     [28,] 0.1274058577 0.005623691 6.605415e-09 8.2
     [29,] 0.1352296342 0.005400734 -2.347626e-05 4.6
     [30,] 0.1377322800 0.005330921 2.992858e-04 3.5
     [31,] 0.1381129175 0.005351387 -2.053358e-06 5.7
     [32,] 0.1411006360 0.005273058 4.315938e-05 4.4
     [33,] 0.1429052994 0.005236800 3.481808e-04 3.5
     [34,] 0.1456245572 0.005269240 -1.945018e-09 8.7
     [35,] 0.1546068729 0.005068069 -4.599247e-07 6.3
     [36,] 0.1546068729 0.005073662 2.723010e-07 6.6
     [37,] 0.1697410365 0.004841818 5.314010e-09 8.3
     [38,] 0.1773275763 0.004659561 5.534044e-05 4.3
     [39,] 0.1900940174 0.004503736 -1.296984e-06 5.9
     [40,] 0.1901476415 0.004534687 5.662356e-09 8.2
     [41,] 0.2147809514 0.004198693 9.791948e-09 8.0
     [42,] 0.2328051066 0.003953993 -9.175810e-08 7.0
     [43,] 0.2520765119 0.003749038 -8.833372e-08 7.1
     [44,] 0.2530829140 0.003752594 3.654361e-09 8.4
     [45,] 0.2539220587 0.003712373 3.289948e-06 5.5
     [46,] 0.2542892786 0.003743439 -1.056649e-09 9.0
     [47,] 0.2600174995 0.003661796 2.348597e-07 6.6
     [48,] 0.2703239068 0.003519992 -1.561648e-04 3.8
     [49,] 0.2716993561 0.003550681 5.130930e-09 8.3
     [50,] 0.2755161961 0.003468040 -4.351715e-04 3.4
     [51,] 0.2901885431 0.003366135 -5.644431e-08 7.2
     [52,] 0.2905790220 0.003344235 1.026676e-07 7.0
     [53,] 0.2905790220 0.003367649 2.260616e-08 7.6
     [54,] 0.2918501987 0.003335528 -1.273388e-06 5.9
     [55,] 0.2965211365 0.003303089 2.253100e-07 6.6
     [56,] 0.2980341749 0.003304623 8.424704e-09 8.1
     [57,] 0.3005567833 0.003239225 4.660033e-05 4.3
     [58,] 0.3223198463 0.003061345 -1.151308e-05 4.9
     [59,] 0.3250712721 0.003023433 -2.470887e-04 3.6
     [60,] 0.3250712721 0.003041464 3.187611e-06 5.5
     [61,] 0.3318884124 0.003008371 -4.960989e-09 8.3
     [62,] 0.4253339570 0.002329958 1.828230e-08 7.7
     [63,] 0.4395037092 0.002224527 -4.535852e-06 5.3
     [64,] 0.4508046241 0.002167703 -4.594550e-08 7.3
     > plot(rL[,1:2], type = "b", main = "inaccurate pchisq/qchisq pairs")
     >
     > plot(rL[,1:2], type = "b", log = "x", ylim = range(0, rL[,"df"]),
     + xaxt = "n",
     + main = "inaccurate pchisq/qchisq pairs"); abline(h = 0, lty=2)
     > ## aha -- a perfect line !!
     > lines(res[,1:2], col = adjustcolor(1, 0.5))
     > eaxis(1); axis(1, at = 1/2)
     >
     > d <- as.data.frame(res)
     > plot (df ~ log(p), data = d, type = "b", cex=1/4, col="gray")
     > points(df ~ log(p), data = as.data.frame(rL), col=2, cex = 1/2)
     >
     > summary(fm <- lm (df ~ log(p), data = d, weights = -log(abs(rE))))
    
     Call:
     lm(formula = df ~ log(p), data = d, weights = -log(abs(rE)))
    
     Weighted Residuals:
     Min 1Q Median 3Q Max
     -6.924e-04 -1.443e-04 -2.096e-05 7.786e-05 1.079e-03
    
     Coefficients:
     Estimate Std. Error t value Pr(>|t|)
     (Intercept) 5.168e-06 1.149e-05 0.45 0.654
     log(p) -2.725e-03 3.683e-06 -739.99 <2e-16 ***
     ---
     Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
    
     Residual standard error: 0.0002575 on 89 degrees of freedom
     Multiple R-squared: 0.9998, Adjusted R-squared: 0.9998
     F-statistic: 5.476e+05 on 1 and 89 DF, p-value: < 2.2e-16
    
     > ## R^2 = 0.9998
     >
     > p0 <- 2^seq(-50,-1, by=1/8)
     > dN <- data.frame(p = p0,
     + df = predict(fm, newdata = data.frame(p = p0)))
     > rE <- with(dN, 1- pchisq(qchisq(p, df),df)/p)
     > dN <- cbind(dN, rE = rE, nDig = round(-log10(abs(rE)), 1))
     > print(dN, digits=10)
     p df rE nDig
     1 8.881784197e-16 0.094454797738 -6.103206185e-07 6.2
     2 9.685654347e-16 0.094218673664 -2.417772682e-07 6.6
     3 1.056228096e-15 0.093982549590 1.482101845e-07 6.8
     4 1.151824906e-15 0.093746425517 5.596452101e-07 6.3
     5 1.256073967e-15 0.093510301443 9.925312783e-07 6.0
     6 1.369758374e-15 0.093274177369 -5.456117484e-07 6.3
     7 1.493732098e-15 0.093038053295 -6.476825187e-08 7.2
     8 1.628926404e-15 0.092801929221 4.375367337e-07 6.4
     9 1.776356839e-15 0.092565805147 9.613066765e-07 6.0
     10 1.937130869e-15 0.092329681073 -4.656782389e-07 6.3
     11 2.112456192e-15 0.092093557000 1.060769096e-07 7.0
     12 2.303649813e-15 0.091857432926 6.993074607e-07 6.2
     13 2.512147934e-15 0.091621308852 -6.429918680e-07 6.2
     14 2.739516748e-15 0.091385184778 -1.755328105e-09 8.8
     15 2.987464197e-15 0.091149060704 6.609670441e-07 6.2
     16 3.257852808e-15 0.090912936630 -5.966163394e-07 6.2
     17 3.552713679e-15 0.090676812556 1.141328021e-07 6.9
     18 3.874261739e-15 0.090440688483 8.463782111e-07 6.1
     19 4.224912384e-15 0.090204564409 -3.264588970e-07 6.5
     20 4.607299625e-15 0.089968440335 4.538340435e-07 6.3
     21 5.024295868e-15 0.089732316261 -6.607811160e-07 6.2
     22 5.479033495e-15 0.089496192187 1.675731415e-07 6.8
     23 5.974928394e-15 0.089260068113 -8.888069507e-07 6.1
     24 6.515705616e-15 0.089023944039 -1.237758629e-08 7.9
     25 7.105427358e-15 0.088787819965 8.855723791e-07 6.1
     26 7.748523477e-15 0.088551695892 -8.599123835e-08 7.1
     27 8.449824769e-15 0.088315571818 8.600545690e-07 6.1
     28 9.214599250e-15 0.088079447744 -5.324092944e-08 7.3
     29 1.004859174e-14 0.087843323670 -9.348371277e-07 6.0
     30 1.095806699e-14 0.087607199596 8.590019329e-08 7.1
     31 1.194985679e-14 0.087371075522 -7.374085143e-07 6.1
     32 1.303141123e-14 0.087134951448 3.314589819e-07 6.5
     33 1.421085472e-14 0.086898827375 -4.335491499e-07 6.4
     34 1.549704695e-14 0.086662703301 6.834622716e-07 6.2
     35 1.689964954e-14 0.086426579227 -2.323222548e-08 7.6
     36 1.842919850e-14 0.086190455153 -6.982056668e-07 6.2
     37 2.009718347e-14 0.085954331079 4.935690683e-07 6.3
     38 2.191613398e-14 0.085718207005 -1.230714122e-07 6.9
     39 2.389971358e-14 0.085482082931 -7.079816475e-07 6.1
     40 2.606282246e-14 0.085245958858 5.585870368e-07 6.3
     41 2.842170943e-14 0.085009834784 3.202904830e-08 7.5
     42 3.099409391e-14 0.084773710710 -4.627895094e-07 6.3
     43 3.379929908e-14 0.084537586636 8.786037373e-07 6.1
     44 3.685839700e-14 0.084301462562 4.421566921e-07 6.4
     45 4.019436694e-14 0.084065338488 3.745824395e-08 7.4
     46 4.383226796e-14 0.083829214414 -3.354886757e-07 6.5
     47 4.779942715e-14 0.083593090340 -6.766811509e-07 6.2
     48 5.212564492e-14 0.083356966267 7.928489877e-07 6.1
     49 5.684341886e-14 0.083120842193 5.100597127e-07 6.3
     50 6.198818782e-14 0.082884718119 2.590340281e-07 6.6
     51 6.759859815e-14 0.082648594045 3.977485741e-08 7.4
     52 7.371679400e-14 0.082412469971 -1.477148934e-07 6.8
     53 8.038873388e-14 0.082176345897 -3.034323055e-07 6.5
     54 8.766453592e-14 0.081940221823 -4.273744663e-07 6.4
     55 9.559885430e-14 0.081704097750 -5.195384647e-07 6.3
     56 1.042512898e-13 0.081467973676 -5.799213925e-07 6.2
     57 1.136868377e-13 0.081231849602 -6.085203379e-07 6.2
     58 1.239763756e-13 0.080995725528 -6.053324058e-07 6.2
     59 1.351971963e-13 0.080759601454 -5.703546833e-07 6.2
     60 1.474335880e-13 0.080523477380 -5.035842794e-07 6.3
     61 1.607774678e-13 0.080287353306 -4.050182891e-07 6.4
     62 1.753290718e-13 0.080051229233 -2.746538159e-07 6.6
     63 1.911977086e-13 0.079815105159 -1.124879638e-07 6.9
     64 2.085025797e-13 0.079578981085 8.148216124e-08 7.1
     65 2.273736754e-13 0.079342857011 3.072594555e-07 6.5
     66 2.479527513e-13 0.079106732937 5.648467992e-07 6.2
     67 2.703943926e-13 0.078870608863 -8.276082903e-07 6.1
     68 2.948671760e-13 0.078634484789 -5.012849573e-07 6.3
     69 3.215549355e-13 0.078398360715 -1.431432968e-07 6.8
     70 3.506581437e-13 0.078162236642 2.468195778e-07 6.6
     71 3.823954172e-13 0.077926112568 6.686065366e-07 6.2
     72 4.170051594e-13 0.077689988494 -5.340348010e-07 6.3
     73 4.547473509e-13 0.077453864420 -4.348594862e-08 7.4
     74 4.959055026e-13 0.077217740346 4.788952396e-07 6.3
     75 5.407887852e-13 0.076981616272 -6.077622940e-07 6.2
     76 5.897343520e-13 0.076745492198 -1.660412186e-08 7.8
     77 6.431098711e-13 0.076509368125 6.063946030e-07 6.2
     78 7.013162874e-13 0.076273244051 -3.642605502e-07 6.4
     79 7.647908344e-13 0.076037119977 3.275302165e-07 6.5
     80 8.340103188e-13 0.075800995903 -5.640570402e-07 6.2
     81 9.094947018e-13 0.075564871829 1.965354515e-07 6.7
     82 9.918110051e-13 0.075328747755 -6.159766142e-07 6.2
     83 1.081577570e-12 0.075092623681 2.134272701e-07 6.7
     84 1.179468704e-12 0.074856499608 -5.200023394e-07 6.3
     85 1.286219742e-12 0.074620375534 3.782225700e-07 6.4
     86 1.402632575e-12 0.074384251460 -2.761173419e-07 6.6
     87 1.529581669e-12 0.074148127386 6.909382196e-07 6.2
     88 1.668020638e-12 0.073912003312 1.156952232e-07 6.9
     89 1.818989404e-12 0.073675879238 -4.174157540e-07 6.4
     90 1.983622010e-12 0.073439755164 6.554521572e-07 6.2
     91 2.163155141e-12 0.073203631090 2.014481738e-07 6.7
     92 2.358937408e-12 0.072967507017 -2.104198829e-07 6.7
     93 2.572439484e-12 0.072731382943 -5.801509786e-07 6.2
     94 2.805265149e-12 0.072495258869 6.355293285e-07 6.2
     95 3.059163338e-12 0.072259134795 3.449181070e-07 6.5
     96 3.336041275e-12 0.072023010721 9.644772814e-08 7.0
     97 3.637978807e-12 0.071786886647 -1.098808029e-07 7.0
     98 3.967244020e-12 0.071550762573 -2.740664691e-07 6.6
     99 4.326310282e-12 0.071314638500 -3.961082631e-07 6.4
     100 4.717874816e-12 0.071078514426 -4.760051799e-07 6.3
     101 5.144878969e-12 0.070842390352 -5.137562191e-07 6.3
     102 5.610530299e-12 0.070606266278 -5.093603785e-07 6.3
     103 6.118326675e-12 0.070370142204 -4.628166750e-07 6.3
     104 6.672082550e-12 0.070134018130 -3.741241081e-07 6.4
     105 7.275957614e-12 0.069897894056 -2.432817023e-07 6.6
     106 7.934488041e-12 0.069661769983 -7.028846949e-08 7.2
     107 8.652620563e-12 0.069425645909 1.448565687e-07 6.8
     108 9.435749632e-12 0.069189521835 4.021543881e-07 6.4
     109 1.028975794e-11 0.068953397761 7.016059603e-07 6.2
     110 1.122106060e-11 0.068717273687 -4.176447372e-07 6.4
     111 1.223665335e-11 0.068481149613 -2.873865945e-08 7.5
     112 1.334416510e-11 0.068245025539 4.023232466e-07 6.4
     113 1.455191523e-11 0.068008901466 -5.698258618e-07 6.2
     114 1.586897608e-11 0.067772777392 -4.930799502e-08 7.3
     115 1.730524113e-11 0.067536653318 5.133677458e-07 6.3
     116 1.887149926e-11 0.067300529244 -3.116849310e-07 6.5
     117 2.057951587e-11 0.067064405170 3.404481462e-07 6.5
     118 2.244212120e-11 0.066828281096 -3.848111123e-07 6.4
     119 2.447330670e-11 0.066592157022 3.567795747e-07 6.4
     120 2.668833020e-11 0.066356032948 -2.686901401e-07 6.6
     121 2.910383046e-11 0.066119908875 5.623583754e-07 6.2
     122 3.173795216e-11 0.065883784801 3.667429294e-08 7.4
     123 3.461048225e-11 0.065647660727 -4.365232460e-07 6.4
     124 3.774299853e-11 0.065411536653 5.312784427e-07 6.3
     125 4.115903175e-11 0.065175412579 1.578594558e-07 6.8
     126 4.488424239e-11 0.064939288505 -1.630783120e-07 6.8
     127 4.894661340e-11 0.064703164431 -4.315370583e-07 6.4
     128 5.337666040e-11 0.064467040358 -6.475189818e-07 6.2
     129 5.820766091e-11 0.064230916284 5.516171563e-07 6.3
     130 6.347590433e-11 0.063994792210 4.354002830e-07 6.4
     131 6.922096451e-11 0.063758668136 3.716548337e-07 6.4
     132 7.548599706e-11 0.063522544062 3.603785849e-07 6.4
     133 8.231806350e-11 0.063286419988 4.015693077e-07 6.4
     134 8.976848478e-11 0.063050295914 4.952247737e-07 6.3
     135 9.789322680e-11 0.062814171841 6.413427337e-07 6.2
     136 1.067533208e-10 0.062578047767 -4.864750427e-07 6.3
     137 1.164153218e-10 0.062341923693 -2.302656055e-07 6.6
     138 1.269518087e-10 0.062105799619 7.839833394e-08 7.1
     139 1.384419290e-10 0.061869675545 4.395145157e-07 6.4
     140 1.509719941e-10 0.061633551471 -4.525763320e-07 6.3
     141 1.646361270e-10 0.061397427397 1.860554089e-08 7.7
     142 1.795369696e-10 0.061161303323 5.422315954e-07 6.3
     143 1.957864536e-10 0.060925179250 -1.718042137e-07 6.8
     144 2.135066416e-10 0.060689055176 4.618673999e-07 6.3
     145 2.328306437e-10 0.060452931102 -1.317493019e-07 6.9
     146 2.539036173e-10 0.060216807028 6.119534922e-07 6.2
     147 2.768838580e-10 0.059980682954 1.387356391e-07 6.9
     148 3.019439882e-10 0.059744558880 -2.716851728e-07 6.6
     149 3.292722540e-10 0.059508434806 -6.193156503e-07 6.2
     150 3.590739391e-10 0.059272310733 3.495618646e-07 6.5
     151 3.915729072e-10 0.059036186659 1.222864269e-07 6.9
     152 4.270132832e-10 0.058800062585 -4.221716110e-08 7.4
     153 4.656612873e-10 0.058563938511 -1.439556299e-07 6.8
     154 5.078072346e-10 0.058327814437 -1.829357232e-07 6.7
     155 5.537677160e-10 0.058091690363 -1.591641836e-07 6.8
     156 6.038879765e-10 0.057855566289 -7.264775914e-08 7.1
     157 6.585445080e-10 0.057619442216 7.660678825e-08 7.1
     158 7.181478783e-10 0.057383318142 2.885926981e-07 6.5
     159 7.831458144e-10 0.057147194068 5.633032045e-07 6.2
     160 8.540265665e-10 0.056911069994 -3.010137886e-07 6.5
     161 9.313225746e-10 0.056674945920 1.043142313e-07 7.0
     162 1.015614469e-09 0.056438821846 5.723448235e-07 6.2
     163 1.107535432e-09 0.056202697772 -8.306453525e-08 7.1
     164 1.207775953e-09 0.055966573698 5.155342088e-07 6.3
     165 1.317089016e-09 0.055730449625 1.091064128e-09 9.0
     166 1.436295757e-09 0.055494325551 -4.402707552e-07 6.4
     167 1.566291629e-09 0.055258201477 3.567050869e-07 6.4
     168 1.708053133e-09 0.055022077403 5.624478649e-08 7.2
     169 1.862645149e-09 0.054785953329 -1.711696178e-07 6.8
     170 2.031228938e-09 0.054549829255 -3.255506589e-07 6.5
     171 2.215070864e-09 0.054313705181 -4.069108679e-07 6.4
     172 2.415551906e-09 0.054077581108 -4.152627815e-07 6.4
     173 2.634178032e-09 0.053841457034 -3.506189497e-07 6.5
     174 2.872591513e-09 0.053605332960 -2.129919214e-07 6.7
     175 3.132583258e-09 0.053369208886 -2.394249909e-09 8.6
     176 3.416106266e-09 0.053133084812 2.811614974e-07 6.6
     177 3.725290298e-09 0.052896960738 -4.754652609e-07 6.3
     178 4.062457877e-09 0.052660836664 -4.082860428e-08 7.4
     179 4.430141728e-09 0.052424712591 4.667262948e-07 6.3
     180 4.831103812e-09 0.052188588517 -5.029516048e-08 7.3
     181 5.268356064e-09 0.051952464443 -4.839870040e-07 6.3
     182 5.745183026e-09 0.051716340369 2.526344739e-07 6.6
     183 6.265166516e-09 0.051480216295 -1.969243901e-08 7.7
     184 6.832212532e-09 0.051244092221 -2.087459541e-07 6.7
     185 7.450580597e-09 0.051007968147 -3.145456657e-07 6.5
     186 8.124915754e-09 0.050771844073 -3.371111803e-07 6.5
     187 8.860283457e-09 0.050535720000 -2.764621003e-07 6.6
     188 9.662207623e-09 0.050299595926 -1.326180317e-07 6.9
     189 1.053671213e-08 0.050063471852 9.440140747e-08 7.0
     190 1.149036605e-08 0.049827347778 4.045766026e-07 6.4
     191 1.253033303e-08 0.049591223704 -2.420871066e-07 6.6
     192 1.366442506e-08 0.049355099630 2.395533774e-07 6.6
     193 1.490116119e-08 0.049118975556 -2.252220010e-07 6.6
     194 1.624983151e-08 0.048882851483 4.277948780e-07 6.4
     195 1.772056691e-08 0.048646727409 1.448078844e-07 6.8
     196 1.932441525e-08 0.048410603335 -4.470068493e-08 7.3
     197 2.107342426e-08 0.048174479261 -1.407587547e-07 6.9
     198 2.298073210e-08 0.047938355187 -1.433942325e-07 6.8
     199 2.506066606e-08 0.047702231113 -5.263503899e-08 7.3
     200 2.732885013e-08 0.047466107039 1.314909133e-07 6.9
     201 2.980232239e-08 0.047229982966 4.089557111e-07 6.4
     202 3.249966302e-08 0.046993858892 -2.026429911e-07 6.7
     203 3.544113383e-08 0.046757734818 2.666381839e-07 6.6
     204 3.864883049e-08 0.046521610744 -1.427213570e-07 6.8
     205 4.214684851e-08 0.046285486670 -4.483846767e-07 6.3
     206 4.596146421e-08 0.046049362596 3.109955485e-07 6.5
     207 5.012133212e-08 0.045813238522 2.073633435e-07 6.7
     208 5.465770025e-08 0.045577114448 2.073183537e-07 6.7
     209 5.960464478e-08 0.045340990375 3.108231207e-07 6.5
     210 6.499932603e-08 0.045104866301 -4.225693764e-07 6.4
     211 7.088226765e-08 0.044868742227 -1.068421438e-07 7.0
     212 7.729766099e-08 0.044632618153 3.123190967e-07 6.5
     213 8.429369702e-08 0.044396494079 -8.979926758e-08 7.0
     214 9.192292842e-08 0.044160370005 -3.780712474e-07 6.4
     215 1.002426642e-07 0.043924245931 3.616102325e-07 6.4
     216 1.093154005e-07 0.043688121858 2.956315699e-07 6.5
     217 1.192092896e-07 0.043451997784 3.433583751e-07 6.5
     218 1.299986521e-07 0.043215873710 -3.936604087e-07 6.4
     219 1.417645353e-07 0.042979749636 -1.134241445e-07 6.9
     220 1.545953220e-07 0.042743625562 2.803691416e-07 6.6
     221 1.685873940e-07 0.042507501488 -9.497759912e-08 7.0
     222 1.838458568e-07 0.042271377414 -3.463650020e-07 6.5
     223 2.004853285e-07 0.042035253341 3.982654885e-07 6.4
     224 2.186308010e-07 0.041799129267 3.893573700e-07 6.4
     225 2.384185791e-07 0.041563005193 -3.573736593e-07 6.4
     226 2.599973041e-07 0.041326881119 -1.135260472e-07 6.9
     227 2.835290706e-07 0.041090757045 2.539798655e-07 6.6
     228 3.091906439e-07 0.040854632971 -1.007498764e-07 7.0
     229 3.371747881e-07 0.040618508897 -3.214330988e-07 6.5
     230 3.676917137e-07 0.040382384823 -4.081432583e-07 6.4
     231 4.009706570e-07 0.040146260750 -3.609537691e-07 6.4
     232 4.372616020e-07 0.039910136676 -1.799379994e-07 6.7
     233 4.768371582e-07 0.039674012602 1.348307247e-07 6.9
     234 5.199946082e-07 0.039437888528 -2.309643412e-07 6.6
     235 5.670581412e-07 0.039201764454 3.563334678e-07 6.4
     236 6.183812879e-07 0.038965640380 2.734302492e-07 6.6
     237 6.743495762e-07 0.038729516306 3.344816584e-07 6.5
     238 7.353834273e-07 0.038493392233 -2.537711326e-07 6.6
     239 8.019413140e-07 0.038257268159 1.001817561e-07 7.0
     240 8.745232040e-07 0.038021144085 -1.848126916e-07 6.7
     241 9.536743164e-07 0.037785020011 -3.156868371e-07 6.5
     242 1.039989216e-06 0.037548895937 -2.925440488e-07 6.5
     243 1.134116282e-06 0.037312771863 -1.154876221e-07 6.9
     244 1.236762576e-06 0.037076647789 2.153792392e-07 6.7
     245 1.348699152e-06 0.036840523716 -5.632338751e-08 7.2
     246 1.470766855e-06 0.036604399642 -1.638943248e-07 6.8
     247 1.603882628e-06 0.036368275568 -1.074536702e-07 7.0
     248 1.749046408e-06 0.036132151494 1.128785959e-07 6.9
     249 1.907348633e-06 0.035896027420 -2.381848818e-07 6.6
     250 2.079978433e-06 0.035659903346 3.148260044e-07 6.5
     251 2.268232565e-06 0.035423779272 3.067288504e-07 6.5
     252 2.473525152e-06 0.035187655198 -2.467805615e-07 6.6
     253 2.697398305e-06 0.034951531125 9.809279378e-08 7.0
     254 2.941533709e-06 0.034715407051 -9.217531383e-08 7.0
     255 3.207765256e-06 0.034479282977 -9.840820714e-08 7.0
     256 3.498092816e-06 0.034243158903 7.923717715e-08 7.1
     257 3.814697266e-06 0.034007034829 -2.523280838e-07 6.6
     258 4.159956866e-06 0.033770910755 2.978638209e-07 6.5
     259 4.536465130e-06 0.033534786681 -3.332980778e-07 6.5
     260 4.947050303e-06 0.033298662608 -8.305711496e-08 7.1
     261 5.394796609e-06 0.033062538534 -3.110326594e-07 6.5
     262 5.883067419e-06 0.032826414460 3.314533775e-07 6.5
     263 6.415530512e-06 0.032590290386 -1.553354063e-07 6.8
     264 6.996185632e-06 0.032354166312 2.279407690e-07 6.6
     265 7.629394531e-06 0.032118042238 1.638951825e-07 6.8
     266 8.319913732e-06 0.031881918164 3.135417774e-07 6.5
     267 9.072930260e-06 0.031645794091 3.656237890e-08 7.4
     268 9.894100606e-06 0.031409670017 -1.661173443e-08 7.8
     269 1.078959322e-05 0.031173545943 1.537749293e-07 6.8
     270 1.176613484e-05 0.030937421869 -7.675076596e-08 7.1
     271 1.283106102e-05 0.030701297795 -7.365044086e-08 7.1
     272 1.399237126e-05 0.030465173721 1.628068312e-07 6.8
     273 1.525878906e-05 0.030229049647 2.398761112e-08 7.6
     274 1.663982746e-05 0.029992925573 1.285369383e-07 6.9
     275 1.814586052e-05 0.029756801500 -1.216179157e-07 6.9
     276 1.978820121e-05 0.029520677426 -1.184344594e-07 6.9
     277 2.157918644e-05 0.029284553352 1.377655910e-07 6.9
     278 2.353226967e-05 0.029048429278 6.475006253e-08 7.2
     279 2.566212205e-05 0.028812305204 2.546562226e-07 6.6
     280 2.798474253e-05 0.028576181130 1.358057637e-07 6.9
     281 3.051757812e-05 0.028340057056 -2.762893581e-07 6.6
     282 3.327965493e-05 0.028103932983 1.553088241e-07 6.8
     283 3.629172104e-05 0.027867808909 -2.519743056e-07 6.6
     284 3.957640242e-05 0.027631684835 1.836182478e-07 6.7
     285 4.315837288e-05 0.027395560761 -1.887623498e-07 6.7
     286 4.706453935e-05 0.027159436687 -2.586992773e-07 6.6
     287 5.132424410e-05 0.026923312613 -2.666510612e-08 7.6
     288 5.596948506e-05 0.026687188539 -2.210357075e-08 7.7
     289 6.103515625e-05 0.026451064466 -2.296375117e-07 6.6
     290 6.655930986e-05 0.026214940392 -1.155410669e-07 6.9
     291 7.258344208e-05 0.025978816318 -1.934344240e-07 6.7
     292 7.915280485e-05 0.025742692244 5.977598905e-08 7.2
     293 8.631674575e-05 0.025506568170 1.410127927e-07 6.9
     294 9.412907870e-05 0.025270444096 6.554035215e-08 7.2
     295 1.026484882e-04 0.025034320022 -1.513968153e-07 6.8
     296 1.119389701e-04 0.024798195948 -7.984736650e-09 8.1
     297 1.220703125e-04 0.024562071875 1.377293413e-08 7.9
     298 1.331186197e-04 0.024325947801 -7.096807053e-08 7.1
     299 1.451668842e-04 0.024089823727 2.236347612e-07 6.7
     300 1.583056097e-04 0.023853699653 -3.406038118e-08 7.5
     301 1.726334915e-04 0.023617575579 1.071465702e-07 7.0
     302 1.882581574e-04 0.023381451505 1.915567694e-07 6.7
     303 2.052969764e-04 0.023145327431 -2.153880145e-07 6.7
     304 2.238779402e-04 0.022909203358 -1.943811789e-07 6.7
     305 2.441406250e-04 0.022673079284 -1.853585263e-07 6.7
     306 2.662372394e-04 0.022436955210 -1.734718758e-07 6.8
     307 2.903337683e-04 0.022200831136 -1.439218900e-07 6.8
     308 3.166112194e-04 0.021964707062 -8.196069534e-08 7.1
     309 3.452669830e-04 0.021728582988 2.710548919e-08 7.6
     310 3.765163148e-04 0.021492458914 1.979137831e-07 6.7
     311 4.105939528e-04 0.021256334841 3.784053582e-08 7.4
     312 4.477558805e-04 0.021020210767 -2.082512274e-08 7.7
     313 4.882812500e-04 0.020784086693 3.635452028e-08 7.4
     314 5.324744788e-04 0.020547962619 -1.675743422e-07 6.8
     315 5.806675366e-04 0.020311838545 1.696075650e-07 6.8
     316 6.332224388e-04 0.020075714471 -9.599203343e-08 7.0
     317 6.905339660e-04 0.019839590397 -1.676104155e-07 6.8
     318 7.530326296e-04 0.019603466323 -3.122730075e-08 7.5
     319 8.211879055e-04 0.019367342250 -3.779823499e-08 7.4
     320 8.955117609e-04 0.019131218176 -1.576477535e-07 6.8
     321 9.765625000e-04 0.018895094102 -6.902106886e-09 8.2
     322 1.064948958e-03 0.018658970028 7.899459131e-08 7.1
     323 1.161335073e-03 0.018422845954 1.293463390e-07 6.9
     324 1.266444878e-03 0.018186721880 -1.651599366e-07 6.8
     325 1.381067932e-03 0.017950597806 -9.328012252e-08 7.0
     326 1.506065259e-03 0.017714473733 3.008480842e-08 7.5
     327 1.642375811e-03 0.017478349659 -8.903183435e-08 7.1
     328 1.791023522e-03 0.017242225585 -8.893825099e-08 7.1
     329 1.953125000e-03 0.017006101511 5.866337671e-08 7.2
     330 2.129897915e-03 0.016769977437 7.502565913e-08 7.1
     331 2.322670146e-03 0.016533853363 3.812254845e-09 8.4
     332 2.532889755e-03 0.016297729289 -1.115639656e-07 7.0
     333 2.762135864e-03 0.016061605216 6.307704159e-08 7.2
     334 3.012130518e-03 0.015825481142 -1.675984129e-08 7.8
     335 3.284751622e-03 0.015589357068 -1.223428781e-08 7.9
     336 3.582047044e-03 0.015353232994 1.188536293e-07 6.9
     337 3.906250000e-03 0.015117108920 -1.217006735e-07 6.9
     338 4.259795831e-03 0.014880984846 -1.314145703e-07 6.9
     339 4.645340293e-03 0.014644860772 -1.288061258e-07 6.9
     340 5.065779510e-03 0.014408736698 -5.759581589e-08 7.2
     341 5.524271728e-03 0.014172612625 -1.110605841e-07 7.0
     342 6.024261037e-03 0.013936488551 2.552848555e-08 7.6
     343 6.569503244e-03 0.013700364477 -7.038977068e-08 7.2
     344 7.164094088e-03 0.013464240403 -8.014783059e-08 7.1
     345 7.812500000e-03 0.013228116329 6.514376105e-08 7.2
     346 8.519591661e-03 0.012991992255 -1.226700341e-08 7.9
     347 9.290680586e-03 0.012755868181 3.829306428e-09 8.4
     348 1.013155902e-02 0.012519744108 -1.720011800e-08 7.8
     349 1.104854346e-02 0.012283620034 2.105846508e-08 7.7
     350 1.204852207e-02 0.012047495960 1.170782316e-08 7.9
     351 1.313900649e-02 0.011811371886 6.422403376e-08 7.2
     352 1.432818818e-02 0.011575247812 9.486125396e-08 7.0
     353 1.562500000e-02 0.011339123738 3.894984368e-08 7.4
     354 1.703918332e-02 0.011102999664 3.208461985e-08 7.5
     355 1.858136117e-02 0.010866875591 3.155170891e-08 7.5
     356 2.026311804e-02 0.010630751517 1.297326291e-08 7.9
     357 2.209708691e-02 0.010394627443 -3.001926907e-08 7.5
     358 2.409704415e-02 0.010158503369 7.470777241e-08 7.1
     359 2.627801298e-02 0.009922379295 2.881776751e-08 7.5
     360 2.865637635e-02 0.009686255221 4.355808669e-08 7.4
     361 3.125000000e-02 0.009450131147 3.486256517e-08 7.5
     362 3.407836665e-02 0.009214007073 -4.540193843e-08 7.3
     363 3.716272234e-02 0.008977883000 6.057274970e-08 7.2
     364 4.052623608e-02 0.008741758926 -4.748039517e-08 7.3
     365 4.419417382e-02 0.008505634852 -4.947995325e-08 7.3
     366 4.819408829e-02 0.008269510778 5.303493644e-09 8.3
     367 5.255602595e-02 0.008033386704 2.750297434e-09 8.6
     368 5.731275270e-02 0.007797262630 7.649789580e-09 8.1
     369 6.250000000e-02 0.007561138556 2.575244529e-08 7.6
     370 6.815673329e-02 0.007325014483 2.588700820e-08 7.6
     371 7.432544469e-02 0.007088890409 -3.873515997e-08 7.4
     372 8.105247217e-02 0.006852766335 -2.868914595e-08 7.5
     373 8.838834765e-02 0.006616642261 8.820984831e-09 8.1
     374 9.638817659e-02 0.006380518187 -1.249980452e-08 7.9
     375 1.051120519e-01 0.006144394113 -2.542298283e-08 7.6
     376 1.146255054e-01 0.005908270039 -3.411814475e-08 7.5
     377 1.250000000e-01 0.005672145966 1.097271574e-08 8.0
     378 1.363134666e-01 0.005436021892 -5.883856513e-09 8.2
     379 1.486508894e-01 0.005199897818 -1.778869496e-08 7.7
     380 1.621049443e-01 0.004963773744 -2.477514438e-08 7.6
     381 1.767766953e-01 0.004727649670 -2.316977810e-08 7.6
     382 1.927763532e-01 0.004491525596 -1.391213433e-08 7.9
     383 2.102241038e-01 0.004255401522 4.032074008e-09 8.4
     384 2.292510108e-01 0.004019277448 1.322844057e-10 9.9
     385 2.500000000e-01 0.003783153375 1.667496141e-09 8.8
     386 2.726269332e-01 0.003547029301 -6.843543954e-09 8.2
     387 2.973017788e-01 0.003310905227 -5.830278704e-09 8.2
     388 3.242098887e-01 0.003074781153 -4.568749379e-09 8.3
     389 3.535533906e-01 0.002838657079 -6.372713468e-09 8.2
     390 3.855527064e-01 0.002602533005 4.293532641e-09 8.4
     391 4.204482076e-01 0.002366408931 4.466793713e-09 8.4
     392 4.585020216e-01 0.002130284858 -3.886739153e-09 8.4
     393 5.000000000e-01 0.001894160784 1.495896629e-09 8.8
     >
     > ## } ## only when we find inaccurate regions
     > showProc.time()
     Time (user system elapsed): 0.097 0 0.098
     >
     >
     > ## Oops: another qgamma() / qchisq() problem: mostly NaN's == all solved now
     > curve(qgamma(x, 20), 1e-16, 1e-10, log='x')
     > curve(qgamma(x, 20), 1e-300, .99 , log='xy') # and add the critical region from above:
     > abline(v=c(1e-16, 1e-10), col="light blue")
     > curve(qgamma(x, 20), 1e-26, 1e-07, log='x')
     > ##-> now using log=TRUE in same region:
     > curve(qgamma(x, 20, log=TRUE), -38, -16)## no problem!!
     > curve(qgamma(exp(x), 20), add=TRUE, col="green3", n=2001)
     > ## had problem here, but no longer !
     >
     > ##--> Further fix for qgamma: when 'x' is very small: use "log=TRUE of log(x)"!
     >
     > ## had bug (gave NaN), but no longer:
     > (q_12 <- qgamma(1e-12, 20))
     [1] 2.330042
     > all.equal(1e-12, pgamma(q_12, 20), tol=0)# show rel.err (Lnx 64-bit: 4.04e-16)
     [1] "Mean relative difference: 4.038968e-16"
     > stopifnot(
     + all.equal(1e-12, pgamma(q_12, 20), tolerance = 1e-14)
     + )
     >
     >
     > ## --- Nice graphic : --- but amazingly *S..L..O..W*
     >
     > p.qgammaSml <- function(from= 1e-110, to = 1e-5, ylim = c(0.4, 1000),
     + n = 201, k.lab = 3,
     + a1 = c(10, seq(10.1,20, by=.2), 21:105),
     + a2 = seq(110,330, by=10),
     + a3 = seq(350,1600, by=50))
     + {
     + ## Purpose: nice qgamma() lines ``for small x'' aka p
     + ## ----------------------------------------------------------------------
     + ## Author: Martin Maechler, Date: 22 Mar 2004, 14:23
     + x <- exp(seq(log(from), log(to), length = n))
     +
     + op <- par(las=1, lab = c(10,10, 7), xaxs = "i", mex = 0.8)
     + on.exit(par(op))
     + plot(x, qgamma(x, a1[1]), log="xy", ylim=ylim, type='l', xaxt = "n",
     + main = paste("qgamma(x, a) for very small x, a in [",
     + formatC(a1[1]),", ",formatC(max(a1,a2,a3)),"] - log-log", sep=''),
     + sub = R.version.string)
     + lab.x <- pretty(log10(c(from,to)), 20)
     + axis(1, at=10^lab.x, lab = paste("10^",formatC(lab.x),sep=''))
     + if(is.nan(qgamma(1e-12, 20)))
     + text(1e-60, 20, "all NaN", cex = 2)
     + if(!is.finite(qgamma(1e-140, 155)))
     + text(1e-240, 5, "all +Inf", cex = 2)
     +
     + lines.txt <- function(a.s, col = par("col")) {
     + col <- rep(col, length=length(a.s))
     + for(i in seq(along=a.s)) {
     + qx <- qgamma(x, (a <- a.s[i]))
     + if(i %% k.lab == 0 &&
     + any(ifi <- is.finite(qx) & qx >= ylim[1])) {
     + ik <- (i%%(2*k.lab))/k.lab # = 0 or 1
     + j <- quantile(which(ifi), c(.02,(1:3)/4+ ik/10, .98))
     + ## "segments" around the labels :
     + i0 <- 1
     + for(jj in j) {
     + ii <- i0:(jj-1)
     + i2 <- jj + -1:1
     + lines(x[ii], qx[ii], col=col[i])
     + lines(x[i2], qx[i2], col=col[i], type = 'c')
     + i0 <- jj+1
     + }
     + text(x[j], qx[j], formatC(a), col= "gray40", cex = 0.8)
     + }
     + else
     + lines(x, qx, col=col[i])
     +
     + }
     + }
     + oo <- options(warn = -1)
     + lines.txt(a1[-1])
     + lines.txt(a2, col= 2)
     + lines.txt(a3, col= rainbow(length(a3), .8, .8,
     + start = (max(a3)-min(a3))/(1+max(a3))))
     + invisible(options(oo))
     + }
     >
     > showProc.time()
     Time (user system elapsed): 0.023 0 0.023
     >
     > p.qgammaSml()
     > p.qgammaSml(1e-300)
     > p.qgammaSml(1e-300,1e-50, a2= seq(100,360, by=4), a3=seq(350,1500, by=10))
     >
     > showProc.time()
     Time (user system elapsed): 1.127 0.047 1.96
     >
     > ## The "upper" problematic corner:
     > p.qgammaSml(1e-19, 1e-3, a2=NULL,a3=NULL, ylim=c(.1,20))
     > p.qgammaSml(1e-19, 1e-3, a2=seq(1,12, by=.04), ylim=c(.1,20),a3=NULL,k.lab=10)
     > ## now shows the problem (quite well):
     > ## could it be in pgamma()'s inaccuracy, leading to qgamma() bias ?
     > aa <- c(seq(9, 22, by=0.25),seq(22.3,40,by=0.4))
     > caa <- formatC(range(aa))
     > sfsmisc::mult.fig(2)
     > curve(pgamma(x, sh=aa[1]), 0.5, 20, log = 'xy', ylim = c(1e-60, .2),
     + main = sprintf("pgamma(x, a) for a in [%s,%s]", caa[1],caa[2]))
     > for(sh in aa) curve(pgamma(x, sh), add = TRUE, col=2)
     > abline(h=c(1e-15), col="light blue", lty=2)
     >
     > curve(pgamma(x, sh=aa[1]), 0.5, 20, log = 'xy', ylim = c(1e-15, .8),
     + main = sprintf("pgamma(x, a) for a in [%s,%s]", caa[1],caa[2]))
     > for(sh in aa) curve(pgamma(x, sh), add = TRUE, col=2)
     > ## the "border curve" between "Pearson" and "Continued fraction (upper tail)"
     > ## in pgamma.c :
     > curve(pgamma(max(1,x), x), add = TRUE, col=4)
     > ## ==> pgamma() is perfect here {series expansion up to eps_C accuracy}!
     >
     > aa <- c(seq(9, 22, by=0.25),seq(22.3,40.4,by=0.4))
     > p.qgammaSml(1e-24, 1e-5, a1=aa, a2=NULL,a3=NULL, ylim=c(.8,8))
     > ## -------- save the above?
     > aa1 <- c(aa,seq(40.5,90, by=0.5))
     > p.qgammaSml(1e-60, 1e-5, a1=aa1, a2=NULL,a3=NULL, ylim=c(.9, 16))
     > aa2 <- c(aa1, seq(91,150, by= 1))
     > p.qgammaSml(1e-90, 1e-5, a1=aa2, a2=NULL,a3=NULL, ylim=c(.9, 35))
     > aa3 <- c(aa2, seq(150,250, by= 2), seq(253, 400, by=5))
     > p.qgammaSml(1e-200, 1e-5, a1=aa3, a2=NULL,a3=NULL, ylim=c(.9, 100))
     > p.qgammaSml(1e-200, 1e-5, a1=aa3, a2=NULL,a3=NULL, ylim=c(.9, 200),k.lab=9e9)
     > p.qgammaSml(1e-60, 1e-5, a1=aa3, a2=NULL,a3=NULL, ylim=c(.9, 200),k.lab=9e9)
     >
     > showProc.time()
     Time (user system elapsed): 2.832 0.032 4.235
     >
     > ## lower a \> 10
     >
     > curve(qgamma(x, 19), 1e-14, 1e-9, log='x')
     > curve(qgamma(x, 18), 1e-14, 1e-9, log='x')
     > curve(qgamma(x, 15), 1e-11, 5e-9, log='x')
     > curve(qgamma(x, 13), 5e-10, 1e-8, log='x')
     > curve(qgamma(x, 11), 1e-8, 5e-8, log='x')
     > curve(qgamma(x, 10.5), 4.2e-8, 6e-8, log='x')
     > curve(qgamma(x, 10.3), 6e-8, 7e-8, log='x')
     > curve(qgamma(x, 10.2), 7.1e-8, 7.6e-8, log='x')
     > curve(qgamma(x, 10.15),7.7e-8, 7.9e-8, log='x')
     > curve(qgamma(x, 10.14),7.88e-8,7.92e-8, log='x',n=10001)
     >
     > ## no more problems for smaller a!! here:
     > curve(qgamma(x, 10.13), 1e-10, 5e-4, log='x',n=20001)
     > curve(qgamma(x, 10.12), 1e-10, 5e-4, log='x',n=20001)
     > curve(qgamma(x, 10.1), 1e-10, 5e-4, log='x',n=20001)
     >
     > showProc.time()
     Time (user system elapsed): 0.464 0.011 0.788
     >
     > ##--- the "+Inf" / premature "0" case:
     > curve(qgamma(x, 155, log=TRUE), -1500, 0, log='y', n=2001,col=2)
     > curve(qgamma(x, 1e3, log=TRUE), -1500, 0, log='y', n=2001,col=2)
     > ## now works, but slowly and with kink
     > curve(qgamma (x, 1e5, log=TRUE), -3e5, 0, log='y', n=2001,col=2,lwd=3)
     > curve(qgammaAppr(x, 1e5, log=TRUE), add = TRUE, n=2001, col="blue",lwd=.4)
     > ## --- curves are almost "identical"
     > ## ===> the kink *does* come from the initial approx... hmm
     >
     > ## still "identical"
     > curve(qgamma (x, 1e4, log=TRUE), -3e4, 0, log='y', n=2001,col=2)
     > curve(qgammaAppr(x, 1e4, log=TRUE), add = TRUE, n=2001, col="tomato3")
     >
     > ## now see some difference (approx. has kink at ~ -165)
     > curve(qgamma (x, 100, log=TRUE), -200, 0, log='y', n=2001,col=2)
     > curve(qgammaAppr(x, 100, log=TRUE), add = TRUE, n=2001, col="tomato3")
     > ##
     > (kk <- 100 * 2/1.24)# 161.29
     [1] 161.2903
     > curve(qgamma (x, 100, log=TRUE), -1.1*kk, -.95*kk, log='y', n=2001,col=2)
     > curve(qgammaAppr(x, 100, log=TRUE), add = TRUE, n=2001, col="tomato3")
     > abline(v = -kk, col='blue', lty=2)# exactly: kink is at a * 2 / 1.24 = a / .62
     > curve(qgammaAppr(x - 100/.62, 100,log=TRUE), -1e-3, +1e-3)
     >
     > showProc.time()
     Time (user system elapsed): 0.114 0.008 0.25
     >
     > p.qgammaLog <- function(alpha, xl.f = 1.5, xr.f = 0.4, n = 2001)
     + {
     + ## Purpose:
     + ## ----------------------------------------------------------------------
     + ## Arguments:
     + ## ----------------------------------------------------------------------
     + ## Author: Martin Maechler, Date: 30 Mar 2004, 18:44
     + kk <- -alpha / .62 # = (alpha * 2) / (-1.24)
     + curve(qgamma(x, alpha, log=TRUE), xl.f*kk, xr.f*kk, log='y',
     + n=n, col=2, lwd=3.6, lty = 4,
     + main= paste("qgamma(x, alpha=",formatC(alpha,digits=10),", log = TRUE)"))
     + lines(kk, qgamma(kk, alpha, log=TRUE), type = 'h', lty = 3)
     + curve(qgamma (exp(x), alpha), add = TRUE, col="orange", n=n, lwd= 2)
     + curve(qgammaAppr(x, alpha, log=TRUE), add = TRUE, col=3, n=n,lwd = .4)
     + }
     >
     > showProc.time()
     Time (user system elapsed): 0.001 0 0.001
     >
     > p.qgammaLog(25)
     > p.qgammaLog(16)# ~ [-25, -20]
     > p.qgammaLog(12, 1.2, 0.8)# small problem remaining
     > p.qgammaLog(11, 1.2, 0.8)# even smaller
     > p.qgammaLog(10.5, 1.1, 0.9)# even smaller
     > p.qgammaLog(10.25, 1.1, 0.9)# even smaller
     > ## 2019-08: __nothing__ visible from here on:
     > p.qgammaLog(10.18, 1.02, 0.98)# even smaller
     > p.qgammaLog(10.15, 1.02, 0.98)# even smaller
     > p.qgammaLog(10.14, 1.001, 0.999)# even smaller
     > p.qgammaLog(10.139, 1.0002, 0.9998)#
     > p.qgammaLog(10.138, 1.0002, 0.9998)#
     > p.qgammaLog(10.137, 1.00001, 0.99999)#
     > p.qgammaLog(10.13699, 1.0000001, 0.9999999)#
     > p.qgammaLog(10.1369899, 1.0000001, 0.9999999)#
     > p.qgammaLog(10.1369894, 1.0000001, 0.9999999)#
     > p.qgammaLog(10.1369893, 1.0000001, 0.9999999)# even smaller at -16.34998
     >
     > showProc.time()
     Time (user system elapsed): 0.407 0.009 0.827
     >
     > ##-- here is the boundary --- for 64-bit AMD Opteron ---
     > ## and for 32-bit AMD Athlon
     >
     > p.qgammaLog(10.1369892, 1.0000001, 0.9999999)# no more
     > p.qgammaLog(10.136989, 1.0000001, 0.9999999)#
     > p.qgammaLog(10.136988, 1.0000001, 0.9999999)#
     > p.qgammaLog(10.136985, 1.0000001, 0.9999999)#
     > p.qgammaLog(10.13698, 1.0000001, 0.9999999)#
     > p.qgammaLog(10.13697, 1.0000001, 0.9999999)#
     > p.qgammaLog(10.13695, 1.0000001, 0.9999999)#
     > p.qgammaLog(10.1368, 1.000001, 0.999999)#
     > p.qgammaLog(10.1365, 1.000001, 0.999999)#
     > p.qgammaLog(10.136, 1.000001, 0.999999)#
     > p.qgammaLog(10.125, 1.1, 0.9)# --- see it now
     > p.qgammaLog(10, 1.2, 0.8)
     > p.qgammaLog(9)
     >
     > showProc.time()
     Time (user system elapsed): 0.311 0 0.627
     >
     > ## For large alpha: show difference to see problem better
     > ## ---> for alpha >= 10, the x problem starts *roughly* at x = -0.8*alpha
     > ##
     >
     > sfsmisc::mult.fig(2)
     > curve(qgammaAppr(x, 5, log=TRUE), - 8.1, -8, n=2001)
     > curve(qgammaAppr(x- 5/.62, 5, log=TRUE), -1e-15, 0)
     >
     > ## is the kink from pgamma() ? : no: this looks fine,
     > curve(pgamma(x, 1e5, log=TRUE), 1, 2e5, log='x', n=2001,col=2)
     > ## and this does too:
     > curve( dgamma(x, 1e5), .5e5, 2e5); par(new=TRUE)
     > curve( dgamma(x, 1e5, log=TRUE), .5e5, 2e5, col=2, yaxt="n")
     > axis(4,col.axis=2); par(new=TRUE)
     > curve( pgamma(x, 1e5), .5e5, 2e5, n=2001, col=3); par(new=TRUE)
     > curve( pgamma(x, 1e5, log=TRUE), .5e5, 2e5, n=2001, col=4); par(new=TRUE)
     > curve(-pgamma(x, 1e5, log=TRUE,lower=FALSE), .5e5, 2e5, n=2001, col=4)
     > ## all looking nice
     >
     >
     > x <- 10^seq(2,6, length=4001)
     > qx <- qgamma(pgamma(x, 1e5, log=TRUE), 1e5, log=TRUE)
     > plot(x, qx, type ='l', col=2, asp = 1); abline(0,1, lty=3)
     >
     > showProc.time()
     Time (user system elapsed): 0.056 0.004 0.124
     > <0c>
     > ###------------- Approximations of qgamma() ------
     > ##
     >
     > ## source("/u/maechler/R/MM/NUMERICS/dpq-functions/qchisqAppr.R")
     > ##--> qchisqAppr()
     > ##--> qchisqWH [ = Wilson Hilferty ]
     > ##--> qchisqKG [ = Kennedy & Gentle's improvements "a la AS 91" ]
     > ## dyn.load("/u/maechler/R/MM/NUMERICS/dpq-functions/qchisq_appr.so")
     >
     > ## Consider the two different implementations of
     > ## lgamma1p(a) := lgamma(1+a) == log(gamma(1+a) == log(a*gamma(a)) "stable":
     >
     > if(!exists("lseq", mode="function"))
     + lseq <- if(requireNamespace("sfsmisc")) sfsmisc::lseq else
     + function(from, to, length) exp(seq(log(from), log(to), length.out = length))
     >
     > if(require("Rmpfr")) { ##---------------- MPFR numbers -------------------------
     +
     + .mpfr.all.eq <- Rmpfr::all.equal
     + AllEq <- function(target, current, ...)
     + .mpfr.all.eq(target, current, ...,
     + formatFUN = function(x, ...) Rmpfr::format(x, digits = 9))
     +
     + print(gammaE <- Const("gamma",200)); pi. <- Const("pi",200)
     + print(a0 <- (gammaE^2 + pi.^2/6)/2)
     + print(psi2.1 <- -2*zeta(mpfr(3,200)))# == psigamma(1,2) =~ -2.4041138
     + print(a1 <- (psi2.1 - gammaE*(pi.^2/2 + gammaE^2))/6)
     +
     + x <- lseq(1e-30, 0.8, length = if(doExtras) 1000 else 125)
     + x. <- mpfr(x, 200)
     + xct. <- log(x. * gamma(x.)) ## using MPFR arithmetic .. no overflow ...
     + xc2. <- log(x.) + lgamma(x.)## (ditto)
     + print(AllEq(xct., xc2., tol = 0)) # 3.15779......e-57
     + xct <- as.numeric(xct.)
     + stopifnot(exprs = {
     + AllEq(xct., xc2., tol = 1e-45)
     + AllEq(xct , xc2., tol = 1e-15)
     + ##
     + all.equal(lgamma1p(x), lgamma1p(x, tol= 1e-16), tol=0)
     + ## -> no difference; i.e., default tol = 1e-14 seems fine enough!
     + })
     + showProc.time()
     +
     + m.appr <- cbind(log(x*gamma(x)), lgamma(1+x), log(x) + lgamma(x),
     + lgamma1p.(x, k=1, cut=3e-6),
     + lgamma1p.(x, k=2, cut=1e-4),
     + lgamma1p.(x, k=3, cut=8e-4),
     + lgamma1p(x))#, tol= 1e-14), # = default
     +
     + eMat <- m.appr - xct # absolute error
     + ## Relative errors:
     + str(reMat. <- m.appr/xct. - 1)
     + str(reMat <- as(reMat., "array")) # as(., "matrix") fails in older versions
     +
     + matplot(x, eMat , log="x", type="l", lty=1) #-> problematic log(x) + lgamma(x) for "large"
     + matplot(x, abs( eMat), log="xy", type="l", lty=1) #-> but good for small; lgamma1p is much better
     + matplot(x, abs(reMat), log="xy", type="l", lty=1)
     + abline(v= 3.47548562941137e-08, col = "gray80", lwd=3)#<- the cutoff value of lgamma1p()
     + ##---> should use earlier cutoff!
     + ## zoom in:
     +
     + matplot(x, abs(reMat), log="xy", type="l", col=1:7, lty=1,
     + lwd=2, xlim=c(8e-9, 1e-3), ylim = c(1e-18, 1e-7), axes=FALSE, frame=TRUE,
     + main = expression(lgamma1p(x) == log(Gamma(x+1)) ~~~ "approximations"
     + ~~~ abs(rel.Err(.))))
     + eaxis(1); eaxis(2)
     + abline(v= 3.47548562941137e-08, col = "gray80", lwd=3)#<- the cutoff value of lgamma1p()
     + abline(h= c(1,2,4)*.Machine$double.eps, lty=3, col="skyblue")
     + legend("topright", col=1:7, lty=1,lwd=2,
     + c("log(x*gamma(x))", "lgamma(1+x)", "log(x) + lgamma(x)",
     + "lgamma1p.(x, k=1, c=3e-6)",
     + "lgamma1p.(x, k=2, c=1e-4)",
     + "lgamma1p.(x, k=3, c=8e-4)",
     + "lgamma1p(x)"), bty="n", ncol=2)
     + abline(v = c(3e-6, 1e-4, 8e-4), col=4:6, lty=2, lwd=1/2)
     +
     + ## FIXME: do the same for the lgaamma1p_series()
     +
     + ## rm(x., xct., xc2., reMat., eMat, AllEq)
     + detach("package:Rmpfr")
     + showProc.time()
     +
     + } ## if( MPFR ) ----------------------------------------------------------------
     Loading required package: Rmpfr
     Loading required package: gmp
    
     Attaching package: 'gmp'
    
     The following objects are masked from 'package:base':
    
     %*%, apply, crossprod, matrix, tcrossprod
    
     C code of R package 'Rmpfr': GMP using 64 bits per limb
    
    
     Attaching package: 'Rmpfr'
    
     The following object is masked from 'package:gmp':
    
     outer
    
     The following object is masked from 'package:DPQ':
    
     log1mexp
    
     The following objects are masked from 'package:stats':
    
     dbinom, dgamma, dnbinom, dnorm, dpois, dt, pnorm
    
     The following objects are masked from 'package:base':
    
     cbind, pmax, pmin, rbind
    
     1 'mpfr' number of precision 200 bits
     [1] 0.57721566490153286060651209008240243104215933593992359880576723
     1 'mpfr' number of precision 200 bits
     [1] 0.98905599532797255539539565150063470793918352072821409044319567
     1 'mpfr' number of precision 200 bits
     [1] -2.404113806319188570799476323022899981529972584680997763584544
     1 'mpfr' number of precision 200 bits
     [1] -0.90747907608088628901656016735627511492861144907256376094133062
     Error in target == current : comparison of these types is not implemented
     Calls: print ... .mpfr.all.eq -> .mpfr.all.eq -> .local -> all.equal.numeric
     Execution halted
    Running the tests in ‘tests/stirlerr-tst.R’ failed.
    Complete output:
     > #### Testing stirlerr(), bd0(), ebd0(), dpois_raw(), ...
     > #### ===============================================
     >
     > require(DPQ)
     Loading required package: DPQ
     > for(pkg in c("Rmpfr", "DPQmpfr"))
     + if(!requireNamespace(pkg)) {
     + cat("no CRAN package", sQuote(pkg), " ---> no tests here.\n")
     + q("no")
     + }
     Loading required namespace: Rmpfr
     Loading required namespace: DPQmpfr
     > require("Rmpfr")
     Loading required package: Rmpfr
     Loading required package: gmp
    
     Attaching package: 'gmp'
    
     The following objects are masked from 'package:base':
    
     %*%, apply, crossprod, matrix, tcrossprod
    
     C code of R package 'Rmpfr': GMP using 64 bits per limb
    
    
     Attaching package: 'Rmpfr'
    
     The following object is masked from 'package:gmp':
    
     outer
    
     The following object is masked from 'package:DPQ':
    
     log1mexp
    
     The following objects are masked from 'package:stats':
    
     dbinom, dgamma, dnbinom, dnorm, dpois, dt, pnorm
    
     The following objects are masked from 'package:base':
    
     cbind, pmax, pmin, rbind
    
     >
     > cutoffs <- c(15,35,80,500) # cut points, n=*, in the above "algorithm"
     > ##
     > n <- c(seq(1,15, by=1/4),seq(16, 25, by=1/2), 26:30, seq(32,50, by=2), seq(55,1000, by=5),
     + 20*c(51:99), 50*(40:80), 150*(27:48), 500*(15:20))
     > st.n <- stirlerr(n)# rather use.halves=TRUE, just here , use.halves=FALSE)
     > plot(st.n ~ n, log="xy", type="b") ## looks good now
     > nM <- mpfr(n, 2048)
     > st.nM <- stirlerr(nM, use.halves=FALSE) ## << on purpose
     > all.equal(asNumeric(st.nM), st.n)# TRUE
     [1] TRUE
     > all.equal(st.nM, as(st.n,"mpfr"))# .. difference: 1.05884..............................e-15
     Error in target == current : comparison of these types is not implemented
     Calls: all.equal -> all.equal -> .local -> all.equal.numeric
     Execution halted
Flavor: r-devel-linux-x86_64-debian-gcc

Version: 0.5-0
Check: examples
Result: ERROR
    Running examples in ‘DPQ-Ex.R’ failed
    The error most likely occurred in:
    
    > ### Name: ppoisson
    > ### Title: Direct Computation of 'ppois()' Poisson Distribution
    > ### Probabilities
    > ### Aliases: ppoisErr ppoisD
    > ### Keywords: distribution
    >
    > ### ** Examples
    >
    > (lams <- outer(c(1,2,5), 10^(0:3)))# 10^4 is already slow!
     [,1] [,2] [,3] [,4]
    [1,] 1 10 100 1000
    [2,] 2 20 200 2000
    [3,] 5 50 500 5000
    > system.time(e1 <- sapply(lams, ppoisErr))
     user system elapsed
     0.012 0.000 0.041
    > e1 / .Machine$double.eps
     [1] 0.0 0.5 -1.0 1.0 5.5 1.5 -4.0 -3.0 1.0 -1.0 2.0 2.0
    >
    > ## Try another 'ppFUN' :---------------------------------
    > ## this relies on the fact that it's *only* used on an 'x' of the form 0:M :
    > ppD0 <- function(x, lambda, all.from.0=TRUE)
    + cumsum(dpois(if(all.from.0) 0:x else x, lambda=lambda))
    > ## and test it:
    > p0 <- ppD0 ( 1000, lambda=10)
    > p1 <- ppois(0:1000, lambda=10)
    > stopifnot(all.equal(p0,p1, tol=8*.Machine$double.eps))
    >
    > system.time(p0.slow <- ppoisD(0:1000, lambda=10, all.from.0=FALSE))# not very slow, here
     user system elapsed
     0.006 0.000 0.018
    > p0.1 <- ppoisD(1000, lambda=10)
    > if(requireNamespace("Rmpfr")) {
    + ppoisMpfr <- function(x, lambda) cumsum(Rmpfr::dpois(x, lambda=lambda))
    + p0.best <- ppoisMpfr(0:1000, lambda = Rmpfr::mpfr(10, precBits = 256))
    + AllEq. <- Rmpfr::all.equal
    + AllEq <- function(target, current, ...)
    + AllEq.(target, current, ...,
    + formatFUN = function(x, ...) Rmpfr::format(x, digits = 9))
    + print(AllEq(p0.best, p0, tol = 0)) # 2.06e-18
    + print(AllEq(p0.best, p0.slow, tol = 0)) # the "worst" (4.44e-17)
    + print(AllEq(p0.best, p0.1, tol = 0)) # 1.08e-18
    + }
    Error in target == current : comparison of these types is not implemented
    Calls: print ... AllEq -> AllEq. -> AllEq. -> .local -> all.equal.numeric
    Execution halted
Flavor: r-devel-linux-x86_64-fedora-clang

Version: 0.5-0
Check: tests
Result: ERROR
     Running ‘chisq-nonc-ex.R’ [43s/124s]
     Running ‘dnbinom-tst.R’ [24s/63s]
     Running ‘dnchisq-tst.R’
     Running ‘hyper-dist-ex.R’ [34s/96s]
     Running ‘pnbeta-tst.R’
     Running ‘pnt-prec.R’ [36s/78s]
     Running ‘ppois-ex.R’
     Running ‘qPoisBinom-ex.R’
     Running ‘qbeta-dist.R’ [15s/37s]
     Running ‘qbeta-tst.R’
     Running ‘qgamma-ex.R’ [15s/36s]
     Running ‘stirlerr-tst.R’
     Running ‘t-nonc-tst.R’ [7s/19s]
     Running ‘wienergerm-pchisq-tst.R’
     Running ‘wienergerm_nchisq.R’ [9s/21s]
    Running the tests in ‘tests/dnbinom-tst.R’ failed.
    Complete output:
     > #### Testing 1) dbinom_raw(), dnbinomR() and dnbinom.mu()
     > #### 2) log1pmx(), logcf() etc
     > require(DPQ)
     Loading required package: DPQ
     > source(system.file(package="Matrix", "test-tools-1.R", mustWork=TRUE))
     Loading required package: tools
     > ## -> showProc.time(), assertError()
     >
     > (doExtras <- DPQ:::doExtras() && !grepl("valgrind", R.home()))
     [1] FALSE
     >
     > if(!dev.interactive(orNone=TRUE)) pdf("wienergerm-accuracy.pdf")
     >
     >
     > ### 1. Testing dbinom_raw(), dnbinomR() and dnbinom.mu() >>> ../R/dbinom-nbinom.R <<<
     > ### ---------- ../man/dbinom_raw.Rd & ../man/dnbinomR.Rd
     >
     > ## "FIXME:" use sfsmisc :: relErrV() already here
     >
     > ### dbinom() vs dbinom.raw() :
     >
     > for(n in 1:20) {
     + cat("n=",n," ")
     + for(x in 0:n)
     + cat(".")
     + for(p in c(0, .1, .5, .8, 1)) {
     + stopifnot(all.equal(dbinom_raw(x, n, p, q=1-p, log=FALSE),
     + dbinom (x, n, p, log=FALSE)),
     + all.equal(dbinom_raw(x, n, p, q=1-p, log =TRUE),
     + dbinom (x, n, p, log =TRUE)))
     + }
     + cat("\n")
     + }
     n= 1 ..
     n= 2 ...
     n= 3 ....
     n= 4 .....
     n= 5 ......
     n= 6 .......
     n= 7 ........
     n= 8 .........
     n= 9 ..........
     n= 10 ...........
     n= 11 ............
     n= 12 .............
     n= 13 ..............
     n= 14 ...............
     n= 15 ................
     n= 16 .................
     n= 17 ..................
     n= 18 ...................
     n= 19 ....................
     n= 20 .....................
     > showProc.time()
     Time (user system elapsed): 2.196 0.102 4.125
     >
     > ### dnbinom*() :
     > stopifnot(exprs = {
     + dnbinomR(0, 1, 1) == 1
     + })
     >
     > ### exploring 'eps' == "true" tests must be done with Rmpfr !!
     >
     > ### 2. Testing log1pmx(), logcf() etc
     > ### ----------
     >
     > ### 2a: logcf()
     > ## == =======
     > x <- c((-20:3)/4, (25:31)/32) # close (but not too close) to upper bound 1
     >
     > (lC <- logcf (x, i=2, d=3, eps=1e-9))
     [1] 0.2225364 0.2273165 0.2323964 0.2378089 0.2435921 0.2497910 0.2564582
     [8] 0.2636564 0.2714610 0.2799634 0.2892758 0.2995378 0.3109259 0.3236668
     [15] 0.3380584 0.3545014 0.3735507 0.3960035 0.4230578 0.4566237 0.5000000
     [22] 0.5595850 0.6503112 0.8221318 0.8574109 0.8989257 0.9490572 1.0118259
     [29] 1.0948318 1.2152882 1.4286636
     > lCt <- logcf (x, i=2, d=3, eps=1e-9, trace=TRUE) ; stopifnot(identical(lCt, lC))
     it= 0: ==> |b2|=162720
     it= 1: ==> |b2|=1.68458e+08
     it= 2: ==> |b2|=3.02689e+11
     it= 3: ==> |b2|=8.40216e+14
     it= 4: ==> |b2|=3.33607e+18
     it= 5: ==> |b2|=1.79478e+22
     it= 6: ==> |b2|=1.25703e+26
     it= 7: ==> |b2|=1.11146e+30
     it= 8: ==> |b2|=1.21086e+34
     it= 9: ==> |b2|=1.5936e+38
     it=10: ==> |b2|=2.49268e+42
     logcf(*) used 11 iterations.
     it= 0: ==> |b2|=151400
     it= 1: ==> |b2|=1.519e+08
     it= 2: ==> |b2|=2.64707e+11
     it= 3: ==> |b2|=7.12814e+14
     it= 4: ==> |b2|=2.74588e+18
     it= 5: ==> |b2|=1.4333e+22
     it= 6: ==> |b2|=9.73998e+25
     it= 7: ==> |b2|=8.35605e+29
     it= 8: ==> |b2|=8.83286e+33
     it= 9: ==> |b2|=1.12795e+38
     it=10: ==> |b2|=1.71192e+42
     logcf(*) used 11 iterations.
     it= 0: ==> |b2|=140480
     it= 1: ==> |b2|=1.36437e+08
     it= 2: ==> |b2|=2.30332e+11
     it= 3: ==> |b2|=6.0102e+14
     it= 4: ==> |b2|=2.24367e+18
     it= 5: ==> |b2|=1.135e+22
     it= 6: ==> |b2|=7.47503e+25
     it= 7: ==> |b2|=6.21522e+29
     it= 8: ==> |b2|=6.3674e+33
     it= 9: ==> |b2|=7.88061e+37
     it=10: ==> |b2|=1.15921e+42
     logcf(*) used 11 iterations.
     it= 0: ==> |b2|=129960
     it= 1: ==> |b2|=1.22034e+08
     it= 2: ==> |b2|=1.99336e+11
     it= 3: ==> |b2|=5.03394e+14
     it= 4: ==> |b2|=1.81889e+18
     it= 5: ==> |b2|=8.90621e+21
     it= 6: ==> |b2|=5.67763e+25
     it= 7: ==> |b2|=4.56957e+29
     it= 8: ==> |b2|=4.53158e+33
     it= 9: ==> |b2|=5.429e+37
     logcf(*) used 10 iterations.
     it= 0: ==> |b2|=119840
     it= 1: ==> |b2|=1.08655e+08
     it= 2: ==> |b2|=1.71497e+11
     it= 3: ==> |b2|=4.18587e+14
     it= 4: ==> |b2|=1.46194e+18
     it= 5: ==> |b2|=6.91963e+21
     it= 6: ==> |b2|=4.26415e+25
     it= 7: ==> |b2|=3.31759e+29
     it= 8: ==> |b2|=3.18042e+33
     it= 9: ==> |b2|=3.68336e+37
     logcf(*) used 10 iterations.
     it= 0: ==> |b2|=110120
     it= 1: ==> |b2|=9.62638e+07
     it= 2: ==> |b2|=1.46601e+11
     it= 3: ==> |b2|=3.45334e+14
     it= 4: ==> |b2|=1.16411e+18
     it= 5: ==> |b2|=5.31835e+21
     it= 6: ==> |b2|=3.16349e+25
     it= 7: ==> |b2|=2.37577e+29
     it= 8: ==> |b2|=2.19845e+33
     it= 9: ==> |b2|=2.45771e+37
     logcf(*) used 10 iterations.
     it= 0: ==> |b2|=100800
     it= 1: ==> |b2|=8.48232e+07
     it= 2: ==> |b2|=1.24442e+11
     it= 3: ==> |b2|=2.82452e+14
     it= 4: ==> |b2|=9.17519e+17
     it= 5: ==> |b2|=4.03952e+21
     it= 6: ==> |b2|=2.3156e+25
     it= 7: ==> |b2|=1.67591e+29
     it= 8: ==> |b2|=1.49457e+33
     logcf(*) used 9 iterations.
     it= 0: ==> |b2|=91880
     it= 1: ==> |b2|=7.42974e+07
     it= 2: ==> |b2|=1.04819e+11
     it= 3: ==> |b2|=2.28837e+14
     it= 4: ==> |b2|=7.15064e+17
     it= 5: ==> |b2|=3.02848e+21
     it= 6: ==> |b2|=1.67007e+25
     it= 7: ==> |b2|=1.1628e+29
     it= 8: ==> |b2|=9.97611e+32
     logcf(*) used 9 iterations.
     it= 0: ==> |b2|=83360
     it= 1: ==> |b2|=6.46501e+07
     it= 2: ==> |b2|=8.75389e+10
     it= 3: ==> |b2|=1.83464e+14
     it= 4: ==> |b2|=5.50387e+17
     it= 5: ==> |b2|=2.23803e+21
     it= 6: ==> |b2|=1.18496e+25
     it= 7: ==> |b2|=7.92152e+28
     it= 8: ==> |b2|=6.52535e+32
     logcf(*) used 9 iterations.
     it= 0: ==> |b2|=75240
     it= 1: ==> |b2|=5.58449e+07
     it= 2: ==> |b2|=7.24171e+10
     it= 3: ==> |b2|=1.45381e+14
     it= 4: ==> |b2|=4.17809e+17
     it= 5: ==> |b2|=1.6276e+21
     it= 6: ==> |b2|=8.25594e+24
     it= 7: ==> |b2|=5.28764e+28
     logcf(*) used 8 iterations.
     it= 0: ==> |b2|=67520
     it= 1: ==> |b2|=4.78456e+07
     it= 2: ==> |b2|=5.92745e+10
     it= 3: ==> |b2|=1.13708e+14
     it= 4: ==> |b2|=3.12287e+17
     it= 5: ==> |b2|=1.16261e+21
     it= 6: ==> |b2|=5.6361e+24
     it= 7: ==> |b2|=3.44989e+28
     logcf(*) used 8 iterations.
     it= 0: ==> |b2|=60200
     it= 1: ==> |b2|=4.06158e+07
     it= 2: ==> |b2|=4.79397e+10
     it= 3: ==> |b2|=8.76351e+13
     it= 4: ==> |b2|=2.2937e+17
     it= 5: ==> |b2|=8.13827e+20
     it= 6: ==> |b2|=3.76013e+24
     it= 7: ==> |b2|=2.19363e+28
     logcf(*) used 8 iterations.
     it= 0: ==> |b2|=53280
     it= 1: ==> |b2|=3.41194e+07
     it= 2: ==> |b2|=3.82483e+10
     it= 3: ==> |b2|=6.64186e+13
     it= 4: ==> |b2|=1.6515e+17
     it= 5: ==> |b2|=5.56707e+20
     it= 6: ==> |b2|=2.44378e+24
     logcf(*) used 7 iterations.
     it= 0: ==> |b2|=46760
     it= 1: ==> |b2|=2.83198e+07
     it= 2: ==> |b2|=3.0043e+10
     it= 3: ==> |b2|=4.93794e+13
     it= 4: ==> |b2|=1.16224e+17
     it= 5: ==> |b2|=3.70875e+20
     it= 6: ==> |b2|=1.54119e+24
     logcf(*) used 7 iterations.
     it= 0: ==> |b2|=40640
     it= 1: ==> |b2|=2.3181e+07
     it= 2: ==> |b2|=2.31738e+10
     it= 3: ==> |b2|=3.59e+13
     it= 4: ==> |b2|=7.96488e+16
     it= 5: ==> |b2|=2.39588e+20
     logcf(*) used 6 iterations.
     it= 0: ==> |b2|=34920
     it= 1: ==> |b2|=1.86664e+07
     it= 2: ==> |b2|=1.74976e+10
     it= 3: ==> |b2|=2.5422e+13
     it= 4: ==> |b2|=5.29017e+16
     it= 5: ==> |b2|=1.49263e+20
     logcf(*) used 6 iterations.
     it= 0: ==> |b2|=29600
     it= 1: ==> |b2|=1.474e+07
     it= 2: ==> |b2|=1.28785e+10
     it= 3: ==> |b2|=1.74436e+13
     it= 4: ==> |b2|=3.38438e+16
     logcf(*) used 5 iterations.
     it= 0: ==> |b2|=24680
     it= 1: ==> |b2|=1.13653e+07
     it= 2: ==> |b2|=9.18785e+09
     it= 3: ==> |b2|=1.1517e+13
     it= 4: ==> |b2|=2.06815e+16
     logcf(*) used 5 iterations.
     it= 0: ==> |b2|=20160
     it= 1: ==> |b2|=8.50608e+06
     it= 2: ==> |b2|=6.30386e+09
     it= 3: ==> |b2|=7.24564e+12
     logcf(*) used 4 iterations.
     it= 0: ==> |b2|=16040
     it= 1: ==> |b2|=6.12601e+06
     it= 2: ==> |b2|=4.11202e+09
     logcf(*) used 3 iterations.
     logcf(*) used 0 iterations.
     it= 0: ==> |b2|=9000
     it= 1: ==> |b2|=2.65815e+06
     it= 2: ==> |b2|=1.38218e+09
     logcf(*) used 3 iterations.
     it= 0: ==> |b2|=6080
     it= 1: ==> |b2|=1.49776e+06
     it= 2: ==> |b2|=6.50656e+08
     it= 3: ==> |b2|=4.39124e+11
     it= 4: ==> |b2|=4.24985e+14
     logcf(*) used 5 iterations.
     it= 0: ==> |b2|=3560
     it= 1: ==> |b2|=671330
     it= 2: ==> |b2|=2.24237e+08
     it= 3: ==> |b2|=1.16565e+11
     it= 4: ==> |b2|=8.69636e+13
     it= 5: ==> |b2|=8.80714e+16
     it= 6: ==> |b2|=1.16246e+20
     it= 7: ==> |b2|=1.93847e+23
     it= 8: ==> |b2|=3.98491e+26
     logcf(*) used 9 iterations.
     it= 0: ==> |b2|=3273.12
     it= 1: ==> |b2|=589700
     it= 2: ==> |b2|=1.88377e+08
     it= 3: ==> |b2|=9.36959e+10
     it= 4: ==> |b2|=6.68994e+13
     it= 5: ==> |b2|=6.48488e+16
     it= 6: ==> |b2|=8.19327e+19
     it= 7: ==> |b2|=1.30789e+23
     it= 8: ==> |b2|=2.57381e+26
     it= 9: ==> |b2|=6.12129e+29
     logcf(*) used 10 iterations.
     it= 0: ==> |b2|=2992.5
     it= 1: ==> |b2|=512650
     it= 2: ==> |b2|=1.55894e+08
     it= 3: ==> |b2|=7.3859e+10
     it= 4: ==> |b2|=5.02475e+13
     it= 5: ==> |b2|=4.64164e+16
     it= 6: ==> |b2|=5.58911e+19
     it= 7: ==> |b2|=8.50347e+22
     it= 8: ==> |b2|=1.595e+26
     it= 9: ==> |b2|=3.61574e+29
     it=10: ==> |b2|=9.74479e+32
     logcf(*) used 11 iterations.
     it= 0: ==> |b2|=2718.12
     it= 1: ==> |b2|=440109
     it= 2: ==> |b2|=1.26644e+08
     it= 3: ==> |b2|=5.68225e+10
     it= 4: ==> |b2|=3.66244e+13
     it= 5: ==> |b2|=3.20598e+16
     it= 6: ==> |b2|=3.65864e+19
     it= 7: ==> |b2|=5.27587e+22
     it= 8: ==> |b2|=9.37997e+25
     it= 9: ==> |b2|=2.01557e+29
     it=10: ==> |b2|=5.14924e+32
     it=11: ==> |b2|=1.54257e+36
     logcf(*) used 12 iterations.
     it= 0: ==> |b2|=2450
     it= 1: ==> |b2|=372006
     it= 2: ==> |b2|=1.00485e+08
     it= 3: ==> |b2|=4.23633e+10
     it= 4: ==> |b2|=2.56713e+13
     it= 5: ==> |b2|=2.11343e+16
     it= 6: ==> |b2|=2.26869e+19
     it= 7: ==> |b2|=3.07772e+22
     it= 8: ==> |b2|=5.14811e+25
     it= 9: ==> |b2|=1.04082e+29
     it=10: ==> |b2|=2.50192e+32
     it=11: ==> |b2|=7.05238e+35
     it=12: ==> |b2|=2.30384e+39
     logcf(*) used 13 iterations.
     it= 0: ==> |b2|=2188.12
     it= 1: ==> |b2|=308271
     it= 2: ==> |b2|=7.72745e+07
     it= 3: ==> |b2|=3.02658e+10
     it= 4: ==> |b2|=1.70531e+13
     it= 5: ==> |b2|=1.30605e+16
     it= 6: ==> |b2|=1.30466e+19
     it= 7: ==> |b2|=1.64734e+22
     it= 8: ==> |b2|=2.56499e+25
     it= 9: ==> |b2|=4.82765e+28
     it=10: ==> |b2|=1.08039e+32
     it=11: ==> |b2|=2.83535e+35
     it=12: ==> |b2|=8.62389e+38
     it=13: ==> |b2|=3.00926e+42
     it=14: ==> |b2|=1.19409e+46
     it=15: ==> |b2|=5.34632e+49
     logcf(*) used 16 iterations.
     it= 0: ==> |b2|=1932.5
     it= 1: ==> |b2|=248832
     it= 2: ==> |b2|=5.68734e+07
     it= 3: ==> |b2|=2.03226e+10
     it= 4: ==> |b2|=1.04577e+13
     it= 5: ==> |b2|=7.32086e+15
     it= 6: ==> |b2|=6.68834e+18
     it= 7: ==> |b2|=7.72653e+21
     it= 8: ==> |b2|=1.10096e+25
     it= 9: ==> |b2|=1.89662e+28
     it=10: ==> |b2|=3.88536e+31
     it=11: ==> |b2|=9.33474e+34
     it=12: ==> |b2|=2.59938e+38
     it=13: ==> |b2|=8.30457e+41
     it=14: ==> |b2|=3.01718e+45
     it=15: ==> |b2|=1.23692e+49
     it=16: ==> |b2|=5.68258e+52
     it=17: ==> |b2|=2.90768e+56
     it=18: ==> |b2|=1.64796e+60
     logcf(*) used 19 iterations.
     it= 0: ==> |b2|=1683.12
     it= 1: ==> |b2|=193619
     it= 2: ==> |b2|=3.91439e+07
     it= 3: ==> |b2|=1.23338e+10
     it= 4: ==> |b2|=5.59551e+12
     it= 5: ==> |b2|=3.4562e+15
     it= 6: ==> |b2|=2.78868e+18
     it= 7: ==> |b2|=2.84748e+21
     it= 8: ==> |b2|=3.58854e+24
     it= 9: ==> |b2|=5.4701e+27
     it=10: ==> |b2|=9.91885e+30
     it=11: ==> |b2|=2.10987e+34
     it=12: ==> |b2|=5.20269e+37
     it=13: ==> |b2|=1.47211e+41
     it=14: ==> |b2|=4.73732e+44
     it=15: ==> |b2|=1.72036e+48
     it=16: ==> |b2|=7.00164e+51
     it=17: ==> |b2|=3.17394e+55
     it=18: ==> |b2|=1.59374e+59
     it=19: ==> |b2|=8.8205e+62
     it=20: ==> |b2|=5.35623e+66
     it=21: ==> |b2|=3.5541e+70
     it=22: ==> |b2|=2.56724e+74
     it=23: ==> |b2|=2.01172e+78 Lrg |b2|
     it=24: ==> |b2|=147221
     it=25: ==> |b2|=1.34508e+09
     it=26: ==> |b2|=1.32142e+13
     it=27: ==> |b2|=1.39232e+17
     logcf(*) used 28 iterations.
     > (lR <- logcfR(x, i=2, d=3, eps=1e-9))
     [1] 0.2225364 0.2273165 0.2323964 0.2378089 0.2435921 0.2497910 0.2564582
     [8] 0.2636564 0.2714610 0.2799634 0.2892758 0.2995378 0.3109259 0.3236668
     [15] 0.3380584 0.3545014 0.3735507 0.3960035 0.4230578 0.4566237 0.5000000
     [22] 0.5595850 0.6503112 0.8221318 0.8574109 0.8989257 0.9490572 1.0118259
     [29] 1.0948318 1.2152882 1.4286636
     > all.equal(lC, lR, tol = 0) # to see if ..
     [1] TRUE
     > stopifnot(all.equal(lC, lR, tol = 4e-16))
     > lRt <- logcfR(x, i=2, d=3, eps=1e-9, trace=TRUE) ; stopifnot(identical(lRt, lR))
     logcf(x[ 1]= -5, i=2, d=3, eps=1e-09) logcf(*) end: after 11 iterations.
     logcf(x[ 2]= -4.75, i=2, d=3, eps=1e-09) logcf(*) end: after 11 iterations.
     logcf(x[ 3]= -4.5, i=2, d=3, eps=1e-09) logcf(*) end: after 11 iterations.
     logcf(x[ 4]= -4.25, i=2, d=3, eps=1e-09) logcf(*) end: after 10 iterations.
     logcf(x[ 5]= -4, i=2, d=3, eps=1e-09) logcf(*) end: after 10 iterations.
     logcf(x[ 6]= -3.75, i=2, d=3, eps=1e-09) logcf(*) end: after 10 iterations.
     logcf(x[ 7]= -3.5, i=2, d=3, eps=1e-09) logcf(*) end: after 9 iterations.
     logcf(x[ 8]= -3.25, i=2, d=3, eps=1e-09) logcf(*) end: after 9 iterations.
     logcf(x[ 9]= -3, i=2, d=3, eps=1e-09) logcf(*) end: after 9 iterations.
     logcf(x[10]= -2.75, i=2, d=3, eps=1e-09) logcf(*) end: after 8 iterations.
     logcf(x[11]= -2.5, i=2, d=3, eps=1e-09) logcf(*) end: after 8 iterations.
     logcf(x[12]= -2.25, i=2, d=3, eps=1e-09) logcf(*) end: after 8 iterations.
     logcf(x[13]= -2, i=2, d=3, eps=1e-09) logcf(*) end: after 7 iterations.
     logcf(x[14]= -1.75, i=2, d=3, eps=1e-09) logcf(*) end: after 7 iterations.
     logcf(x[15]= -1.5, i=2, d=3, eps=1e-09) logcf(*) end: after 6 iterations.
     logcf(x[16]= -1.25, i=2, d=3, eps=1e-09) logcf(*) end: after 6 iterations.
     logcf(x[17]= -1, i=2, d=3, eps=1e-09) logcf(*) end: after 5 iterations.
     logcf(x[18]= -0.75, i=2, d=3, eps=1e-09) logcf(*) end: after 5 iterations.
     logcf(x[19]= -0.5, i=2, d=3, eps=1e-09) logcf(*) end: after 4 iterations.
     logcf(x[20]= -0.25, i=2, d=3, eps=1e-09) logcf(*) end: after 3 iterations.
     logcf(x[21]= 0, i=2, d=3, eps=1e-09) logcf(*) end: after 0 iterations.
     logcf(x[22]= 0.25, i=2, d=3, eps=1e-09) logcf(*) end: after 3 iterations.
     logcf(x[23]= 0.5, i=2, d=3, eps=1e-09) logcf(*) end: after 5 iterations.
     logcf(x[24]= 0.75, i=2, d=3, eps=1e-09) logcf(*) end: after 9 iterations.
     logcf(x[25]= 0.78125, i=2, d=3, eps=1e-09) logcf(*) end: after 10 iterations.
     logcf(x[26]= 0.8125, i=2, d=3, eps=1e-09) logcf(*) end: after 11 iterations.
     logcf(x[27]= 0.84375, i=2, d=3, eps=1e-09) logcf(*) end: after 12 iterations.
     logcf(x[28]= 0.875, i=2, d=3, eps=1e-09) logcf(*) end: after 13 iterations.
     logcf(x[29]= 0.90625, i=2, d=3, eps=1e-09) logcf(*) end: after 16 iterations.
     logcf(x[30]= 0.9375, i=2, d=3, eps=1e-09) logcf(*) end: after 19 iterations.
     logcf(x[31]= 0.96875, i=2, d=3, eps=1e-09) logcf(*) end: after 28 iterations.
     > lRt2 <- logcfR(x, i=2, d=3, eps=1e-9, trace= 2) ; stopifnot(identical(lRt2,lR))
     logcf(x[ 1]= -5, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 162720 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.0168627
     it= 2: ==> B2= 1.68458e+08 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00303811
     it= 3: ==> B2= 3.02689e+11 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000541327
     it= 4: ==> B2= 8.40216e+14 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 9.60626e-05
     it= 5: ==> B2= 3.33607e+18 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.70167e-05
     it= 6: ==> B2= 1.79478e+22 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.01154e-06
     it= 7: ==> B2= 1.25703e+26 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 5.32664e-07
     it= 8: ==> B2= 1.11146e+30 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 9.4179e-08
     it= 9: ==> B2= 1.21086e+34 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.66472e-08
     it=10: ==> B2= 1.5936e+38 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.942e-09
     it=11: ==> B2= 2.49268e+42 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 5.19854e-10
     logcf(*) end: after 11 iterations.
     logcf(x[ 2]= -4.75, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 151400 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.0157061
     it= 2: ==> B2= 1.519e+08 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00271234
     it= 3: ==> B2= 2.64707e+11 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000463242
     it= 4: ==> B2= 7.12814e+14 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 7.88006e-05
     it= 5: ==> B2= 2.74588e+18 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.33808e-05
     it= 6: ==> B2= 1.4333e+22 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.26999e-06
     it= 7: ==> B2= 9.73998e+25 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.84872e-07
     it= 8: ==> B2= 8.35605e+29 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 6.52292e-08
     it= 9: ==> B2= 8.83286e+33 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.10523e-08
     it=10: ==> B2= 1.12795e+38 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.8723e-09
     it=11: ==> B2= 1.71192e+42 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.1713e-10
     logcf(*) end: after 11 iterations.
     logcf(x[ 3]= -4.5, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 140480 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.014539
     it= 2: ==> B2= 1.36437e+08 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00239872
     it= 3: ==> B2= 2.30332e+11 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000391409
     it= 4: ==> B2= 6.0102e+14 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 6.36152e-05
     it= 5: ==> B2= 2.24367e+18 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.03211e-05
     it= 6: ==> B2= 1.135e+22 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.67293e-06
     it= 7: ==> B2= 7.47503e+25 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.71005e-07
     it= 8: ==> B2= 6.21522e+29 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 4.38843e-08
     it= 9: ==> B2= 6.3674e+33 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 7.10431e-09
     it=10: ==> B2= 7.88061e+37 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.14987e-09
     it=11: ==> B2= 1.15921e+42 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.86085e-10
     logcf(*) end: after 11 iterations.
     logcf(x[ 4]= -4.25, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 129960 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.0133641
     it= 2: ==> B2= 1.22034e+08 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00209863
     it= 3: ==> B2= 1.99336e+11 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000325959
     it= 4: ==> B2= 5.03394e+14 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 5.04302e-05
     it= 5: ==> B2= 1.81889e+18 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 7.78852e-06
     it= 6: ==> B2= 8.90621e+21 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.20172e-06
     it= 7: ==> B2= 5.67763e+25 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.85309e-07
     it= 8: ==> B2= 4.56957e+29 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.85641e-08
     it= 9: ==> B2= 4.53158e+33 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 4.40173e-09
     it=10: ==> B2= 5.429e+37 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 6.78171e-10
     logcf(*) end: after 10 iterations.
     logcf(x[ 5]= -4, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 119840 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.0121853
     it= 2: ==> B2= 1.08655e+08 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00181353
     it= 3: ==> B2= 1.71497e+11 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000266983
     it= 4: ==> B2= 4.18587e+14 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.91528e-05
     it= 5: ==> B2= 1.46194e+18 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 5.73167e-06
     it= 6: ==> B2= 6.91963e+21 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 8.38266e-07
     it= 7: ==> B2= 4.26415e+25 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.22525e-07
     it= 8: ==> B2= 3.31759e+29 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.79016e-08
     it= 9: ==> B2= 3.18042e+33 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.6148e-09
     it=10: ==> B2= 3.68336e+37 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.81854e-10
     logcf(*) end: after 10 iterations.
     logcf(x[ 6]= -3.75, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 110120 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.0110067
     it= 2: ==> B2= 9.62638e+07 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00154491
     it= 3: ==> B2= 1.46601e+11 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00021452
     it= 4: ==> B2= 3.45334e+14 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.96738e-05
     it= 5: ==> B2= 1.16411e+18 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 4.0975e-06
     it= 6: ==> B2= 5.31835e+21 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 5.65252e-07
     it= 7: ==> B2= 3.16349e+25 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 7.79297e-08
     it= 8: ==> B2= 2.37577e+29 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.07396e-08
     it= 9: ==> B2= 2.19845e+33 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.47962e-09
     it=10: ==> B2= 2.45771e+37 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.03808e-10
     logcf(*) end: after 10 iterations.
     logcf(x[ 7]= -3.5, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 100800 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00983372
     it= 2: ==> B2= 8.48232e+07 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00129431
     it= 3: ==> B2= 1.24442e+11 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000168553
     it= 4: ==> B2= 2.82452e+14 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.18673e-05
     it= 5: ==> B2= 9.17519e+17 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.83198e-06
     it= 6: ==> B2= 4.03952e+21 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.66405e-07
     it= 7: ==> B2= 2.3156e+25 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 4.73767e-08
     it= 8: ==> B2= 1.67591e+29 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 6.12337e-09
     it= 9: ==> B2= 1.49457e+33 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 7.91207e-10
     logcf(*) end: after 9 iterations.
     logcf(x[ 8]= -3.25, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 91880 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00867282
     it= 2: ==> B2= 7.42974e+07 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00106327
     it= 3: ==> B2= 1.04819e+11 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000128994
     it= 4: ==> B2= 2.28837e+14 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.55909e-05
     it= 5: ==> B2= 7.15064e+17 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.88109e-06
     it= 6: ==> B2= 3.02848e+21 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.26734e-07
     it= 7: ==> B2= 1.67007e+25 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.7312e-08
     it= 8: ==> B2= 1.1628e+29 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.28859e-09
     it= 9: ==> B2= 9.97611e+32 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.95855e-10
     logcf(*) end: after 9 iterations.
     logcf(x[ 9]= -3, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 83360 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00753175
     it= 2: ==> B2= 6.46501e+07 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000853254
     it= 3: ==> B2= 8.75389e+10 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 9.5671e-05
     it= 4: ==> B2= 1.83464e+14 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.06874e-05
     it= 5: ==> B2= 5.50387e+17 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.19178e-06
     it= 6: ==> B2= 2.23803e+21 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.32765e-07
     it= 7: ==> B2= 1.18496e+25 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.47808e-08
     it= 8: ==> B2= 7.92152e+28 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.64484e-09
     it= 9: ==> B2= 6.52535e+32 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.82987e-10
     logcf(*) end: after 9 iterations.
     logcf(x[10]= -2.75, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 75240 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00641978
     it= 2: ==> B2= 5.58449e+07 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000665641
     it= 3: ==> B2= 7.24171e+10 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 6.83222e-05
     it= 4: ==> B2= 1.45381e+14 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 6.98689e-06
     it= 5: ==> B2= 4.17809e+17 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 7.13234e-07
     it= 6: ==> B2= 1.6276e+21 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 7.27338e-08
     it= 7: ==> B2= 8.25594e+24 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 7.41242e-09
     it= 8: ==> B2= 5.28764e+28 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 7.55083e-10
     logcf(*) end: after 8 iterations.
     logcf(x[11]= -2.5, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 67520 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00534792
     it= 2: ==> B2= 4.78456e+07 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.0005016
     it= 3: ==> B2= 5.92745e+10 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 4.65818e-05
     it= 4: ==> B2= 1.13708e+14 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 4.31004e-06
     it= 5: ==> B2= 3.12287e+17 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.98074e-07
     it= 6: ==> B2= 1.16261e+21 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.67278e-08
     it= 7: ==> B2= 5.6361e+24 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.38642e-09
     it= 8: ==> B2= 3.44989e+28 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.12099e-10
     logcf(*) end: after 8 iterations.
     logcf(x[12]= -2.25, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 60200 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00432911
     it= 2: ==> B2= 4.06158e+07 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00036199
     it= 3: ==> B2= 4.79397e+10 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.99753e-05
     it= 4: ==> B2= 8.76351e+13 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.47309e-06
     it= 5: ==> B2= 2.2937e+17 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.03667e-07
     it= 6: ==> B2= 8.13827e+20 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.67549e-08
     it= 7: ==> B2= 3.76013e+24 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.37743e-09
     it= 8: ==> B2= 2.19363e+28 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.13188e-10
     logcf(*) end: after 8 iterations.
     logcf(x[13]= -2, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 53280 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00337838
     it= 2: ==> B2= 3.41194e+07 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00024722
     it= 3: ==> B2= 3.82483e+10 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.79187e-05
     it= 4: ==> B2= 6.64186e+13 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.29399e-06
     it= 5: ==> B2= 1.6515e+17 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 9.32713e-08
     it= 6: ==> B2= 5.56707e+20 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 6.71575e-09
     it= 7: ==> B2= 2.44378e+24 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 4.83216e-10
     logcf(*) end: after 7 iterations.
     logcf(x[14]= -1.75, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 46760 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00251277
     it= 2: ==> B2= 2.83198e+07 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000157067
     it= 3: ==> B2= 3.0043e+10 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 9.72602e-06
     it= 4: ==> B2= 4.93794e+13 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 6.00029e-07
     it= 5: ==> B2= 1.16224e+17 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.69475e-08
     it= 6: ==> B2= 3.70875e+20 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.27255e-09
     it= 7: ==> B2= 1.54119e+24 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.39681e-10
     logcf(*) end: after 7 iterations.
     logcf(x[15]= -1.5, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 40640 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.001751
     it= 2: ==> B2= 2.3181e+07 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 9.04805e-05
     it= 3: ==> B2= 2.31738e+10 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 4.63221e-06
     it= 4: ==> B2= 3.59e+13 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.36258e-07
     it= 5: ==> B2= 7.96488e+16 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.20265e-08
     it= 6: ==> B2= 2.39588e+20 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 6.11497e-10
     logcf(*) end: after 6 iterations.
     logcf(x[16]= -1.25, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 34920 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00111245
     it= 2: ==> B2= 1.86664e+07 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 4.53708e-05
     it= 3: ==> B2= 1.74976e+10 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.8334e-06
     it= 4: ==> B2= 2.5422e+13 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 7.38016e-08
     it= 5: ==> B2= 5.29017e+16 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.96486e-09
     it= 6: ==> B2= 1.49263e+20 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.18968e-10
     logcf(*) end: after 6 iterations.
     logcf(x[17]= -1, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 29600 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000614941
     it= 2: ==> B2= 1.474e+07 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.84647e-05
     it= 3: ==> B2= 1.28785e+10 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 5.49306e-07
     it= 4: ==> B2= 1.74436e+13 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.62767e-08
     it= 5: ==> B2= 3.38438e+16 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 4.81303e-10
     logcf(*) end: after 5 iterations.
     logcf(x[18]= -0.75, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 24680 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000270385
     it= 2: ==> B2= 1.13653e+07 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 5.33194e-06
     it= 3: ==> B2= 9.18785e+09 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.04153e-07
     it= 4: ==> B2= 1.1517e+13 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.02617e-09
     it= 5: ==> B2= 2.06815e+16 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.93312e-11
     logcf(*) end: after 5 iterations.
     logcf(x[19]= -0.5, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 20160 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 7.69704e-05
     it= 2: ==> B2= 8.50608e+06 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 8.02466e-07
     it= 3: ==> B2= 6.30386e+09 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 8.28445e-09
     it= 4: ==> B2= 7.24564e+12 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 8.51583e-11
     logcf(*) end: after 4 iterations.
     logcf(x[20]= -0.25, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 16040 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 7.26392e-06
     it= 2: ==> B2= 6.12601e+06 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.30773e-08
     it= 3: ==> B2= 4.11202e+09 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 7.25571e-11
     logcf(*) end: after 3 iterations.
     logcf(x[21]= 0, i=2, d=3, eps=1e-09) iterations:
     logcf(*) end: after 0 iterations.
     logcf(x[22]= 0.25, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 9000 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.04918e-05
     it= 2: ==> B2= 2.65815e+06 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.08623e-07
     it= 3: ==> B2= 1.38218e+09 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 5.68393e-10
     logcf(*) end: after 3 iterations.
     logcf(x[23]= 0.5, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 6080 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000659523
     it= 2: ==> B2= 1.49776e+06 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.00942e-05
     it= 3: ==> B2= 6.50656e+08 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 6.02264e-07
     it= 4: ==> B2= 4.39124e+11 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.79273e-08
     it= 5: ==> B2= 4.24985e+14 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 5.3174e-10
     logcf(*) end: after 5 iterations.
     logcf(x[24]= 0.75, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 3560 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00856402
     it= 2: ==> B2= 671330 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00100335
     it= 3: ==> B2= 2.24237e+08 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000114482
     it= 4: ==> B2= 1.16565e+11 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.2922e-05
     it= 5: ==> B2= 8.69636e+13 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.45089e-06
     it= 6: ==> B2= 8.80714e+16 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.62421e-07
     it= 7: ==> B2= 1.16246e+20 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.81486e-08
     it= 8: ==> B2= 1.93847e+23 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.02539e-09
     it= 9: ==> B2= 3.98491e+26 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.25837e-10
     logcf(*) end: after 9 iterations.
     logcf(x[25]= 0.78125, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 3273.12 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.0116683
     it= 2: ==> B2= 589700 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00162549
     it= 3: ==> B2= 1.88377e+08 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000220072
     it= 4: ==> B2= 9.36959e+10 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.94426e-05
     it= 5: ==> B2= 6.68994e+13 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.9163e-06
     it= 6: ==> B2= 6.48488e+16 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 5.19244e-07
     it= 7: ==> B2= 8.19327e+19 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 6.87066e-08
     it= 8: ==> B2= 1.30789e+23 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 9.07922e-09
     it= 9: ==> B2= 2.57381e+26 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.19866e-09
     it=10: ==> B2= 6.12129e+29 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.58143e-10
     logcf(*) end: after 10 iterations.
     logcf(x[26]= 0.8125, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 2992.5 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.0159401
     it= 2: ==> B2= 512650 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00265674
     it= 3: ==> B2= 1.55894e+08 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000429426
     it= 4: ==> B2= 7.3859e+10 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 6.84941e-05
     it= 5: ==> B2= 5.02475e+13 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.08546e-05
     it= 6: ==> B2= 4.64164e+16 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.71404e-06
     it= 7: ==> B2= 5.58911e+19 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.70071e-07
     it= 8: ==> B2= 8.50347e+22 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 4.2492e-08
     it= 9: ==> B2= 1.595e+26 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 6.6788e-09
     it=10: ==> B2= 3.61574e+29 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.04899e-09
     it=11: ==> B2= 9.74479e+32 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.64668e-10
     logcf(*) end: after 11 iterations.
     logcf(x[27]= 0.84375, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 2718.12 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.0218736
     it= 2: ==> B2= 440109 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00440022
     it= 3: ==> B2= 1.26644e+08 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000856838
     it= 4: ==> B2= 5.68225e+10 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000164362
     it= 5: ==> B2= 3.66244e+13 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.12964e-05
     it= 6: ==> B2= 3.20598e+16 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 5.93505e-06
     it= 7: ==> B2= 3.65864e+19 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.12276e-06
     it= 8: ==> B2= 5.27587e+22 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.12056e-07
     it= 9: ==> B2= 9.37997e+25 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 4.00067e-08
     it=10: ==> B2= 2.01557e+29 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 7.54162e-09
     it=11: ==> B2= 5.14924e+32 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.42081e-09
     it=12: ==> B2= 1.54257e+36 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.67552e-10
     logcf(*) end: after 12 iterations.
     logcf(x[28]= 0.875, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 2450 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.0302147
     it= 2: ==> B2= 372006 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00742718
     it= 3: ==> B2= 1.00485e+08 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00176572
     it= 4: ==> B2= 4.23633e+10 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000412688
     it= 5: ==> B2= 2.56713e+13 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 9.56191e-05
     it= 6: ==> B2= 2.11343e+16 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.20491e-05
     it= 7: ==> B2= 2.26869e+19 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 5.0698e-06
     it= 8: ==> B2= 3.07772e+22 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.16356e-06
     it= 9: ==> B2= 5.14811e+25 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.66706e-07
     it=10: ==> B2= 1.04082e+29 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 6.10778e-08
     it=11: ==> B2= 2.50192e+32 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.39778e-08
     it=12: ==> B2= 7.05238e+35 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.19721e-09
     it=13: ==> B2= 2.30384e+39 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 7.31016e-10
     logcf(*) end: after 13 iterations.
     logcf(x[29]= 0.90625, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 2188.12 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.0421192
     it= 2: ==> B2= 308271 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.0128742
     it= 3: ==> B2= 7.72745e+07 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00381265
     it= 4: ==> B2= 3.02658e+10 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00110791
     it= 5: ==> B2= 1.70531e+13 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000318587
     it= 6: ==> B2= 1.30605e+16 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 9.10705e-05
     it= 7: ==> B2= 1.30466e+19 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.5941e-05
     it= 8: ==> B2= 1.64734e+22 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 7.37247e-06
     it= 9: ==> B2= 2.56499e+25 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.09206e-06
     it=10: ==> B2= 4.82765e+28 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 5.93012e-07
     it=11: ==> B2= 1.08039e+32 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.6796e-07
     it=12: ==> B2= 2.83535e+35 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 4.75428e-08
     it=13: ==> B2= 8.62389e+38 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.34511e-08
     it=14: ==> B2= 3.00926e+42 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.80425e-09
     it=15: ==> B2= 1.19409e+46 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.07559e-09
     it=16: ==> B2= 5.34632e+49 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.04032e-10
     logcf(*) end: after 16 iterations.
     logcf(x[30]= 0.9375, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 1932.5 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.0594391
     it= 2: ==> B2= 248832 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.02317
     it= 3: ==> B2= 5.68734e+07 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00882488
     it= 4: ==> B2= 2.03226e+10 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00329775
     it= 5: ==> B2= 1.04577e+13 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00121712
     it= 6: ==> B2= 7.32086e+15 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000445765
     it= 7: ==> B2= 6.68834e+18 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00016248
     it= 8: ==> B2= 7.72653e+21 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 5.90414e-05
     it= 9: ==> B2= 1.10096e+25 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.14101e-05
     it=10: ==> B2= 1.89662e+28 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 7.7527e-06
     it=11: ==> B2= 3.88536e+31 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.80437e-06
     it=12: ==> B2= 9.33474e+34 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.01363e-06
     it=13: ==> B2= 2.59938e+38 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.6615e-07
     it=14: ==> B2= 8.30457e+41 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.32201e-07
     it=15: ==> B2= 3.01718e+45 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 4.77137e-08
     it=16: ==> B2= 1.23692e+49 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.72154e-08
     it=17: ==> B2= 5.68258e+52 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 6.20986e-09
     it=18: ==> B2= 2.90768e+56 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.23951e-09
     it=19: ==> B2= 1.64796e+60 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 8.07503e-10
     logcf(*) end: after 19 iterations.
     logcf(x[31]= 0.96875, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 1683.12 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.0852619
     it= 2: ==> B2= 193619 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.0440308
     it= 3: ==> B2= 3.91439e+07 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.0227933
     it= 4: ==> B2= 1.23338e+10 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.0116823
     it= 5: ==> B2= 5.59551e+12 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00592272
     it= 6: ==> B2= 3.4562e+15 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00297607
     it= 7: ==> B2= 2.78868e+18 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00148555
     it= 8: ==> B2= 2.84748e+21 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00073801
     it= 9: ==> B2= 3.58854e+24 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000365396
     it=10: ==> B2= 5.4701e+27 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000180472
     it=11: ==> B2= 9.91885e+30 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 8.89791e-05
     it=12: ==> B2= 2.10987e+34 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 4.38122e-05
     it=13: ==> B2= 5.20269e+37 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.15512e-05
     it=14: ==> B2= 1.47211e+41 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.05929e-05
     it=15: ==> B2= 4.73732e+44 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 5.20351e-06
     it=16: ==> B2= 1.72036e+48 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.55486e-06
     it=17: ==> B2= 7.00164e+51 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.25391e-06
     it=18: ==> B2= 3.17394e+55 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 6.15209e-07
     it=19: ==> B2= 1.59374e+59 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.0176e-07
     it=20: ==> B2= 8.8205e+62 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.47979e-07
     it=21: ==> B2= 5.35623e+66 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 7.25526e-08
     it=22: ==> B2= 3.5541e+70 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.55657e-08
     it=23: ==> B2= 2.56724e+74 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.7432e-08
     it=24: ==> B2= 2.01172e+78 Lrg m.B2
     --> crit. |A2*b1 - a1*B2|/|b1*B2| = 8.54292e-09
     it=25: ==> B2= 147221 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 4.18617e-09
     it=26: ==> B2= 1.34508e+09 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.05108e-09
     it=27: ==> B2= 1.32142e+13 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.00487e-09
     it=28: ==> B2= 1.39232e+17 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 4.92273e-10
     logcf(*) end: after 28 iterations.
     >
     > lR. <- logcfR.(x, i=2, d=3, eps=1e-9)
     > lR.t <- logcfR.(x, i=2, d=3, eps=1e-9, trace=TRUE) ; stopifnot(identical(lR.t, lR.))
     logcf(x[], i=2, d=3, eps=1e-09) iterations:
     it= 1: needIt: 30 TRUE, and 1 F.; length(x[<todo>])=30, m.B2= 23307.2
     it= 2: needIt: 30 TRUE; length(x[<todo>])=30, m.B2= 1.01159e+07
     it= 3: needIt: 30 TRUE; length(x[<todo>])=30, m.B2= 7.70605e+09
     it= 4: needIt: 28 TRUE, and 2 F.; length(x[<todo>])=28, m.B2= 1.00852e+13
     it= 5: needIt: 27 TRUE, and 1 F.; length(x[<todo>])=27, m.B2= 1.76419e+16
     it= 6: needIt: 24 TRUE, and 3 F.; length(x[<todo>])=24, m.B2= 4.75316e+19
     it= 7: needIt: 22 TRUE, and 2 F.; length(x[<todo>])=22, m.B2= 1.2798e+23
     it= 8: needIt: 20 TRUE, and 2 F.; length(x[<todo>])=20, m.B2= 3.63581e+26
     it= 9: needIt: 17 TRUE, and 3 F.; length(x[<todo>])=17, m.B2= 6.8674e+29
     it=10: needIt: 13 TRUE, and 4 F.; length(x[<todo>])=13, m.B2= 1.03776e+33
     it=11: needIt: 9 TRUE, and 4 F.; length(x[<todo>])= 9, m.B2= 3.09233e+35
     it=12: needIt: 5 TRUE, and 4 F.; length(x[<todo>])= 5, m.B2= 2.27357e+35
     it=13: needIt: 4 TRUE, and 1 F.; length(x[<todo>])= 4, m.B2= 4.04868e+38
     it=14: needIt: 3 TRUE, and 1 F.; length(x[<todo>])= 3, m.B2= 7.16537e+41
     it=15: needIt: 3 TRUE; length(x[<todo>])= 3, m.B2= 2.57468e+45
     it=16: needIt: 3 TRUE; length(x[<todo>])= 3, m.B2= 1.04393e+49
     it=17: needIt: 2 TRUE, and 1 F.; length(x[<todo>])= 2, m.B2= 1.99468e+52
     it=18: needIt: 2 TRUE; length(x[<todo>])= 2, m.B2= 9.60666e+55
     it=19: needIt: 2 TRUE; length(x[<todo>])= 2, m.B2= 5.12487e+59
     it=20: needIt: 1 TRUE, and 1 F.; length(x[<todo>])= 1, m.B2= 8.8205e+62
     it=21: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 5.35623e+66
     it=22: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 3.5541e+70
     it=23: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 2.56724e+74
     it=24: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 2.01172e+78 Lrg m.B2
     it=25: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 147221
     it=26: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 1.34508e+09
     it=27: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 1.32142e+13
     it=28: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 1.39232e+17
     logcf(*) end: after 28 iterations.
     >
     > all.equal(lC, lR., tol = 0) # TRUE !! (every where ?)
     [1] TRUE
     > all.equal(lR, lR., tol = 0) # TRUE !! " "
     [1] TRUE
     > stopifnot(all.equal(lC, lR., tol = 1e-15))
     > ## (even though they used eps=1e-9 .. i.e., are not *so* accurate)
     > showProc.time()
     Time (user system elapsed): 0.052 0.014 0.145
     >
     > ##--- now with improved logcfR.() {<< will become the new logcfR() at least for MPFR !}:
     >
     > ##require(Rmpfr) may be not, see if NS loading (via "::") is sufficient:
     > requireNamespace("Rmpfr") || quit("no")
     Loading required namespace: Rmpfr
     [1] TRUE
     > ## ----- ----------
     > xM <- Rmpfr::mpfr(x, 512)
     > (ct.14 <- system.time(lR.14 <- logcfR.(xM, i=2, d=3, eps=1e-20, trace=TRUE))) # 0.55 sec
     logcf(x[], i=2, d=3, eps=1e-20) iterations:
     it= 1: needIt: 30 TRUE, and 1 F.; length(x[<todo>])=30, m.B2= 23307.2
     it= 2: needIt: 30 TRUE; length(x[<todo>])=30, m.B2= 1.01159e+07
     it= 3: needIt: 30 TRUE; length(x[<todo>])=30, m.B2= 7.70605e+09
     it= 4: needIt: 30 TRUE; length(x[<todo>])=30, m.B2= 9.10781e+12
     it= 5: needIt: 30 TRUE; length(x[<todo>])=30, m.B2= 1.54287e+16
     it= 6: needIt: 30 TRUE; length(x[<todo>])=30, m.B2= 3.54543e+19
     it= 7: needIt: 30 TRUE; length(x[<todo>])=30, m.B2= 1.06137e+23
     it= 8: needIt: 29 TRUE, and 1 F.; length(x[<todo>])=29, m.B2= 4.19177e+26
     it= 9: needIt: 28 TRUE, and 1 F.; length(x[<todo>])=28, m.B2= 2.26761e+30
     it=10: needIt: 27 TRUE, and 1 F.; length(x[<todo>])=27, m.B2= 1.33011e+34
     it=11: needIt: 27 TRUE; length(x[<todo>])=27, m.B2= 9.0823e+37
     it=12: needIt: 26 TRUE, and 1 F.; length(x[<todo>])=26, m.B2= 7.15387e+41
     it=13: needIt: 25 TRUE, and 1 F.; length(x[<todo>])=25, m.B2= 6.21918e+45
     it=14: needIt: 24 TRUE, and 1 F.; length(x[<todo>])=24, m.B2= 9.51187e+49
     it=15: needIt: 23 TRUE, and 1 F.; length(x[<todo>])=23, m.B2= 1.04428e+54
     it=16: needIt: 22 TRUE, and 1 F.; length(x[<todo>])=22, m.B2= 1.19866e+58
     it=17: needIt: 21 TRUE, and 1 F.; length(x[<todo>])=21, m.B2= 1.40641e+62
     it=18: needIt: 20 TRUE, and 1 F.; length(x[<todo>])=20, m.B2= 1.64566e+66
     it=19: needIt: 19 TRUE, and 1 F.; length(x[<todo>])=19, m.B2= 1.86787e+70
     it=20: needIt: 17 TRUE, and 2 F.; length(x[<todo>])=17, m.B2= 9.5095e+73
     it=21: needIt: 15 TRUE, and 2 F.; length(x[<todo>])=15, m.B2= 2.07684e+78 Lrg m.B2
     it=22: needIt: 14 TRUE, and 1 F.; length(x[<todo>])=14, m.B2= 122830
     it=23: needIt: 11 TRUE, and 3 F.; length(x[<todo>])=11, m.B2= 3.76273e+08
     it=24: needIt: 10 TRUE, and 1 F.; length(x[<todo>])=10, m.B2= 7.77428e+11
     it=25: needIt: 7 TRUE, and 3 F.; length(x[<todo>])= 7, m.B2= 4.17254e+13
     it=26: needIt: 6 TRUE, and 1 F.; length(x[<todo>])= 6, m.B2= 1.55243e+15
     it=27: needIt: 5 TRUE, and 1 F.; length(x[<todo>])= 5, m.B2= 2.47748e+15
     it=28: needIt: 4 TRUE, and 1 F.; length(x[<todo>])= 4, m.B2= 1.06982e+19
     it=29: needIt: 4 TRUE; length(x[<todo>])= 4, m.B2= 1.40477e+23
     it=30: needIt: 4 TRUE; length(x[<todo>])= 4, m.B2= 1.9693e+27
     it=31: needIt: 3 TRUE, and 1 F.; length(x[<todo>])= 3, m.B2= 7.6538e+30
     it=32: needIt: 3 TRUE; length(x[<todo>])= 3, m.B2= 1.16488e+35
     it=33: needIt: 3 TRUE; length(x[<todo>])= 3, m.B2= 1.88175e+39
     it=34: needIt: 3 TRUE; length(x[<todo>])= 3, m.B2= 3.22081e+43
     it=35: needIt: 3 TRUE; length(x[<todo>])= 3, m.B2= 5.83159e+47
     it=36: needIt: 3 TRUE; length(x[<todo>])= 3, m.B2= 1.11521e+52
     it=37: needIt: 2 TRUE, and 1 F.; length(x[<todo>])= 2, m.B2= 3.51533e+55
     it=38: needIt: 2 TRUE; length(x[<todo>])= 2, m.B2= 7.10714e+59
     it=39: needIt: 2 TRUE; length(x[<todo>])= 2, m.B2= 1.51138e+64
     it=40: needIt: 2 TRUE; length(x[<todo>])= 2, m.B2= 3.37644e+68
     it=41: needIt: 2 TRUE; length(x[<todo>])= 2, m.B2= 7.91477e+72
     it=42: needIt: 2 TRUE; length(x[<todo>])= 2, m.B2= 1.94455e+77 Lrg m.B2
     it=43: needIt: 2 TRUE; length(x[<todo>])= 2, m.B2= 43197.4
     it=44: needIt: 2 TRUE; length(x[<todo>])= 2, m.B2= 1.16214e+09
     it=45: needIt: 1 TRUE, and 1 F.; length(x[<todo>])= 1, m.B2= 2.11103e+12
     it=46: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 5.83147e+16
     it=47: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 1.68004e+21
     it=48: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 5.04365e+25
     it=49: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 1.57649e+30
     it=50: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 5.12638e+34
     it=51: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 1.7329e+39
     it=52: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 6.08495e+43
     it=53: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 2.21796e+48
     it=54: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 8.38622e+52
     it=55: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 3.28706e+57
     it=56: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 1.33476e+62
     it=57: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 5.61156e+66
     it=58: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 2.44114e+71
     it=59: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 1.0982e+76
     it=60: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 5.10636e+80 Lrg m.B2
     it=61: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 2.1182e+08
     it=62: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 1.05047e+13
     it=63: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 5.37602e+17
     logcf(*) end: after 63 iterations.
     user system elapsed
     2.250 0.233 7.224
     > (ct14 <- system.time(lR14 <- logcfR (xM, i=2, d=3, eps=1e-20, trace=TRUE))) # 4 sec
     logcf(x[ 1]= -5, i=2, d=3, eps=1e-20) logcf(*) end: after 26 iterations.
     logcf(x[ 2]= -4.75, i=2, d=3, eps=1e-20) logcf(*) end: after 25 iterations.
     logcf(x[ 3]= -4.5, i=2, d=3, eps=1e-20) logcf(*) end: after 24 iterations.
     logcf(x[ 4]= -4.25, i=2, d=3, eps=1e-20) logcf(*) end: after 24 iterations.
     logcf(x[ 5]= -4, i=2, d=3, eps=1e-20) logcf(*) end: after 23 iterations.
     logcf(x[ 6]= -3.75, i=2, d=3, eps=1e-20) logcf(*) end: after 22 iterations.
     logcf(x[ 7]= -3.5, i=2, d=3, eps=1e-20) logcf(*) end: after 22 iterations.
     logcf(x[ 8]= -3.25, i=2, d=3, eps=1e-20) logcf(*) end: after 21 iterations.
     logcf(x[ 9]= -3, i=2, d=3, eps=1e-20) logcf(*) end: after 20 iterations.
     logcf(x[10]= -2.75, i=2, d=3, eps=1e-20) logcf(*) end: after 19 iterations.
     logcf(x[11]= -2.5, i=2, d=3, eps=1e-20) logcf(*) end: after 19 iterations.
     logcf(x[12]= -2.25, i=2, d=3, eps=1e-20) logcf(*) end: after 18 iterations.
     logcf(x[13]= -2, i=2, d=3, eps=1e-20) logcf(*) end: after 17 iterations.
     logcf(x[14]= -1.75, i=2, d=3, eps=1e-20) logcf(*) end: after 16 iterations.
     logcf(x[15]= -1.5, i=2, d=3, eps=1e-20) logcf(*) end: after 15 iterations.
     logcf(x[16]= -1.25, i=2, d=3, eps=1e-20) logcf(*) end: after 14 iterations.
     logcf(x[17]= -1, i=2, d=3, eps=1e-20) logcf(*) end: after 12 iterations.
     logcf(x[18]= -0.75, i=2, d=3, eps=1e-20) logcf(*) end: after 11 iterations.
     logcf(x[19]= -0.5, i=2, d=3, eps=1e-20) logcf(*) end: after 9 iterations.
     logcf(x[20]= -0.25, i=2, d=3, eps=1e-20) logcf(*) end: after 7 iterations.
     logcf(x[21]= 0, i=2, d=3, eps=1e-20) logcf(*) end: after 0 iterations.
     logcf(x[22]= 0.25, i=2, d=3, eps=1e-20) logcf(*) end: after 8 iterations.
     logcf(x[23]= 0.5, i=2, d=3, eps=1e-20) logcf(*) end: after 13 iterations.
     logcf(x[24]= 0.75, i=2, d=3, eps=1e-20) logcf(*) end: after 20 iterations.
     logcf(x[25]= 0.78125, i=2, d=3, eps=1e-20) logcf(*) end: after 22 iterations.
     logcf(x[26]= 0.8125, i=2, d=3, eps=1e-20) logcf(*) end: after 24 iterations.
     logcf(x[27]= 0.84375, i=2, d=3, eps=1e-20) logcf(*) end: after 27 iterations.
     logcf(x[28]= 0.875, i=2, d=3, eps=1e-20) logcf(*) end: after 30 iterations.
     logcf(x[29]= 0.90625, i=2, d=3, eps=1e-20) logcf(*) end: after 36 iterations.
     logcf(x[30]= 0.9375, i=2, d=3, eps=1e-20) logcf(*) end: after 44 iterations.
     logcf(x[31]= 0.96875, i=2, d=3, eps=1e-20) logcf(*) end: after 63 iterations.
     user system elapsed
     16.153 0.020 45.982
     >
     > all.equal(lR.14, lR14, tol=0) # TRUE
     [1] TRUE
     > identical(lR.14, lR14) # TRUE !! (not sure if on all platforms!)
     [1] TRUE
     >
     > SS <- function(ch, digits=7)
     + sub(paste0("([0-9]{1,",digits,"})[0-9]*e"), "\\1e", ch)
     > ## double prec <--> MPFR: vvvv (same eps)
     > lR.9 <- logcfR.(xM, 2,3, eps=1e-9)
     > ## show:
     > SS(Rmpfr::all.equal(Rmpfr::roundMpfr(lR.9, 64), lR, tol=0))# .. 5.1138e-16
     Error in target == current : comparison of these types is not implemented
     Calls: SS ... <Anonymous> -> <Anonymous> -> .local -> all.equal.numeric
     Execution halted
    Running the tests in ‘tests/qgamma-ex.R’ failed.
    Complete output:
     > library(DPQ)
     >
     > ###---> Automatically find places where qgamma() is not so precise (PR#2214) :
     > ### For PR#2214, had '1e-8' below and found quite a bit
     > ## see /u/maechler/R/MM/NUMERICS/dpq-functions/beta-gamma-etc/qgamma-ex.R ..
     >
     > ## FIXME: Timing ! --- partly these matplot() partly get quite slow ~?
     > source(system.file(package="Matrix", "test-tools-1.R", mustWork=TRUE))
     Loading required package: tools
     > ##--> showProc.time(), assertError(), relErrV(), ...
     > showProc.time()
     Time (user system elapsed): 2.047 0.072 5.457
     >
     > (doExtras <- DPQ:::doExtras())
     [1] FALSE
     > (sdir <- system.file("safe", package="DPQ")) ## save directory (to read from)
     [1] "/data/gannet/ripley/R/packages/tests-clang/DPQ.Rcheck/DPQ/safe"
     >
     > ### Nowadays finds cases in a special region for really small p and cutoff 1e-11 :
     > set.seed(47)
     > n <- if(doExtras) 100 else 32
     > res <- cbind(p=1,df=1,rE=1)[-1,]
     > for(M in 1:(if(doExtras) 20 else 10))
     + for(p in runif(n)) for(df in rlnorm(n)) {
     + r <- 1- pchisq(qchisq(p, df),df)/p
     + if(abs(r) > 1e-11) res <- rbind(res, c(p,df,r))
     + }
     >
     > ### use df in U[0,1]: finds two cases with bound 1e-11
     > for(p in runif(n)/2) for(df in runif(n)) {
     + qq <- qchisq(p, df)
     + if(qq > 0 && p > 0) {
     + r <- 1- pchisq(qq, df) / p
     + if(abs(r) > 1e-11) res <- rbind(res, c(p,df,r))
     + }
     + }
     >
     > ### now df very close to 0 : ==> finds more cases
     > for(p in sort(c(runif(64)/2, exp(-(1+rlnorm(256))))))
     + for(df in 2^-rlnorm(256, mean=2, sdlog=1.5)) {
     + qq <- qchisq(p, df)
     + if(qq > 0 && p > 0) {
     + r <- 1- pchisq(qq, df) / p
     + if(abs(r) > 1e-11) res <- rbind(res, c(p,df,r))
     + }
     + }
     > showProc.time()
     Time (user system elapsed): 1.072 0.049 1.928
     >
     > require(graphics)
     > if(!dev.interactive(orNone=TRUE)) pdf("qgamma-appr.pdf")
     > eaxis <- sfsmisc::eaxis
     >
     > showProc.time()
     Time (user system elapsed): 0.086 0.001 0.088
     > ## if(nrow(res) > 0) {
     > cat("Found inaccurate examples where pchisq(qchisq(p, df),df) != p\n")
     Found inaccurate examples where pchisq(qchisq(p, df),df) != p
     > ## sort in p, then df:
     > res <- res[order(res[,"p"], res[,"df"]), ]
     > rE <- res[,"rE"]
     > if(nrow(res) > 20) { hist(rE, breaks = 30); rug(rE) }
     > plot(res[,1:2])##--> quite interesting : all along one curve
     > ## p <= 1/2 and df <= 1 (about) !!
     > res <- cbind(res, nDig = round(-log10(abs(rE)), 1))
     > print(res, digits=12)
     p df rE nDig
     [1,] 0.000194375438651 0.02334079639198 -4.05718514340e-08 7.4
     [2,] 0.000605300028912 0.02041606754775 -1.99857908001e-11 10.7
     [3,] 0.001012316063255 0.01855615147677 -2.59145555106e-04 3.6
     [4,] 0.001248285290785 0.01838201076117 -2.84196000067e-10 9.5
     [5,] 0.001682388899865 0.01720736646288 5.53088974600e-04 3.3
     [6,] 0.001746787400790 0.01731189518997 -5.86897839217e-08 7.2
     [7,] 0.002664451237518 0.01599317398629 1.48421342013e-04 3.8
     [8,] 0.002664451237518 0.01618024201222 -3.82806282229e-08 7.4
     [9,] 0.003159421860255 0.01557612780310 -7.92117005632e-06 5.1
     [10,] 0.003159421860255 0.01568183691729 -4.52237520765e-08 7.3
     [11,] 0.004055462418244 0.01493858731306 4.15166391654e-06 5.4
     [12,] 0.004400694140827 0.01459101672970 9.07907026434e-04 3.0
     [13,] 0.004458811277768 0.01457506850867 9.03139988533e-05 4.0
     [14,] 0.004481882165743 0.01468883074316 -3.23309491179e-07 6.5
     [15,] 0.004939609905705 0.01440168350452 -2.81810098879e-06 5.6
     [16,] 0.008824465120182 0.01276352706510 1.21107345756e-04 3.9
     [17,] 0.009040265960535 0.01273711629661 1.38964402733e-05 4.9
     [18,] 0.010839089634828 0.01242499920422 2.63413624246e-10 9.6
     [19,] 0.011642124851282 0.01201471267173 1.44956234150e-04 3.8
     [20,] 0.014753716559535 0.01155624353203 1.52962087441e-10 9.8
     [21,] 0.015499213434879 0.01125420134457 -9.69695930770e-05 4.0
     [22,] 0.015499213434879 0.01135920381800 -9.55739012376e-08 7.0
     [23,] 0.018603016576955 0.01071716109330 1.63971046474e-03 2.8
     [24,] 0.018603016576955 0.01073655493589 2.14388784340e-04 3.7
     [25,] 0.022624242394389 0.01033379525113 -3.37865757594e-09 8.5
     [26,] 0.022624242394389 0.01034206121729 -2.76332994265e-08 7.6
     [27,] 0.023730217356634 0.01016252135853 -1.07732682708e-06 6.0
     [28,] 0.032427027472295 0.00942923095016 5.11205522358e-11 10.3
     [29,] 0.044753525441333 0.00839626444749 1.22224173549e-05 4.9
     [30,] 0.081818424963746 0.00686007746204 8.92777740624e-10 9.0
     [31,] 0.081818424963746 0.00689856335721 2.28502772259e-11 10.6
     [32,] 0.082800309102258 0.00681234719059 4.17997558788e-09 8.4
     [33,] 0.083507718914457 0.00680676700443 9.77167236016e-11 10.0
     [34,] 0.090821658072474 0.00655269761981 -7.16033632386e-09 8.1
     [35,] 0.102294760453517 0.00623563107239 3.69438657444e-09 8.4
     [36,] 0.110869751789691 0.00603268830251 -3.44006823028e-10 9.5
     [37,] 0.123950804624116 0.00571305309327 2.84683721041e-10 9.5
     [38,] 0.127405857731893 0.00562369059572 6.60541454867e-09 8.2
     [39,] 0.135229634154169 0.00540073357520 -2.34762594200e-05 4.6
     [40,] 0.137732279982451 0.00533092076413 2.99285844990e-04 3.5
     [41,] 0.138112917548194 0.00535138710974 -2.05335777981e-06 5.7
     [42,] 0.141100635980184 0.00527305771429 4.31593832968e-05 4.4
     [43,] 0.141100635980184 0.00537073537183 -3.00640179418e-10 9.5
     [44,] 0.142905299416015 0.00523680041306 3.48180824883e-04 3.5
     [45,] 0.145624557210331 0.00526923971034 -1.94501770245e-09 8.7
     [46,] 0.154606872884529 0.00506806894407 -4.59924667240e-07 6.3
     [47,] 0.154606872884529 0.00507366168703 2.72301046933e-07 6.6
     [48,] 0.163535630067488 0.00497650928578 3.39664962823e-11 10.5
     [49,] 0.169741036539408 0.00484181845356 5.31400978776e-09 8.3
     [50,] 0.177327576288650 0.00465956102839 5.53404362603e-05 4.3
     [51,] 0.178169157856761 0.00471949961255 4.79807527043e-10 9.3
     [52,] 0.190094017358772 0.00450373552308 -1.29698447116e-06 5.9
     [53,] 0.190147641510530 0.00453468705710 5.66235636157e-09 8.2
     [54,] 0.200112534472267 0.00442273120514 7.20473680715e-11 10.1
     [55,] 0.201518808589718 0.00439936964342 1.58748569845e-11 10.8
     [56,] 0.201518808589718 0.00439976887947 -9.97182336704e-11 10.0
     [57,] 0.210803673024037 0.00427351441034 -1.70232938856e-10 9.8
     [58,] 0.213058614771766 0.00426179831847 1.10152997834e-11 11.0
     [59,] 0.214780951412088 0.00419869272965 9.79194836326e-09 8.0
     [60,] 0.232805106603566 0.00395399315002 -9.17581020055e-08 7.0
     [61,] 0.249102914025652 0.00380019404026 -1.15818465929e-10 9.9
     [62,] 0.249102914025652 0.00382493512126 -1.39670497390e-11 10.9
     [63,] 0.252076511947811 0.00374903834738 -8.83337205604e-08 7.1
     [64,] 0.253082914021191 0.00375259362798 3.65436092498e-09 8.4
     [65,] 0.253922058700076 0.00371237348323 3.28994798726e-06 5.5
     [66,] 0.254289278570932 0.00374343873151 -1.05664899053e-09 9.0
     [67,] 0.260017499519858 0.00366179605930 2.34859742765e-07 6.6
     [68,] 0.270323906831467 0.00351999192121 -1.56164756277e-04 3.8
     [69,] 0.271699356057456 0.00355068132680 5.13092990317e-09 8.3
     [70,] 0.275516196070002 0.00346804047756 -4.35171547588e-04 3.4
     [71,] 0.280722231049885 0.00348224101220 5.48759926389e-10 9.3
     [72,] 0.284601233201101 0.00344936339590 1.57145851887e-10 9.8
     [73,] 0.290188543054775 0.00336613521112 -5.64443074502e-08 7.2
     [74,] 0.290579022038283 0.00334423496113 1.02667567892e-07 7.0
     [75,] 0.290579022038283 0.00336764858994 2.26061565023e-08 7.6
     [76,] 0.291850198713803 0.00333552811650 -1.27338760580e-06 5.9
     [77,] 0.296521136452775 0.00330308865102 2.25309977453e-07 6.6
     [78,] 0.298034174946132 0.00330462333485 8.42470393447e-09 8.1
     [79,] 0.300556783277253 0.00323922530004 4.66003314391e-05 4.3
     [80,] 0.303182283998467 0.00328704590597 -1.46205270113e-11 10.8
     [81,] 0.322319846303892 0.00306134512927 -1.15130830540e-05 4.9
     [82,] 0.322319846303892 0.00310689001755 8.57751647487e-11 10.1
     [83,] 0.325071272052651 0.00302343293053 -2.47088704493e-04 3.6
     [84,] 0.325071272052651 0.00304146419577 3.18761056051e-06 5.5
     [85,] 0.331888412404218 0.00300837121343 -4.96098895297e-09 8.3
     [86,] 0.362278153188527 0.00278204202032 4.53939330569e-10 9.3
     [87,] 0.385389476781711 0.00260981704384 7.37274796769e-10 9.1
     [88,] 0.425333956955001 0.00232995789362 1.82823025607e-08 7.7
     [89,] 0.439503709203564 0.00222452690840 -4.53585193982e-06 5.3
     [90,] 0.439503709203564 0.00224964327069 -3.02331937263e-10 9.5
     [91,] 0.450804624124430 0.00216770324934 -4.59455036239e-08 7.3
     >
     > if(requireNamespace("scatterplot3d")) {
     + scatterplot3d::scatterplot3d(res[,1:3], type ='h') ## quite interesting:
     + ## the inaccurate (p,df) points are on nice monotone curve !!!
     + ## this is *less* revealing
     + scatterplot3d::scatterplot3d(res[,c("p","df","nDig")], type ='h')
     + }
     Loading required namespace: scatterplot3d
     > rL <- res[abs(res[,'rE']) > 1e-9,]
     > rL <- rL[order(rL[,1],rL[,2]),]
     > rL
     p df rE nDig
     [1,] 0.0001943754 0.023340796 -4.057185e-08 7.4
     [2,] 0.0010123161 0.018556151 -2.591456e-04 3.6
     [3,] 0.0016823889 0.017207366 5.530890e-04 3.3
     [4,] 0.0017467874 0.017311895 -5.868978e-08 7.2
     [5,] 0.0026644512 0.015993174 1.484213e-04 3.8
     [6,] 0.0026644512 0.016180242 -3.828063e-08 7.4
     [7,] 0.0031594219 0.015576128 -7.921170e-06 5.1
     [8,] 0.0031594219 0.015681837 -4.522375e-08 7.3
     [9,] 0.0040554624 0.014938587 4.151664e-06 5.4
     [10,] 0.0044006941 0.014591017 9.079070e-04 3.0
     [11,] 0.0044588113 0.014575069 9.031400e-05 4.0
     [12,] 0.0044818822 0.014688831 -3.233095e-07 6.5
     [13,] 0.0049396099 0.014401684 -2.818101e-06 5.6
     [14,] 0.0088244651 0.012763527 1.211073e-04 3.9
     [15,] 0.0090402660 0.012737116 1.389644e-05 4.9
     [16,] 0.0116421249 0.012014713 1.449562e-04 3.8
     [17,] 0.0154992134 0.011254201 -9.696959e-05 4.0
     [18,] 0.0154992134 0.011359204 -9.557390e-08 7.0
     [19,] 0.0186030166 0.010717161 1.639710e-03 2.8
     [20,] 0.0186030166 0.010736555 2.143888e-04 3.7
     [21,] 0.0226242424 0.010333795 -3.378658e-09 8.5
     [22,] 0.0226242424 0.010342061 -2.763330e-08 7.6
     [23,] 0.0237302174 0.010162521 -1.077327e-06 6.0
     [24,] 0.0447535254 0.008396264 1.222242e-05 4.9
     [25,] 0.0828003091 0.006812347 4.179976e-09 8.4
     [26,] 0.0908216581 0.006552698 -7.160336e-09 8.1
     [27,] 0.1022947605 0.006235631 3.694387e-09 8.4
     [28,] 0.1274058577 0.005623691 6.605415e-09 8.2
     [29,] 0.1352296342 0.005400734 -2.347626e-05 4.6
     [30,] 0.1377322800 0.005330921 2.992858e-04 3.5
     [31,] 0.1381129175 0.005351387 -2.053358e-06 5.7
     [32,] 0.1411006360 0.005273058 4.315938e-05 4.4
     [33,] 0.1429052994 0.005236800 3.481808e-04 3.5
     [34,] 0.1456245572 0.005269240 -1.945018e-09 8.7
     [35,] 0.1546068729 0.005068069 -4.599247e-07 6.3
     [36,] 0.1546068729 0.005073662 2.723010e-07 6.6
     [37,] 0.1697410365 0.004841818 5.314010e-09 8.3
     [38,] 0.1773275763 0.004659561 5.534044e-05 4.3
     [39,] 0.1900940174 0.004503736 -1.296984e-06 5.9
     [40,] 0.1901476415 0.004534687 5.662356e-09 8.2
     [41,] 0.2147809514 0.004198693 9.791948e-09 8.0
     [42,] 0.2328051066 0.003953993 -9.175810e-08 7.0
     [43,] 0.2520765119 0.003749038 -8.833372e-08 7.1
     [44,] 0.2530829140 0.003752594 3.654361e-09 8.4
     [45,] 0.2539220587 0.003712373 3.289948e-06 5.5
     [46,] 0.2542892786 0.003743439 -1.056649e-09 9.0
     [47,] 0.2600174995 0.003661796 2.348597e-07 6.6
     [48,] 0.2703239068 0.003519992 -1.561648e-04 3.8
     [49,] 0.2716993561 0.003550681 5.130930e-09 8.3
     [50,] 0.2755161961 0.003468040 -4.351715e-04 3.4
     [51,] 0.2901885431 0.003366135 -5.644431e-08 7.2
     [52,] 0.2905790220 0.003344235 1.026676e-07 7.0
     [53,] 0.2905790220 0.003367649 2.260616e-08 7.6
     [54,] 0.2918501987 0.003335528 -1.273388e-06 5.9
     [55,] 0.2965211365 0.003303089 2.253100e-07 6.6
     [56,] 0.2980341749 0.003304623 8.424704e-09 8.1
     [57,] 0.3005567833 0.003239225 4.660033e-05 4.3
     [58,] 0.3223198463 0.003061345 -1.151308e-05 4.9
     [59,] 0.3250712721 0.003023433 -2.470887e-04 3.6
     [60,] 0.3250712721 0.003041464 3.187611e-06 5.5
     [61,] 0.3318884124 0.003008371 -4.960989e-09 8.3
     [62,] 0.4253339570 0.002329958 1.828230e-08 7.7
     [63,] 0.4395037092 0.002224527 -4.535852e-06 5.3
     [64,] 0.4508046241 0.002167703 -4.594550e-08 7.3
     > plot(rL[,1:2], type = "b", main = "inaccurate pchisq/qchisq pairs")
     >
     > plot(rL[,1:2], type = "b", log = "x", ylim = range(0, rL[,"df"]),
     + xaxt = "n",
     + main = "inaccurate pchisq/qchisq pairs"); abline(h = 0, lty=2)
     > ## aha -- a perfect line !!
     > lines(res[,1:2], col = adjustcolor(1, 0.5))
     > eaxis(1); axis(1, at = 1/2)
     >
     > d <- as.data.frame(res)
     > plot (df ~ log(p), data = d, type = "b", cex=1/4, col="gray")
     > points(df ~ log(p), data = as.data.frame(rL), col=2, cex = 1/2)
     >
     > summary(fm <- lm (df ~ log(p), data = d, weights = -log(abs(rE))))
    
     Call:
     lm(formula = df ~ log(p), data = d, weights = -log(abs(rE)))
    
     Weighted Residuals:
     Min 1Q Median 3Q Max
     -6.924e-04 -1.443e-04 -2.096e-05 7.786e-05 1.079e-03
    
     Coefficients:
     Estimate Std. Error t value Pr(>|t|)
     (Intercept) 5.168e-06 1.149e-05 0.45 0.654
     log(p) -2.725e-03 3.683e-06 -739.99 <2e-16 ***
     ---
     Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
    
     Residual standard error: 0.0002575 on 89 degrees of freedom
     Multiple R-squared: 0.9998, Adjusted R-squared: 0.9998
     F-statistic: 5.476e+05 on 1 and 89 DF, p-value: < 2.2e-16
    
     > ## R^2 = 0.9998
     >
     > p0 <- 2^seq(-50,-1, by=1/8)
     > dN <- data.frame(p = p0,
     + df = predict(fm, newdata = data.frame(p = p0)))
     > rE <- with(dN, 1- pchisq(qchisq(p, df),df)/p)
     > dN <- cbind(dN, rE = rE, nDig = round(-log10(abs(rE)), 1))
     > print(dN, digits=10)
     p df rE nDig
     1 8.881784197e-16 0.094454797738 -6.103206185e-07 6.2
     2 9.685654347e-16 0.094218673664 -2.417772682e-07 6.6
     3 1.056228096e-15 0.093982549590 1.482101845e-07 6.8
     4 1.151824906e-15 0.093746425517 5.596452101e-07 6.3
     5 1.256073967e-15 0.093510301443 9.925312783e-07 6.0
     6 1.369758374e-15 0.093274177369 -5.456117484e-07 6.3
     7 1.493732098e-15 0.093038053295 -6.476825187e-08 7.2
     8 1.628926404e-15 0.092801929221 4.375367337e-07 6.4
     9 1.776356839e-15 0.092565805147 9.613066765e-07 6.0
     10 1.937130869e-15 0.092329681073 -4.656782389e-07 6.3
     11 2.112456192e-15 0.092093557000 1.060769096e-07 7.0
     12 2.303649813e-15 0.091857432926 6.993074607e-07 6.2
     13 2.512147934e-15 0.091621308852 -6.429918680e-07 6.2
     14 2.739516748e-15 0.091385184778 -1.755328105e-09 8.8
     15 2.987464197e-15 0.091149060704 6.609670441e-07 6.2
     16 3.257852808e-15 0.090912936630 -5.966163394e-07 6.2
     17 3.552713679e-15 0.090676812556 1.141328021e-07 6.9
     18 3.874261739e-15 0.090440688483 8.463782111e-07 6.1
     19 4.224912384e-15 0.090204564409 -3.264588970e-07 6.5
     20 4.607299625e-15 0.089968440335 4.538340435e-07 6.3
     21 5.024295868e-15 0.089732316261 -6.607811160e-07 6.2
     22 5.479033495e-15 0.089496192187 1.675731415e-07 6.8
     23 5.974928394e-15 0.089260068113 -8.888069507e-07 6.1
     24 6.515705616e-15 0.089023944039 -1.237758629e-08 7.9
     25 7.105427358e-15 0.088787819965 8.855723791e-07 6.1
     26 7.748523477e-15 0.088551695892 -8.599123835e-08 7.1
     27 8.449824769e-15 0.088315571818 8.600545690e-07 6.1
     28 9.214599250e-15 0.088079447744 -5.324092944e-08 7.3
     29 1.004859174e-14 0.087843323670 -9.348371277e-07 6.0
     30 1.095806699e-14 0.087607199596 8.590019329e-08 7.1
     31 1.194985679e-14 0.087371075522 -7.374085143e-07 6.1
     32 1.303141123e-14 0.087134951448 3.314589819e-07 6.5
     33 1.421085472e-14 0.086898827375 -4.335491499e-07 6.4
     34 1.549704695e-14 0.086662703301 6.834622716e-07 6.2
     35 1.689964954e-14 0.086426579227 -2.323222548e-08 7.6
     36 1.842919850e-14 0.086190455153 -6.982056668e-07 6.2
     37 2.009718347e-14 0.085954331079 4.935690683e-07 6.3
     38 2.191613398e-14 0.085718207005 -1.230714122e-07 6.9
     39 2.389971358e-14 0.085482082931 -7.079816475e-07 6.1
     40 2.606282246e-14 0.085245958858 5.585870368e-07 6.3
     41 2.842170943e-14 0.085009834784 3.202904830e-08 7.5
     42 3.099409391e-14 0.084773710710 -4.627895094e-07 6.3
     43 3.379929908e-14 0.084537586636 8.786037373e-07 6.1
     44 3.685839700e-14 0.084301462562 4.421566921e-07 6.4
     45 4.019436694e-14 0.084065338488 3.745824395e-08 7.4
     46 4.383226796e-14 0.083829214414 -3.354886757e-07 6.5
     47 4.779942715e-14 0.083593090340 -6.766811509e-07 6.2
     48 5.212564492e-14 0.083356966267 7.928489877e-07 6.1
     49 5.684341886e-14 0.083120842193 5.100597127e-07 6.3
     50 6.198818782e-14 0.082884718119 2.590340281e-07 6.6
     51 6.759859815e-14 0.082648594045 3.977485741e-08 7.4
     52 7.371679400e-14 0.082412469971 -1.477148934e-07 6.8
     53 8.038873388e-14 0.082176345897 -3.034323055e-07 6.5
     54 8.766453592e-14 0.081940221823 -4.273744663e-07 6.4
     55 9.559885430e-14 0.081704097750 -5.195384647e-07 6.3
     56 1.042512898e-13 0.081467973676 -5.799213925e-07 6.2
     57 1.136868377e-13 0.081231849602 -6.085203379e-07 6.2
     58 1.239763756e-13 0.080995725528 -6.053324058e-07 6.2
     59 1.351971963e-13 0.080759601454 -5.703546833e-07 6.2
     60 1.474335880e-13 0.080523477380 -5.035842794e-07 6.3
     61 1.607774678e-13 0.080287353306 -4.050182891e-07 6.4
     62 1.753290718e-13 0.080051229233 -2.746538159e-07 6.6
     63 1.911977086e-13 0.079815105159 -1.124879638e-07 6.9
     64 2.085025797e-13 0.079578981085 8.148216124e-08 7.1
     65 2.273736754e-13 0.079342857011 3.072594555e-07 6.5
     66 2.479527513e-13 0.079106732937 5.648467992e-07 6.2
     67 2.703943926e-13 0.078870608863 -8.276082903e-07 6.1
     68 2.948671760e-13 0.078634484789 -5.012849573e-07 6.3
     69 3.215549355e-13 0.078398360715 -1.431432968e-07 6.8
     70 3.506581437e-13 0.078162236642 2.468195778e-07 6.6
     71 3.823954172e-13 0.077926112568 6.686065366e-07 6.2
     72 4.170051594e-13 0.077689988494 -5.340348010e-07 6.3
     73 4.547473509e-13 0.077453864420 -4.348594862e-08 7.4
     74 4.959055026e-13 0.077217740346 4.788952396e-07 6.3
     75 5.407887852e-13 0.076981616272 -6.077622940e-07 6.2
     76 5.897343520e-13 0.076745492198 -1.660412186e-08 7.8
     77 6.431098711e-13 0.076509368125 6.063946030e-07 6.2
     78 7.013162874e-13 0.076273244051 -3.642605502e-07 6.4
     79 7.647908344e-13 0.076037119977 3.275302165e-07 6.5
     80 8.340103188e-13 0.075800995903 -5.640570402e-07 6.2
     81 9.094947018e-13 0.075564871829 1.965354515e-07 6.7
     82 9.918110051e-13 0.075328747755 -6.159766142e-07 6.2
     83 1.081577570e-12 0.075092623681 2.134272701e-07 6.7
     84 1.179468704e-12 0.074856499608 -5.200023394e-07 6.3
     85 1.286219742e-12 0.074620375534 3.782225700e-07 6.4
     86 1.402632575e-12 0.074384251460 -2.761173419e-07 6.6
     87 1.529581669e-12 0.074148127386 6.909382196e-07 6.2
     88 1.668020638e-12 0.073912003312 1.156952232e-07 6.9
     89 1.818989404e-12 0.073675879238 -4.174157540e-07 6.4
     90 1.983622010e-12 0.073439755164 6.554521572e-07 6.2
     91 2.163155141e-12 0.073203631090 2.014481738e-07 6.7
     92 2.358937408e-12 0.072967507017 -2.104198829e-07 6.7
     93 2.572439484e-12 0.072731382943 -5.801509786e-07 6.2
     94 2.805265149e-12 0.072495258869 6.355293285e-07 6.2
     95 3.059163338e-12 0.072259134795 3.449181070e-07 6.5
     96 3.336041275e-12 0.072023010721 9.644772814e-08 7.0
     97 3.637978807e-12 0.071786886647 -1.098808029e-07 7.0
     98 3.967244020e-12 0.071550762573 -2.740664691e-07 6.6
     99 4.326310282e-12 0.071314638500 -3.961082631e-07 6.4
     100 4.717874816e-12 0.071078514426 -4.760051799e-07 6.3
     101 5.144878969e-12 0.070842390352 -5.137562191e-07 6.3
     102 5.610530299e-12 0.070606266278 -5.093603785e-07 6.3
     103 6.118326675e-12 0.070370142204 -4.628166750e-07 6.3
     104 6.672082550e-12 0.070134018130 -3.741241081e-07 6.4
     105 7.275957614e-12 0.069897894056 -2.432817023e-07 6.6
     106 7.934488041e-12 0.069661769983 -7.028846949e-08 7.2
     107 8.652620563e-12 0.069425645909 1.448565687e-07 6.8
     108 9.435749632e-12 0.069189521835 4.021543881e-07 6.4
     109 1.028975794e-11 0.068953397761 7.016059603e-07 6.2
     110 1.122106060e-11 0.068717273687 -4.176447372e-07 6.4
     111 1.223665335e-11 0.068481149613 -2.873865945e-08 7.5
     112 1.334416510e-11 0.068245025539 4.023232466e-07 6.4
     113 1.455191523e-11 0.068008901466 -5.698258618e-07 6.2
     114 1.586897608e-11 0.067772777392 -4.930799502e-08 7.3
     115 1.730524113e-11 0.067536653318 5.133677458e-07 6.3
     116 1.887149926e-11 0.067300529244 -3.116849310e-07 6.5
     117 2.057951587e-11 0.067064405170 3.404481462e-07 6.5
     118 2.244212120e-11 0.066828281096 -3.848111123e-07 6.4
     119 2.447330670e-11 0.066592157022 3.567795747e-07 6.4
     120 2.668833020e-11 0.066356032948 -2.686901401e-07 6.6
     121 2.910383046e-11 0.066119908875 5.623583754e-07 6.2
     122 3.173795216e-11 0.065883784801 3.667429294e-08 7.4
     123 3.461048225e-11 0.065647660727 -4.365232460e-07 6.4
     124 3.774299853e-11 0.065411536653 5.312784427e-07 6.3
     125 4.115903175e-11 0.065175412579 1.578594558e-07 6.8
     126 4.488424239e-11 0.064939288505 -1.630783120e-07 6.8
     127 4.894661340e-11 0.064703164431 -4.315370583e-07 6.4
     128 5.337666040e-11 0.064467040358 -6.475189818e-07 6.2
     129 5.820766091e-11 0.064230916284 5.516171563e-07 6.3
     130 6.347590433e-11 0.063994792210 4.354002830e-07 6.4
     131 6.922096451e-11 0.063758668136 3.716548337e-07 6.4
     132 7.548599706e-11 0.063522544062 3.603785849e-07 6.4
     133 8.231806350e-11 0.063286419988 4.015693077e-07 6.4
     134 8.976848478e-11 0.063050295914 4.952247737e-07 6.3
     135 9.789322680e-11 0.062814171841 6.413427337e-07 6.2
     136 1.067533208e-10 0.062578047767 -4.864750427e-07 6.3
     137 1.164153218e-10 0.062341923693 -2.302656055e-07 6.6
     138 1.269518087e-10 0.062105799619 7.839833394e-08 7.1
     139 1.384419290e-10 0.061869675545 4.395145157e-07 6.4
     140 1.509719941e-10 0.061633551471 -4.525763320e-07 6.3
     141 1.646361270e-10 0.061397427397 1.860554089e-08 7.7
     142 1.795369696e-10 0.061161303323 5.422315954e-07 6.3
     143 1.957864536e-10 0.060925179250 -1.718042137e-07 6.8
     144 2.135066416e-10 0.060689055176 4.618673999e-07 6.3
     145 2.328306437e-10 0.060452931102 -1.317493019e-07 6.9
     146 2.539036173e-10 0.060216807028 6.119534922e-07 6.2
     147 2.768838580e-10 0.059980682954 1.387356391e-07 6.9
     148 3.019439882e-10 0.059744558880 -2.716851728e-07 6.6
     149 3.292722540e-10 0.059508434806 -6.193156503e-07 6.2
     150 3.590739391e-10 0.059272310733 3.495618646e-07 6.5
     151 3.915729072e-10 0.059036186659 1.222864269e-07 6.9
     152 4.270132832e-10 0.058800062585 -4.221716110e-08 7.4
     153 4.656612873e-10 0.058563938511 -1.439556299e-07 6.8
     154 5.078072346e-10 0.058327814437 -1.829357232e-07 6.7
     155 5.537677160e-10 0.058091690363 -1.591641836e-07 6.8
     156 6.038879765e-10 0.057855566289 -7.264775914e-08 7.1
     157 6.585445080e-10 0.057619442216 7.660678825e-08 7.1
     158 7.181478783e-10 0.057383318142 2.885926981e-07 6.5
     159 7.831458144e-10 0.057147194068 5.633032045e-07 6.2
     160 8.540265665e-10 0.056911069994 -3.010137886e-07 6.5
     161 9.313225746e-10 0.056674945920 1.043142313e-07 7.0
     162 1.015614469e-09 0.056438821846 5.723448235e-07 6.2
     163 1.107535432e-09 0.056202697772 -8.306453525e-08 7.1
     164 1.207775953e-09 0.055966573698 5.155342088e-07 6.3
     165 1.317089016e-09 0.055730449625 1.091064128e-09 9.0
     166 1.436295757e-09 0.055494325551 -4.402707552e-07 6.4
     167 1.566291629e-09 0.055258201477 3.567050869e-07 6.4
     168 1.708053133e-09 0.055022077403 5.624478649e-08 7.2
     169 1.862645149e-09 0.054785953329 -1.711696178e-07 6.8
     170 2.031228938e-09 0.054549829255 -3.255506589e-07 6.5
     171 2.215070864e-09 0.054313705181 -4.069108679e-07 6.4
     172 2.415551906e-09 0.054077581108 -4.152627815e-07 6.4
     173 2.634178032e-09 0.053841457034 -3.506189497e-07 6.5
     174 2.872591513e-09 0.053605332960 -2.129919214e-07 6.7
     175 3.132583258e-09 0.053369208886 -2.394249909e-09 8.6
     176 3.416106266e-09 0.053133084812 2.811614974e-07 6.6
     177 3.725290298e-09 0.052896960738 -4.754652609e-07 6.3
     178 4.062457877e-09 0.052660836664 -4.082860428e-08 7.4
     179 4.430141728e-09 0.052424712591 4.667262948e-07 6.3
     180 4.831103812e-09 0.052188588517 -5.029516048e-08 7.3
     181 5.268356064e-09 0.051952464443 -4.839870040e-07 6.3
     182 5.745183026e-09 0.051716340369 2.526344739e-07 6.6
     183 6.265166516e-09 0.051480216295 -1.969243901e-08 7.7
     184 6.832212532e-09 0.051244092221 -2.087459541e-07 6.7
     185 7.450580597e-09 0.051007968147 -3.145456657e-07 6.5
     186 8.124915754e-09 0.050771844073 -3.371111803e-07 6.5
     187 8.860283457e-09 0.050535720000 -2.764621003e-07 6.6
     188 9.662207623e-09 0.050299595926 -1.326180317e-07 6.9
     189 1.053671213e-08 0.050063471852 9.440140747e-08 7.0
     190 1.149036605e-08 0.049827347778 4.045766026e-07 6.4
     191 1.253033303e-08 0.049591223704 -2.420871066e-07 6.6
     192 1.366442506e-08 0.049355099630 2.395533774e-07 6.6
     193 1.490116119e-08 0.049118975556 -2.252220010e-07 6.6
     194 1.624983151e-08 0.048882851483 4.277948780e-07 6.4
     195 1.772056691e-08 0.048646727409 1.448078844e-07 6.8
     196 1.932441525e-08 0.048410603335 -4.470068493e-08 7.3
     197 2.107342426e-08 0.048174479261 -1.407587547e-07 6.9
     198 2.298073210e-08 0.047938355187 -1.433942325e-07 6.8
     199 2.506066606e-08 0.047702231113 -5.263503899e-08 7.3
     200 2.732885013e-08 0.047466107039 1.314909133e-07 6.9
     201 2.980232239e-08 0.047229982966 4.089557111e-07 6.4
     202 3.249966302e-08 0.046993858892 -2.026429911e-07 6.7
     203 3.544113383e-08 0.046757734818 2.666381839e-07 6.6
     204 3.864883049e-08 0.046521610744 -1.427213570e-07 6.8
     205 4.214684851e-08 0.046285486670 -4.483846767e-07 6.3
     206 4.596146421e-08 0.046049362596 3.109955485e-07 6.5
     207 5.012133212e-08 0.045813238522 2.073633435e-07 6.7
     208 5.465770025e-08 0.045577114448 2.073183537e-07 6.7
     209 5.960464478e-08 0.045340990375 3.108231207e-07 6.5
     210 6.499932603e-08 0.045104866301 -4.225693764e-07 6.4
     211 7.088226765e-08 0.044868742227 -1.068421438e-07 7.0
     212 7.729766099e-08 0.044632618153 3.123190967e-07 6.5
     213 8.429369702e-08 0.044396494079 -8.979926758e-08 7.0
     214 9.192292842e-08 0.044160370005 -3.780712474e-07 6.4
     215 1.002426642e-07 0.043924245931 3.616102325e-07 6.4
     216 1.093154005e-07 0.043688121858 2.956315699e-07 6.5
     217 1.192092896e-07 0.043451997784 3.433583751e-07 6.5
     218 1.299986521e-07 0.043215873710 -3.936604087e-07 6.4
     219 1.417645353e-07 0.042979749636 -1.134241445e-07 6.9
     220 1.545953220e-07 0.042743625562 2.803691416e-07 6.6
     221 1.685873940e-07 0.042507501488 -9.497759912e-08 7.0
     222 1.838458568e-07 0.042271377414 -3.463650020e-07 6.5
     223 2.004853285e-07 0.042035253341 3.982654885e-07 6.4
     224 2.186308010e-07 0.041799129267 3.893573700e-07 6.4
     225 2.384185791e-07 0.041563005193 -3.573736593e-07 6.4
     226 2.599973041e-07 0.041326881119 -1.135260472e-07 6.9
     227 2.835290706e-07 0.041090757045 2.539798655e-07 6.6
     228 3.091906439e-07 0.040854632971 -1.007498764e-07 7.0
     229 3.371747881e-07 0.040618508897 -3.214330988e-07 6.5
     230 3.676917137e-07 0.040382384823 -4.081432583e-07 6.4
     231 4.009706570e-07 0.040146260750 -3.609537691e-07 6.4
     232 4.372616020e-07 0.039910136676 -1.799379994e-07 6.7
     233 4.768371582e-07 0.039674012602 1.348307247e-07 6.9
     234 5.199946082e-07 0.039437888528 -2.309643412e-07 6.6
     235 5.670581412e-07 0.039201764454 3.563334678e-07 6.4
     236 6.183812879e-07 0.038965640380 2.734302492e-07 6.6
     237 6.743495762e-07 0.038729516306 3.344816584e-07 6.5
     238 7.353834273e-07 0.038493392233 -2.537711326e-07 6.6
     239 8.019413140e-07 0.038257268159 1.001817561e-07 7.0
     240 8.745232040e-07 0.038021144085 -1.848126916e-07 6.7
     241 9.536743164e-07 0.037785020011 -3.156868371e-07 6.5
     242 1.039989216e-06 0.037548895937 -2.925440488e-07 6.5
     243 1.134116282e-06 0.037312771863 -1.154876221e-07 6.9
     244 1.236762576e-06 0.037076647789 2.153792392e-07 6.7
     245 1.348699152e-06 0.036840523716 -5.632338751e-08 7.2
     246 1.470766855e-06 0.036604399642 -1.638943248e-07 6.8
     247 1.603882628e-06 0.036368275568 -1.074536702e-07 7.0
     248 1.749046408e-06 0.036132151494 1.128785959e-07 6.9
     249 1.907348633e-06 0.035896027420 -2.381848818e-07 6.6
     250 2.079978433e-06 0.035659903346 3.148260044e-07 6.5
     251 2.268232565e-06 0.035423779272 3.067288504e-07 6.5
     252 2.473525152e-06 0.035187655198 -2.467805615e-07 6.6
     253 2.697398305e-06 0.034951531125 9.809279378e-08 7.0
     254 2.941533709e-06 0.034715407051 -9.217531383e-08 7.0
     255 3.207765256e-06 0.034479282977 -9.840820714e-08 7.0
     256 3.498092816e-06 0.034243158903 7.923717715e-08 7.1
     257 3.814697266e-06 0.034007034829 -2.523280838e-07 6.6
     258 4.159956866e-06 0.033770910755 2.978638209e-07 6.5
     259 4.536465130e-06 0.033534786681 -3.332980778e-07 6.5
     260 4.947050303e-06 0.033298662608 -8.305711496e-08 7.1
     261 5.394796609e-06 0.033062538534 -3.110326594e-07 6.5
     262 5.883067419e-06 0.032826414460 3.314533775e-07 6.5
     263 6.415530512e-06 0.032590290386 -1.553354063e-07 6.8
     264 6.996185632e-06 0.032354166312 2.279407690e-07 6.6
     265 7.629394531e-06 0.032118042238 1.638951825e-07 6.8
     266 8.319913732e-06 0.031881918164 3.135417774e-07 6.5
     267 9.072930260e-06 0.031645794091 3.656237890e-08 7.4
     268 9.894100606e-06 0.031409670017 -1.661173443e-08 7.8
     269 1.078959322e-05 0.031173545943 1.537749293e-07 6.8
     270 1.176613484e-05 0.030937421869 -7.675076596e-08 7.1
     271 1.283106102e-05 0.030701297795 -7.365044086e-08 7.1
     272 1.399237126e-05 0.030465173721 1.628068312e-07 6.8
     273 1.525878906e-05 0.030229049647 2.398761112e-08 7.6
     274 1.663982746e-05 0.029992925573 1.285369383e-07 6.9
     275 1.814586052e-05 0.029756801500 -1.216179157e-07 6.9
     276 1.978820121e-05 0.029520677426 -1.184344594e-07 6.9
     277 2.157918644e-05 0.029284553352 1.377655910e-07 6.9
     278 2.353226967e-05 0.029048429278 6.475006253e-08 7.2
     279 2.566212205e-05 0.028812305204 2.546562226e-07 6.6
     280 2.798474253e-05 0.028576181130 1.358057637e-07 6.9
     281 3.051757812e-05 0.028340057056 -2.762893581e-07 6.6
     282 3.327965493e-05 0.028103932983 1.553088241e-07 6.8
     283 3.629172104e-05 0.027867808909 -2.519743056e-07 6.6
     284 3.957640242e-05 0.027631684835 1.836182478e-07 6.7
     285 4.315837288e-05 0.027395560761 -1.887623498e-07 6.7
     286 4.706453935e-05 0.027159436687 -2.586992773e-07 6.6
     287 5.132424410e-05 0.026923312613 -2.666510612e-08 7.6
     288 5.596948506e-05 0.026687188539 -2.210357075e-08 7.7
     289 6.103515625e-05 0.026451064466 -2.296375117e-07 6.6
     290 6.655930986e-05 0.026214940392 -1.155410669e-07 6.9
     291 7.258344208e-05 0.025978816318 -1.934344240e-07 6.7
     292 7.915280485e-05 0.025742692244 5.977598905e-08 7.2
     293 8.631674575e-05 0.025506568170 1.410127927e-07 6.9
     294 9.412907870e-05 0.025270444096 6.554035215e-08 7.2
     295 1.026484882e-04 0.025034320022 -1.513968153e-07 6.8
     296 1.119389701e-04 0.024798195948 -7.984736650e-09 8.1
     297 1.220703125e-04 0.024562071875 1.377293413e-08 7.9
     298 1.331186197e-04 0.024325947801 -7.096807053e-08 7.1
     299 1.451668842e-04 0.024089823727 2.236347612e-07 6.7
     300 1.583056097e-04 0.023853699653 -3.406038118e-08 7.5
     301 1.726334915e-04 0.023617575579 1.071465702e-07 7.0
     302 1.882581574e-04 0.023381451505 1.915567694e-07 6.7
     303 2.052969764e-04 0.023145327431 -2.153880145e-07 6.7
     304 2.238779402e-04 0.022909203358 -1.943811789e-07 6.7
     305 2.441406250e-04 0.022673079284 -1.853585263e-07 6.7
     306 2.662372394e-04 0.022436955210 -1.734718758e-07 6.8
     307 2.903337683e-04 0.022200831136 -1.439218900e-07 6.8
     308 3.166112194e-04 0.021964707062 -8.196069534e-08 7.1
     309 3.452669830e-04 0.021728582988 2.710548919e-08 7.6
     310 3.765163148e-04 0.021492458914 1.979137831e-07 6.7
     311 4.105939528e-04 0.021256334841 3.784053582e-08 7.4
     312 4.477558805e-04 0.021020210767 -2.082512274e-08 7.7
     313 4.882812500e-04 0.020784086693 3.635452028e-08 7.4
     314 5.324744788e-04 0.020547962619 -1.675743422e-07 6.8
     315 5.806675366e-04 0.020311838545 1.696075650e-07 6.8
     316 6.332224388e-04 0.020075714471 -9.599203343e-08 7.0
     317 6.905339660e-04 0.019839590397 -1.676104155e-07 6.8
     318 7.530326296e-04 0.019603466323 -3.122730075e-08 7.5
     319 8.211879055e-04 0.019367342250 -3.779823499e-08 7.4
     320 8.955117609e-04 0.019131218176 -1.576477535e-07 6.8
     321 9.765625000e-04 0.018895094102 -6.902106886e-09 8.2
     322 1.064948958e-03 0.018658970028 7.899459131e-08 7.1
     323 1.161335073e-03 0.018422845954 1.293463390e-07 6.9
     324 1.266444878e-03 0.018186721880 -1.651599366e-07 6.8
     325 1.381067932e-03 0.017950597806 -9.328012252e-08 7.0
     326 1.506065259e-03 0.017714473733 3.008480842e-08 7.5
     327 1.642375811e-03 0.017478349659 -8.903183435e-08 7.1
     328 1.791023522e-03 0.017242225585 -8.893825099e-08 7.1
     329 1.953125000e-03 0.017006101511 5.866337671e-08 7.2
     330 2.129897915e-03 0.016769977437 7.502565913e-08 7.1
     331 2.322670146e-03 0.016533853363 3.812254845e-09 8.4
     332 2.532889755e-03 0.016297729289 -1.115639656e-07 7.0
     333 2.762135864e-03 0.016061605216 6.307704159e-08 7.2
     334 3.012130518e-03 0.015825481142 -1.675984129e-08 7.8
     335 3.284751622e-03 0.015589357068 -1.223428781e-08 7.9
     336 3.582047044e-03 0.015353232994 1.188536293e-07 6.9
     337 3.906250000e-03 0.015117108920 -1.217006735e-07 6.9
     338 4.259795831e-03 0.014880984846 -1.314145703e-07 6.9
     339 4.645340293e-03 0.014644860772 -1.288061258e-07 6.9
     340 5.065779510e-03 0.014408736698 -5.759581589e-08 7.2
     341 5.524271728e-03 0.014172612625 -1.110605841e-07 7.0
     342 6.024261037e-03 0.013936488551 2.552848555e-08 7.6
     343 6.569503244e-03 0.013700364477 -7.038977068e-08 7.2
     344 7.164094088e-03 0.013464240403 -8.014783059e-08 7.1
     345 7.812500000e-03 0.013228116329 6.514376105e-08 7.2
     346 8.519591661e-03 0.012991992255 -1.226700341e-08 7.9
     347 9.290680586e-03 0.012755868181 3.829306428e-09 8.4
     348 1.013155902e-02 0.012519744108 -1.720011800e-08 7.8
     349 1.104854346e-02 0.012283620034 2.105846508e-08 7.7
     350 1.204852207e-02 0.012047495960 1.170782316e-08 7.9
     351 1.313900649e-02 0.011811371886 6.422403376e-08 7.2
     352 1.432818818e-02 0.011575247812 9.486125396e-08 7.0
     353 1.562500000e-02 0.011339123738 3.894984368e-08 7.4
     354 1.703918332e-02 0.011102999664 3.208461985e-08 7.5
     355 1.858136117e-02 0.010866875591 3.155170891e-08 7.5
     356 2.026311804e-02 0.010630751517 1.297326291e-08 7.9
     357 2.209708691e-02 0.010394627443 -3.001926907e-08 7.5
     358 2.409704415e-02 0.010158503369 7.470777241e-08 7.1
     359 2.627801298e-02 0.009922379295 2.881776751e-08 7.5
     360 2.865637635e-02 0.009686255221 4.355808669e-08 7.4
     361 3.125000000e-02 0.009450131147 3.486256517e-08 7.5
     362 3.407836665e-02 0.009214007073 -4.540193843e-08 7.3
     363 3.716272234e-02 0.008977883000 6.057274970e-08 7.2
     364 4.052623608e-02 0.008741758926 -4.748039517e-08 7.3
     365 4.419417382e-02 0.008505634852 -4.947995325e-08 7.3
     366 4.819408829e-02 0.008269510778 5.303493644e-09 8.3
     367 5.255602595e-02 0.008033386704 2.750297434e-09 8.6
     368 5.731275270e-02 0.007797262630 7.649789580e-09 8.1
     369 6.250000000e-02 0.007561138556 2.575244529e-08 7.6
     370 6.815673329e-02 0.007325014483 2.588700820e-08 7.6
     371 7.432544469e-02 0.007088890409 -3.873515997e-08 7.4
     372 8.105247217e-02 0.006852766335 -2.868914595e-08 7.5
     373 8.838834765e-02 0.006616642261 8.820984831e-09 8.1
     374 9.638817659e-02 0.006380518187 -1.249980452e-08 7.9
     375 1.051120519e-01 0.006144394113 -2.542298283e-08 7.6
     376 1.146255054e-01 0.005908270039 -3.411814475e-08 7.5
     377 1.250000000e-01 0.005672145966 1.097271574e-08 8.0
     378 1.363134666e-01 0.005436021892 -5.883856513e-09 8.2
     379 1.486508894e-01 0.005199897818 -1.778869496e-08 7.7
     380 1.621049443e-01 0.004963773744 -2.477514438e-08 7.6
     381 1.767766953e-01 0.004727649670 -2.316977810e-08 7.6
     382 1.927763532e-01 0.004491525596 -1.391213433e-08 7.9
     383 2.102241038e-01 0.004255401522 4.032074008e-09 8.4
     384 2.292510108e-01 0.004019277448 1.322844057e-10 9.9
     385 2.500000000e-01 0.003783153375 1.667496141e-09 8.8
     386 2.726269332e-01 0.003547029301 -6.843543954e-09 8.2
     387 2.973017788e-01 0.003310905227 -5.830278704e-09 8.2
     388 3.242098887e-01 0.003074781153 -4.568749379e-09 8.3
     389 3.535533906e-01 0.002838657079 -6.372713468e-09 8.2
     390 3.855527064e-01 0.002602533005 4.293532641e-09 8.4
     391 4.204482076e-01 0.002366408931 4.466793713e-09 8.4
     392 4.585020216e-01 0.002130284858 -3.886739153e-09 8.4
     393 5.000000000e-01 0.001894160784 1.495896629e-09 8.8
     >
     > ## } ## only when we find inaccurate regions
     > showProc.time()
     Time (user system elapsed): 0.137 0.02 0.344
     >
     >
     > ## Oops: another qgamma() / qchisq() problem: mostly NaN's == all solved now
     > curve(qgamma(x, 20), 1e-16, 1e-10, log='x')
     > curve(qgamma(x, 20), 1e-300, .99 , log='xy') # and add the critical region from above:
     > abline(v=c(1e-16, 1e-10), col="light blue")
     > curve(qgamma(x, 20), 1e-26, 1e-07, log='x')
     > ##-> now using log=TRUE in same region:
     > curve(qgamma(x, 20, log=TRUE), -38, -16)## no problem!!
     > curve(qgamma(exp(x), 20), add=TRUE, col="green3", n=2001)
     > ## had problem here, but no longer !
     >
     > ##--> Further fix for qgamma: when 'x' is very small: use "log=TRUE of log(x)"!
     >
     > ## had bug (gave NaN), but no longer:
     > (q_12 <- qgamma(1e-12, 20))
     [1] 2.330042
     > all.equal(1e-12, pgamma(q_12, 20), tol=0)# show rel.err (Lnx 64-bit: 4.04e-16)
     [1] "Mean relative difference: 4.038968e-16"
     > stopifnot(
     + all.equal(1e-12, pgamma(q_12, 20), tolerance = 1e-14)
     + )
     >
     >
     > ## --- Nice graphic : --- but amazingly *S..L..O..W*
     >
     > p.qgammaSml <- function(from= 1e-110, to = 1e-5, ylim = c(0.4, 1000),
     + n = 201, k.lab = 3,
     + a1 = c(10, seq(10.1,20, by=.2), 21:105),
     + a2 = seq(110,330, by=10),
     + a3 = seq(350,1600, by=50))
     + {
     + ## Purpose: nice qgamma() lines ``for small x'' aka p
     + ## ----------------------------------------------------------------------
     + ## Author: Martin Maechler, Date: 22 Mar 2004, 14:23
     + x <- exp(seq(log(from), log(to), length = n))
     +
     + op <- par(las=1, lab = c(10,10, 7), xaxs = "i", mex = 0.8)
     + on.exit(par(op))
     + plot(x, qgamma(x, a1[1]), log="xy", ylim=ylim, type='l', xaxt = "n",
     + main = paste("qgamma(x, a) for very small x, a in [",
     + formatC(a1[1]),", ",formatC(max(a1,a2,a3)),"] - log-log", sep=''),
     + sub = R.version.string)
     + lab.x <- pretty(log10(c(from,to)), 20)
     + axis(1, at=10^lab.x, lab = paste("10^",formatC(lab.x),sep=''))
     + if(is.nan(qgamma(1e-12, 20)))
     + text(1e-60, 20, "all NaN", cex = 2)
     + if(!is.finite(qgamma(1e-140, 155)))
     + text(1e-240, 5, "all +Inf", cex = 2)
     +
     + lines.txt <- function(a.s, col = par("col")) {
     + col <- rep(col, length=length(a.s))
     + for(i in seq(along=a.s)) {
     + qx <- qgamma(x, (a <- a.s[i]))
     + if(i %% k.lab == 0 &&
     + any(ifi <- is.finite(qx) & qx >= ylim[1])) {
     + ik <- (i%%(2*k.lab))/k.lab # = 0 or 1
     + j <- quantile(which(ifi), c(.02,(1:3)/4+ ik/10, .98))
     + ## "segments" around the labels :
     + i0 <- 1
     + for(jj in j) {
     + ii <- i0:(jj-1)
     + i2 <- jj + -1:1
     + lines(x[ii], qx[ii], col=col[i])
     + lines(x[i2], qx[i2], col=col[i], type = 'c')
     + i0 <- jj+1
     + }
     + text(x[j], qx[j], formatC(a), col= "gray40", cex = 0.8)
     + }
     + else
     + lines(x, qx, col=col[i])
     +
     + }
     + }
     + oo <- options(warn = -1)
     + lines.txt(a1[-1])
     + lines.txt(a2, col= 2)
     + lines.txt(a3, col= rainbow(length(a3), .8, .8,
     + start = (max(a3)-min(a3))/(1+max(a3))))
     + invisible(options(oo))
     + }
     >
     > showProc.time()
     Time (user system elapsed): 0.035 0.002 0.084
     >
     > p.qgammaSml()
     > p.qgammaSml(1e-300)
     > p.qgammaSml(1e-300,1e-50, a2= seq(100,360, by=4), a3=seq(350,1500, by=10))
     >
     > showProc.time()
     Time (user system elapsed): 1.918 0.017 2.783
     >
     > ## The "upper" problematic corner:
     > p.qgammaSml(1e-19, 1e-3, a2=NULL,a3=NULL, ylim=c(.1,20))
     > p.qgammaSml(1e-19, 1e-3, a2=seq(1,12, by=.04), ylim=c(.1,20),a3=NULL,k.lab=10)
     > ## now shows the problem (quite well):
     > ## could it be in pgamma()'s inaccuracy, leading to qgamma() bias ?
     > aa <- c(seq(9, 22, by=0.25),seq(22.3,40,by=0.4))
     > caa <- formatC(range(aa))
     > sfsmisc::mult.fig(2)
     > curve(pgamma(x, sh=aa[1]), 0.5, 20, log = 'xy', ylim = c(1e-60, .2),
     + main = sprintf("pgamma(x, a) for a in [%s,%s]", caa[1],caa[2]))
     > for(sh in aa) curve(pgamma(x, sh), add = TRUE, col=2)
     > abline(h=c(1e-15), col="light blue", lty=2)
     >
     > curve(pgamma(x, sh=aa[1]), 0.5, 20, log = 'xy', ylim = c(1e-15, .8),
     + main = sprintf("pgamma(x, a) for a in [%s,%s]", caa[1],caa[2]))
     > for(sh in aa) curve(pgamma(x, sh), add = TRUE, col=2)
     > ## the "border curve" between "Pearson" and "Continued fraction (upper tail)"
     > ## in pgamma.c :
     > curve(pgamma(max(1,x), x), add = TRUE, col=4)
     > ## ==> pgamma() is perfect here {series expansion up to eps_C accuracy}!
     >
     > aa <- c(seq(9, 22, by=0.25),seq(22.3,40.4,by=0.4))
     > p.qgammaSml(1e-24, 1e-5, a1=aa, a2=NULL,a3=NULL, ylim=c(.8,8))
     > ## -------- save the above?
     > aa1 <- c(aa,seq(40.5,90, by=0.5))
     > p.qgammaSml(1e-60, 1e-5, a1=aa1, a2=NULL,a3=NULL, ylim=c(.9, 16))
     > aa2 <- c(aa1, seq(91,150, by= 1))
     > p.qgammaSml(1e-90, 1e-5, a1=aa2, a2=NULL,a3=NULL, ylim=c(.9, 35))
     > aa3 <- c(aa2, seq(150,250, by= 2), seq(253, 400, by=5))
     > p.qgammaSml(1e-200, 1e-5, a1=aa3, a2=NULL,a3=NULL, ylim=c(.9, 100))
     > p.qgammaSml(1e-200, 1e-5, a1=aa3, a2=NULL,a3=NULL, ylim=c(.9, 200),k.lab=9e9)
     > p.qgammaSml(1e-60, 1e-5, a1=aa3, a2=NULL,a3=NULL, ylim=c(.9, 200),k.lab=9e9)
     >
     > showProc.time()
     Time (user system elapsed): 5.042 0.176 10.4
     >
     > ## lower a \> 10
     >
     > curve(qgamma(x, 19), 1e-14, 1e-9, log='x')
     > curve(qgamma(x, 18), 1e-14, 1e-9, log='x')
     > curve(qgamma(x, 15), 1e-11, 5e-9, log='x')
     > curve(qgamma(x, 13), 5e-10, 1e-8, log='x')
     > curve(qgamma(x, 11), 1e-8, 5e-8, log='x')
     > curve(qgamma(x, 10.5), 4.2e-8, 6e-8, log='x')
     > curve(qgamma(x, 10.3), 6e-8, 7e-8, log='x')
     > curve(qgamma(x, 10.2), 7.1e-8, 7.6e-8, log='x')
     > curve(qgamma(x, 10.15),7.7e-8, 7.9e-8, log='x')
     > curve(qgamma(x, 10.14),7.88e-8,7.92e-8, log='x',n=10001)
     >
     > ## no more problems for smaller a!! here:
     > curve(qgamma(x, 10.13), 1e-10, 5e-4, log='x',n=20001)
     > curve(qgamma(x, 10.12), 1e-10, 5e-4, log='x',n=20001)
     > curve(qgamma(x, 10.1), 1e-10, 5e-4, log='x',n=20001)
     >
     > showProc.time()
     Time (user system elapsed): 0.814 0.02 3.968
     >
     > ##--- the "+Inf" / premature "0" case:
     > curve(qgamma(x, 155, log=TRUE), -1500, 0, log='y', n=2001,col=2)
     > curve(qgamma(x, 1e3, log=TRUE), -1500, 0, log='y', n=2001,col=2)
     > ## now works, but slowly and with kink
     > curve(qgamma (x, 1e5, log=TRUE), -3e5, 0, log='y', n=2001,col=2,lwd=3)
     > curve(qgammaAppr(x, 1e5, log=TRUE), add = TRUE, n=2001, col="blue",lwd=.4)
     > ## --- curves are almost "identical"
     > ## ===> the kink *does* come from the initial approx... hmm
     >
     > ## still "identical"
     > curve(qgamma (x, 1e4, log=TRUE), -3e4, 0, log='y', n=2001,col=2)
     > curve(qgammaAppr(x, 1e4, log=TRUE), add = TRUE, n=2001, col="tomato3")
     >
     > ## now see some difference (approx. has kink at ~ -165)
     > curve(qgamma (x, 100, log=TRUE), -200, 0, log='y', n=2001,col=2)
     > curve(qgammaAppr(x, 100, log=TRUE), add = TRUE, n=2001, col="tomato3")
     > ##
     > (kk <- 100 * 2/1.24)# 161.29
     [1] 161.2903
     > curve(qgamma (x, 100, log=TRUE), -1.1*kk, -.95*kk, log='y', n=2001,col=2)
     > curve(qgammaAppr(x, 100, log=TRUE), add = TRUE, n=2001, col="tomato3")
     > abline(v = -kk, col='blue', lty=2)# exactly: kink is at a * 2 / 1.24 = a / .62
     > curve(qgammaAppr(x - 100/.62, 100,log=TRUE), -1e-3, +1e-3)
     >
     > showProc.time()
     Time (user system elapsed): 0.195 0.009 0.601
     >
     > p.qgammaLog <- function(alpha, xl.f = 1.5, xr.f = 0.4, n = 2001)
     + {
     + ## Purpose:
     + ## ----------------------------------------------------------------------
     + ## Arguments:
     + ## ----------------------------------------------------------------------
     + ## Author: Martin Maechler, Date: 30 Mar 2004, 18:44
     + kk <- -alpha / .62 # = (alpha * 2) / (-1.24)
     + curve(qgamma(x, alpha, log=TRUE), xl.f*kk, xr.f*kk, log='y',
     + n=n, col=2, lwd=3.6, lty = 4,
     + main= paste("qgamma(x, alpha=",formatC(alpha,digits=10),", log = TRUE)"))
     + lines(kk, qgamma(kk, alpha, log=TRUE), type = 'h', lty = 3)
     + curve(qgamma (exp(x), alpha), add = TRUE, col="orange", n=n, lwd= 2)
     + curve(qgammaAppr(x, alpha, log=TRUE), add = TRUE, col=3, n=n,lwd = .4)
     + }
     >
     > showProc.time()
     Time (user system elapsed): 0 0.001 0.001
     >
     > p.qgammaLog(25)
     > p.qgammaLog(16)# ~ [-25, -20]
     > p.qgammaLog(12, 1.2, 0.8)# small problem remaining
     > p.qgammaLog(11, 1.2, 0.8)# even smaller
     > p.qgammaLog(10.5, 1.1, 0.9)# even smaller
     > p.qgammaLog(10.25, 1.1, 0.9)# even smaller
     > ## 2019-08: __nothing__ visible from here on:
     > p.qgammaLog(10.18, 1.02, 0.98)# even smaller
     > p.qgammaLog(10.15, 1.02, 0.98)# even smaller
     > p.qgammaLog(10.14, 1.001, 0.999)# even smaller
     > p.qgammaLog(10.139, 1.0002, 0.9998)#
     > p.qgammaLog(10.138, 1.0002, 0.9998)#
     > p.qgammaLog(10.137, 1.00001, 0.99999)#
     > p.qgammaLog(10.13699, 1.0000001, 0.9999999)#
     > p.qgammaLog(10.1369899, 1.0000001, 0.9999999)#
     > p.qgammaLog(10.1369894, 1.0000001, 0.9999999)#
     > p.qgammaLog(10.1369893, 1.0000001, 0.9999999)# even smaller at -16.34998
     >
     > showProc.time()
     Time (user system elapsed): 0.698 0.03 1.873
     >
     > ##-- here is the boundary --- for 64-bit AMD Opteron ---
     > ## and for 32-bit AMD Athlon
     >
     > p.qgammaLog(10.1369892, 1.0000001, 0.9999999)# no more
     > p.qgammaLog(10.136989, 1.0000001, 0.9999999)#
     > p.qgammaLog(10.136988, 1.0000001, 0.9999999)#
     > p.qgammaLog(10.136985, 1.0000001, 0.9999999)#
     > p.qgammaLog(10.13698, 1.0000001, 0.9999999)#
     > p.qgammaLog(10.13697, 1.0000001, 0.9999999)#
     > p.qgammaLog(10.13695, 1.0000001, 0.9999999)#
     > p.qgammaLog(10.1368, 1.000001, 0.999999)#
     > p.qgammaLog(10.1365, 1.000001, 0.999999)#
     > p.qgammaLog(10.136, 1.000001, 0.999999)#
     > p.qgammaLog(10.125, 1.1, 0.9)# --- see it now
     > p.qgammaLog(10, 1.2, 0.8)
     > p.qgammaLog(9)
     >
     > showProc.time()
     Time (user system elapsed): 0.536 0.025 1.425
     >
     > ## For large alpha: show difference to see problem better
     > ## ---> for alpha >= 10, the x problem starts *roughly* at x = -0.8*alpha
     > ##
     >
     > sfsmisc::mult.fig(2)
     > curve(qgammaAppr(x, 5, log=TRUE), - 8.1, -8, n=2001)
     > curve(qgammaAppr(x- 5/.62, 5, log=TRUE), -1e-15, 0)
     >
     > ## is the kink from pgamma() ? : no: this looks fine,
     > curve(pgamma(x, 1e5, log=TRUE), 1, 2e5, log='x', n=2001,col=2)
     > ## and this does too:
     > curve( dgamma(x, 1e5), .5e5, 2e5); par(new=TRUE)
     > curve( dgamma(x, 1e5, log=TRUE), .5e5, 2e5, col=2, yaxt="n")
     > axis(4,col.axis=2); par(new=TRUE)
     > curve( pgamma(x, 1e5), .5e5, 2e5, n=2001, col=3); par(new=TRUE)
     > curve( pgamma(x, 1e5, log=TRUE), .5e5, 2e5, n=2001, col=4); par(new=TRUE)
     > curve(-pgamma(x, 1e5, log=TRUE,lower=FALSE), .5e5, 2e5, n=2001, col=4)
     > ## all looking nice
     >
     >
     > x <- 10^seq(2,6, length=4001)
     > qx <- qgamma(pgamma(x, 1e5, log=TRUE), 1e5, log=TRUE)
     > plot(x, qx, type ='l', col=2, asp = 1); abline(0,1, lty=3)
     >
     > showProc.time()
     Time (user system elapsed): 0.095 0.007 0.325
     > <0c>
     > ###------------- Approximations of qgamma() ------
     > ##
     >
     > ## source("/u/maechler/R/MM/NUMERICS/dpq-functions/qchisqAppr.R")
     > ##--> qchisqAppr()
     > ##--> qchisqWH [ = Wilson Hilferty ]
     > ##--> qchisqKG [ = Kennedy & Gentle's improvements "a la AS 91" ]
     > ## dyn.load("/u/maechler/R/MM/NUMERICS/dpq-functions/qchisq_appr.so")
     >
     > ## Consider the two different implementations of
     > ## lgamma1p(a) := lgamma(1+a) == log(gamma(1+a) == log(a*gamma(a)) "stable":
     >
     > if(!exists("lseq", mode="function"))
     + lseq <- if(requireNamespace("sfsmisc")) sfsmisc::lseq else
     + function(from, to, length) exp(seq(log(from), log(to), length.out = length))
     >
     > if(require("Rmpfr")) { ##---------------- MPFR numbers -------------------------
     +
     + .mpfr.all.eq <- Rmpfr::all.equal
     + AllEq <- function(target, current, ...)
     + .mpfr.all.eq(target, current, ...,
     + formatFUN = function(x, ...) Rmpfr::format(x, digits = 9))
     +
     + print(gammaE <- Const("gamma",200)); pi. <- Const("pi",200)
     + print(a0 <- (gammaE^2 + pi.^2/6)/2)
     + print(psi2.1 <- -2*zeta(mpfr(3,200)))# == psigamma(1,2) =~ -2.4041138
     + print(a1 <- (psi2.1 - gammaE*(pi.^2/2 + gammaE^2))/6)
     +
     + x <- lseq(1e-30, 0.8, length = if(doExtras) 1000 else 125)
     + x. <- mpfr(x, 200)
     + xct. <- log(x. * gamma(x.)) ## using MPFR arithmetic .. no overflow ...
     + xc2. <- log(x.) + lgamma(x.)## (ditto)
     + print(AllEq(xct., xc2., tol = 0)) # 3.15779......e-57
     + xct <- as.numeric(xct.)
     + stopifnot(exprs = {
     + AllEq(xct., xc2., tol = 1e-45)
     + AllEq(xct , xc2., tol = 1e-15)
     + ##
     + all.equal(lgamma1p(x), lgamma1p(x, tol= 1e-16), tol=0)
     + ## -> no difference; i.e., default tol = 1e-14 seems fine enough!
     + })
     + showProc.time()
     +
     + m.appr <- cbind(log(x*gamma(x)), lgamma(1+x), log(x) + lgamma(x),
     + lgamma1p.(x, k=1, cut=3e-6),
     + lgamma1p.(x, k=2, cut=1e-4),
     + lgamma1p.(x, k=3, cut=8e-4),
     + lgamma1p(x))#, tol= 1e-14), # = default
     +
     + eMat <- m.appr - xct # absolute error
     + ## Relative errors:
     + str(reMat. <- m.appr/xct. - 1)
     + str(reMat <- as(reMat., "array")) # as(., "matrix") fails in older versions
     +
     + matplot(x, eMat , log="x", type="l", lty=1) #-> problematic log(x) + lgamma(x) for "large"
     + matplot(x, abs( eMat), log="xy", type="l", lty=1) #-> but good for small; lgamma1p is much better
     + matplot(x, abs(reMat), log="xy", type="l", lty=1)
     + abline(v= 3.47548562941137e-08, col = "gray80", lwd=3)#<- the cutoff value of lgamma1p()
     + ##---> should use earlier cutoff!
     + ## zoom in:
     +
     + matplot(x, abs(reMat), log="xy", type="l", col=1:7, lty=1,
     + lwd=2, xlim=c(8e-9, 1e-3), ylim = c(1e-18, 1e-7), axes=FALSE, frame=TRUE,
     + main = expression(lgamma1p(x) == log(Gamma(x+1)) ~~~ "approximations"
     + ~~~ abs(rel.Err(.))))
     + eaxis(1); eaxis(2)
     + abline(v= 3.47548562941137e-08, col = "gray80", lwd=3)#<- the cutoff value of lgamma1p()
     + abline(h= c(1,2,4)*.Machine$double.eps, lty=3, col="skyblue")
     + legend("topright", col=1:7, lty=1,lwd=2,
     + c("log(x*gamma(x))", "lgamma(1+x)", "log(x) + lgamma(x)",
     + "lgamma1p.(x, k=1, c=3e-6)",
     + "lgamma1p.(x, k=2, c=1e-4)",
     + "lgamma1p.(x, k=3, c=8e-4)",
     + "lgamma1p(x)"), bty="n", ncol=2)
     + abline(v = c(3e-6, 1e-4, 8e-4), col=4:6, lty=2, lwd=1/2)
     +
     + ## FIXME: do the same for the lgaamma1p_series()
     +
     + ## rm(x., xct., xc2., reMat., eMat, AllEq)
     + detach("package:Rmpfr")
     + showProc.time()
     +
     + } ## if( MPFR ) ----------------------------------------------------------------
     Loading required package: Rmpfr
     Loading required package: gmp
    
     Attaching package: 'gmp'
    
     The following objects are masked from 'package:base':
    
     %*%, apply, crossprod, matrix, tcrossprod
    
     C code of R package 'Rmpfr': GMP using 64 bits per limb
    
    
     Attaching package: 'Rmpfr'
    
     The following object is masked from 'package:gmp':
    
     outer
    
     The following object is masked from 'package:DPQ':
    
     log1mexp
    
     The following objects are masked from 'package:stats':
    
     dbinom, dgamma, dnbinom, dnorm, dpois, dt, pnorm
    
     The following objects are masked from 'package:base':
    
     cbind, pmax, pmin, rbind
    
     1 'mpfr' number of precision 200 bits
     [1] 0.57721566490153286060651209008240243104215933593992359880576723
     1 'mpfr' number of precision 200 bits
     [1] 0.98905599532797255539539565150063470793918352072821409044319567
     1 'mpfr' number of precision 200 bits
     [1] -2.404113806319188570799476323022899981529972584680997763584544
     1 'mpfr' number of precision 200 bits
     [1] -0.90747907608088628901656016735627511492861144907256376094133062
     Error in target == current : comparison of these types is not implemented
     Calls: print ... .mpfr.all.eq -> .mpfr.all.eq -> .local -> all.equal.numeric
     Execution halted
    Running the tests in ‘tests/stirlerr-tst.R’ failed.
    Complete output:
     > #### Testing stirlerr(), bd0(), ebd0(), dpois_raw(), ...
     > #### ===============================================
     >
     > require(DPQ)
     Loading required package: DPQ
     > for(pkg in c("Rmpfr", "DPQmpfr"))
     + if(!requireNamespace(pkg)) {
     + cat("no CRAN package", sQuote(pkg), " ---> no tests here.\n")
     + q("no")
     + }
     Loading required namespace: Rmpfr
     Loading required namespace: DPQmpfr
     > require("Rmpfr")
     Loading required package: Rmpfr
     Loading required package: gmp
    
     Attaching package: 'gmp'
    
     The following objects are masked from 'package:base':
    
     %*%, apply, crossprod, matrix, tcrossprod
    
     C code of R package 'Rmpfr': GMP using 64 bits per limb
    
    
     Attaching package: 'Rmpfr'
    
     The following object is masked from 'package:gmp':
    
     outer
    
     The following object is masked from 'package:DPQ':
    
     log1mexp
    
     The following objects are masked from 'package:stats':
    
     dbinom, dgamma, dnbinom, dnorm, dpois, dt, pnorm
    
     The following objects are masked from 'package:base':
    
     cbind, pmax, pmin, rbind
    
     >
     > cutoffs <- c(15,35,80,500) # cut points, n=*, in the above "algorithm"
     > ##
     > n <- c(seq(1,15, by=1/4),seq(16, 25, by=1/2), 26:30, seq(32,50, by=2), seq(55,1000, by=5),
     + 20*c(51:99), 50*(40:80), 150*(27:48), 500*(15:20))
     > st.n <- stirlerr(n)# rather use.halves=TRUE, just here , use.halves=FALSE)
     > plot(st.n ~ n, log="xy", type="b") ## looks good now
     > nM <- mpfr(n, 2048)
     > st.nM <- stirlerr(nM, use.halves=FALSE) ## << on purpose
     > all.equal(asNumeric(st.nM), st.n)# TRUE
     [1] TRUE
     > all.equal(st.nM, as(st.n,"mpfr"))# .. difference: 1.05884..............................e-15
     Error in target == current : comparison of these types is not implemented
     Calls: all.equal -> all.equal -> .local -> all.equal.numeric
     Execution halted
Flavor: r-devel-linux-x86_64-fedora-clang

Version: 0.5-0
Check: examples
Result: ERROR
    Running examples in ‘DPQ-Ex.R’ failed
    The error most likely occurred in:
    
    > ### Name: ppoisson
    > ### Title: Direct Computation of 'ppois()' Poisson Distribution
    > ### Probabilities
    > ### Aliases: ppoisErr ppoisD
    > ### Keywords: distribution
    >
    > ### ** Examples
    >
    > (lams <- outer(c(1,2,5), 10^(0:3)))# 10^4 is already slow!
     [,1] [,2] [,3] [,4]
    [1,] 1 10 100 1000
    [2,] 2 20 200 2000
    [3,] 5 50 500 5000
    > system.time(e1 <- sapply(lams, ppoisErr))
     user system elapsed
     0.013 0.000 0.013
    > e1 / .Machine$double.eps
     [1] 0.0 0.5 -1.0 1.0 5.5 1.5 -4.0 -3.0 1.0 -1.0 2.0 2.0
    >
    > ## Try another 'ppFUN' :---------------------------------
    > ## this relies on the fact that it's *only* used on an 'x' of the form 0:M :
    > ppD0 <- function(x, lambda, all.from.0=TRUE)
    + cumsum(dpois(if(all.from.0) 0:x else x, lambda=lambda))
    > ## and test it:
    > p0 <- ppD0 ( 1000, lambda=10)
    > p1 <- ppois(0:1000, lambda=10)
    > stopifnot(all.equal(p0,p1, tol=8*.Machine$double.eps))
    >
    > system.time(p0.slow <- ppoisD(0:1000, lambda=10, all.from.0=FALSE))# not very slow, here
     user system elapsed
     0.005 0.000 0.019
    > p0.1 <- ppoisD(1000, lambda=10)
    > if(requireNamespace("Rmpfr")) {
    + ppoisMpfr <- function(x, lambda) cumsum(Rmpfr::dpois(x, lambda=lambda))
    + p0.best <- ppoisMpfr(0:1000, lambda = Rmpfr::mpfr(10, precBits = 256))
    + AllEq. <- Rmpfr::all.equal
    + AllEq <- function(target, current, ...)
    + AllEq.(target, current, ...,
    + formatFUN = function(x, ...) Rmpfr::format(x, digits = 9))
    + print(AllEq(p0.best, p0, tol = 0)) # 2.06e-18
    + print(AllEq(p0.best, p0.slow, tol = 0)) # the "worst" (4.44e-17)
    + print(AllEq(p0.best, p0.1, tol = 0)) # 1.08e-18
    + }
    Error in target == current : comparison of these types is not implemented
    Calls: print ... AllEq -> AllEq. -> AllEq. -> .local -> all.equal.numeric
    Execution halted
Flavor: r-devel-linux-x86_64-fedora-gcc

Version: 0.5-0
Check: tests
Result: ERROR
     Running ‘chisq-nonc-ex.R’ [43s/114s]
     Running ‘dnbinom-tst.R’ [26s/70s]
     Running ‘dnchisq-tst.R’
     Running ‘hyper-dist-ex.R’ [35s/103s]
     Running ‘pnbeta-tst.R’
     Running ‘pnt-prec.R’ [35s/95s]
     Running ‘ppois-ex.R’
     Running ‘qPoisBinom-ex.R’
     Running ‘qbeta-dist.R’ [15s/41s]
     Running ‘qbeta-tst.R’
     Running ‘qgamma-ex.R’ [15s/38s]
     Running ‘stirlerr-tst.R’
     Running ‘t-nonc-tst.R’ [7s/17s]
     Running ‘wienergerm-pchisq-tst.R’
     Running ‘wienergerm_nchisq.R’ [9s/21s]
    Running the tests in ‘tests/dnbinom-tst.R’ failed.
    Complete output:
     > #### Testing 1) dbinom_raw(), dnbinomR() and dnbinom.mu()
     > #### 2) log1pmx(), logcf() etc
     > require(DPQ)
     Loading required package: DPQ
     > source(system.file(package="Matrix", "test-tools-1.R", mustWork=TRUE))
     Loading required package: tools
     > ## -> showProc.time(), assertError()
     >
     > (doExtras <- DPQ:::doExtras() && !grepl("valgrind", R.home()))
     [1] FALSE
     >
     > if(!dev.interactive(orNone=TRUE)) pdf("wienergerm-accuracy.pdf")
     >
     >
     > ### 1. Testing dbinom_raw(), dnbinomR() and dnbinom.mu() >>> ../R/dbinom-nbinom.R <<<
     > ### ---------- ../man/dbinom_raw.Rd & ../man/dnbinomR.Rd
     >
     > ## "FIXME:" use sfsmisc :: relErrV() already here
     >
     > ### dbinom() vs dbinom.raw() :
     >
     > for(n in 1:20) {
     + cat("n=",n," ")
     + for(x in 0:n)
     + cat(".")
     + for(p in c(0, .1, .5, .8, 1)) {
     + stopifnot(all.equal(dbinom_raw(x, n, p, q=1-p, log=FALSE),
     + dbinom (x, n, p, log=FALSE)),
     + all.equal(dbinom_raw(x, n, p, q=1-p, log =TRUE),
     + dbinom (x, n, p, log =TRUE)))
     + }
     + cat("\n")
     + }
     n= 1 ..
     n= 2 ...
     n= 3 ....
     n= 4 .....
     n= 5 ......
     n= 6 .......
     n= 7 ........
     n= 8 .........
     n= 9 ..........
     n= 10 ...........
     n= 11 ............
     n= 12 .............
     n= 13 ..............
     n= 14 ...............
     n= 15 ................
     n= 16 .................
     n= 17 ..................
     n= 18 ...................
     n= 19 ....................
     n= 20 .....................
     > showProc.time()
     Time (user system elapsed): 2.269 0.094 5.335
     >
     > ### dnbinom*() :
     > stopifnot(exprs = {
     + dnbinomR(0, 1, 1) == 1
     + })
     >
     > ### exploring 'eps' == "true" tests must be done with Rmpfr !!
     >
     > ### 2. Testing log1pmx(), logcf() etc
     > ### ----------
     >
     > ### 2a: logcf()
     > ## == =======
     > x <- c((-20:3)/4, (25:31)/32) # close (but not too close) to upper bound 1
     >
     > (lC <- logcf (x, i=2, d=3, eps=1e-9))
     [1] 0.2225364 0.2273165 0.2323964 0.2378089 0.2435921 0.2497910 0.2564582
     [8] 0.2636564 0.2714610 0.2799634 0.2892758 0.2995378 0.3109259 0.3236668
     [15] 0.3380584 0.3545014 0.3735507 0.3960035 0.4230578 0.4566237 0.5000000
     [22] 0.5595850 0.6503112 0.8221318 0.8574109 0.8989257 0.9490572 1.0118259
     [29] 1.0948318 1.2152882 1.4286636
     > lCt <- logcf (x, i=2, d=3, eps=1e-9, trace=TRUE) ; stopifnot(identical(lCt, lC))
     it= 0: ==> |b2|=162720
     it= 1: ==> |b2|=1.68458e+08
     it= 2: ==> |b2|=3.02689e+11
     it= 3: ==> |b2|=8.40216e+14
     it= 4: ==> |b2|=3.33607e+18
     it= 5: ==> |b2|=1.79478e+22
     it= 6: ==> |b2|=1.25703e+26
     it= 7: ==> |b2|=1.11146e+30
     it= 8: ==> |b2|=1.21086e+34
     it= 9: ==> |b2|=1.5936e+38
     it=10: ==> |b2|=2.49268e+42
     logcf(*) used 11 iterations.
     it= 0: ==> |b2|=151400
     it= 1: ==> |b2|=1.519e+08
     it= 2: ==> |b2|=2.64707e+11
     it= 3: ==> |b2|=7.12814e+14
     it= 4: ==> |b2|=2.74588e+18
     it= 5: ==> |b2|=1.4333e+22
     it= 6: ==> |b2|=9.73998e+25
     it= 7: ==> |b2|=8.35605e+29
     it= 8: ==> |b2|=8.83286e+33
     it= 9: ==> |b2|=1.12795e+38
     it=10: ==> |b2|=1.71192e+42
     logcf(*) used 11 iterations.
     it= 0: ==> |b2|=140480
     it= 1: ==> |b2|=1.36437e+08
     it= 2: ==> |b2|=2.30332e+11
     it= 3: ==> |b2|=6.0102e+14
     it= 4: ==> |b2|=2.24367e+18
     it= 5: ==> |b2|=1.135e+22
     it= 6: ==> |b2|=7.47503e+25
     it= 7: ==> |b2|=6.21522e+29
     it= 8: ==> |b2|=6.3674e+33
     it= 9: ==> |b2|=7.88061e+37
     it=10: ==> |b2|=1.15921e+42
     logcf(*) used 11 iterations.
     it= 0: ==> |b2|=129960
     it= 1: ==> |b2|=1.22034e+08
     it= 2: ==> |b2|=1.99336e+11
     it= 3: ==> |b2|=5.03394e+14
     it= 4: ==> |b2|=1.81889e+18
     it= 5: ==> |b2|=8.90621e+21
     it= 6: ==> |b2|=5.67763e+25
     it= 7: ==> |b2|=4.56957e+29
     it= 8: ==> |b2|=4.53158e+33
     it= 9: ==> |b2|=5.429e+37
     logcf(*) used 10 iterations.
     it= 0: ==> |b2|=119840
     it= 1: ==> |b2|=1.08655e+08
     it= 2: ==> |b2|=1.71497e+11
     it= 3: ==> |b2|=4.18587e+14
     it= 4: ==> |b2|=1.46194e+18
     it= 5: ==> |b2|=6.91963e+21
     it= 6: ==> |b2|=4.26415e+25
     it= 7: ==> |b2|=3.31759e+29
     it= 8: ==> |b2|=3.18042e+33
     it= 9: ==> |b2|=3.68336e+37
     logcf(*) used 10 iterations.
     it= 0: ==> |b2|=110120
     it= 1: ==> |b2|=9.62638e+07
     it= 2: ==> |b2|=1.46601e+11
     it= 3: ==> |b2|=3.45334e+14
     it= 4: ==> |b2|=1.16411e+18
     it= 5: ==> |b2|=5.31835e+21
     it= 6: ==> |b2|=3.16349e+25
     it= 7: ==> |b2|=2.37577e+29
     it= 8: ==> |b2|=2.19845e+33
     it= 9: ==> |b2|=2.45771e+37
     logcf(*) used 10 iterations.
     it= 0: ==> |b2|=100800
     it= 1: ==> |b2|=8.48232e+07
     it= 2: ==> |b2|=1.24442e+11
     it= 3: ==> |b2|=2.82452e+14
     it= 4: ==> |b2|=9.17519e+17
     it= 5: ==> |b2|=4.03952e+21
     it= 6: ==> |b2|=2.3156e+25
     it= 7: ==> |b2|=1.67591e+29
     it= 8: ==> |b2|=1.49457e+33
     logcf(*) used 9 iterations.
     it= 0: ==> |b2|=91880
     it= 1: ==> |b2|=7.42974e+07
     it= 2: ==> |b2|=1.04819e+11
     it= 3: ==> |b2|=2.28837e+14
     it= 4: ==> |b2|=7.15064e+17
     it= 5: ==> |b2|=3.02848e+21
     it= 6: ==> |b2|=1.67007e+25
     it= 7: ==> |b2|=1.1628e+29
     it= 8: ==> |b2|=9.97611e+32
     logcf(*) used 9 iterations.
     it= 0: ==> |b2|=83360
     it= 1: ==> |b2|=6.46501e+07
     it= 2: ==> |b2|=8.75389e+10
     it= 3: ==> |b2|=1.83464e+14
     it= 4: ==> |b2|=5.50387e+17
     it= 5: ==> |b2|=2.23803e+21
     it= 6: ==> |b2|=1.18496e+25
     it= 7: ==> |b2|=7.92152e+28
     it= 8: ==> |b2|=6.52535e+32
     logcf(*) used 9 iterations.
     it= 0: ==> |b2|=75240
     it= 1: ==> |b2|=5.58449e+07
     it= 2: ==> |b2|=7.24171e+10
     it= 3: ==> |b2|=1.45381e+14
     it= 4: ==> |b2|=4.17809e+17
     it= 5: ==> |b2|=1.6276e+21
     it= 6: ==> |b2|=8.25594e+24
     it= 7: ==> |b2|=5.28764e+28
     logcf(*) used 8 iterations.
     it= 0: ==> |b2|=67520
     it= 1: ==> |b2|=4.78456e+07
     it= 2: ==> |b2|=5.92745e+10
     it= 3: ==> |b2|=1.13708e+14
     it= 4: ==> |b2|=3.12287e+17
     it= 5: ==> |b2|=1.16261e+21
     it= 6: ==> |b2|=5.6361e+24
     it= 7: ==> |b2|=3.44989e+28
     logcf(*) used 8 iterations.
     it= 0: ==> |b2|=60200
     it= 1: ==> |b2|=4.06158e+07
     it= 2: ==> |b2|=4.79397e+10
     it= 3: ==> |b2|=8.76351e+13
     it= 4: ==> |b2|=2.2937e+17
     it= 5: ==> |b2|=8.13827e+20
     it= 6: ==> |b2|=3.76013e+24
     it= 7: ==> |b2|=2.19363e+28
     logcf(*) used 8 iterations.
     it= 0: ==> |b2|=53280
     it= 1: ==> |b2|=3.41194e+07
     it= 2: ==> |b2|=3.82483e+10
     it= 3: ==> |b2|=6.64186e+13
     it= 4: ==> |b2|=1.6515e+17
     it= 5: ==> |b2|=5.56707e+20
     it= 6: ==> |b2|=2.44378e+24
     logcf(*) used 7 iterations.
     it= 0: ==> |b2|=46760
     it= 1: ==> |b2|=2.83198e+07
     it= 2: ==> |b2|=3.0043e+10
     it= 3: ==> |b2|=4.93794e+13
     it= 4: ==> |b2|=1.16224e+17
     it= 5: ==> |b2|=3.70875e+20
     it= 6: ==> |b2|=1.54119e+24
     logcf(*) used 7 iterations.
     it= 0: ==> |b2|=40640
     it= 1: ==> |b2|=2.3181e+07
     it= 2: ==> |b2|=2.31738e+10
     it= 3: ==> |b2|=3.59e+13
     it= 4: ==> |b2|=7.96488e+16
     it= 5: ==> |b2|=2.39588e+20
     logcf(*) used 6 iterations.
     it= 0: ==> |b2|=34920
     it= 1: ==> |b2|=1.86664e+07
     it= 2: ==> |b2|=1.74976e+10
     it= 3: ==> |b2|=2.5422e+13
     it= 4: ==> |b2|=5.29017e+16
     it= 5: ==> |b2|=1.49263e+20
     logcf(*) used 6 iterations.
     it= 0: ==> |b2|=29600
     it= 1: ==> |b2|=1.474e+07
     it= 2: ==> |b2|=1.28785e+10
     it= 3: ==> |b2|=1.74436e+13
     it= 4: ==> |b2|=3.38438e+16
     logcf(*) used 5 iterations.
     it= 0: ==> |b2|=24680
     it= 1: ==> |b2|=1.13653e+07
     it= 2: ==> |b2|=9.18785e+09
     it= 3: ==> |b2|=1.1517e+13
     it= 4: ==> |b2|=2.06815e+16
     logcf(*) used 5 iterations.
     it= 0: ==> |b2|=20160
     it= 1: ==> |b2|=8.50608e+06
     it= 2: ==> |b2|=6.30386e+09
     it= 3: ==> |b2|=7.24564e+12
     logcf(*) used 4 iterations.
     it= 0: ==> |b2|=16040
     it= 1: ==> |b2|=6.12601e+06
     it= 2: ==> |b2|=4.11202e+09
     logcf(*) used 3 iterations.
     logcf(*) used 0 iterations.
     it= 0: ==> |b2|=9000
     it= 1: ==> |b2|=2.65815e+06
     it= 2: ==> |b2|=1.38218e+09
     logcf(*) used 3 iterations.
     it= 0: ==> |b2|=6080
     it= 1: ==> |b2|=1.49776e+06
     it= 2: ==> |b2|=6.50656e+08
     it= 3: ==> |b2|=4.39124e+11
     it= 4: ==> |b2|=4.24985e+14
     logcf(*) used 5 iterations.
     it= 0: ==> |b2|=3560
     it= 1: ==> |b2|=671330
     it= 2: ==> |b2|=2.24237e+08
     it= 3: ==> |b2|=1.16565e+11
     it= 4: ==> |b2|=8.69636e+13
     it= 5: ==> |b2|=8.80714e+16
     it= 6: ==> |b2|=1.16246e+20
     it= 7: ==> |b2|=1.93847e+23
     it= 8: ==> |b2|=3.98491e+26
     logcf(*) used 9 iterations.
     it= 0: ==> |b2|=3273.12
     it= 1: ==> |b2|=589700
     it= 2: ==> |b2|=1.88377e+08
     it= 3: ==> |b2|=9.36959e+10
     it= 4: ==> |b2|=6.68994e+13
     it= 5: ==> |b2|=6.48488e+16
     it= 6: ==> |b2|=8.19327e+19
     it= 7: ==> |b2|=1.30789e+23
     it= 8: ==> |b2|=2.57381e+26
     it= 9: ==> |b2|=6.12129e+29
     logcf(*) used 10 iterations.
     it= 0: ==> |b2|=2992.5
     it= 1: ==> |b2|=512650
     it= 2: ==> |b2|=1.55894e+08
     it= 3: ==> |b2|=7.3859e+10
     it= 4: ==> |b2|=5.02475e+13
     it= 5: ==> |b2|=4.64164e+16
     it= 6: ==> |b2|=5.58911e+19
     it= 7: ==> |b2|=8.50347e+22
     it= 8: ==> |b2|=1.595e+26
     it= 9: ==> |b2|=3.61574e+29
     it=10: ==> |b2|=9.74479e+32
     logcf(*) used 11 iterations.
     it= 0: ==> |b2|=2718.12
     it= 1: ==> |b2|=440109
     it= 2: ==> |b2|=1.26644e+08
     it= 3: ==> |b2|=5.68225e+10
     it= 4: ==> |b2|=3.66244e+13
     it= 5: ==> |b2|=3.20598e+16
     it= 6: ==> |b2|=3.65864e+19
     it= 7: ==> |b2|=5.27587e+22
     it= 8: ==> |b2|=9.37997e+25
     it= 9: ==> |b2|=2.01557e+29
     it=10: ==> |b2|=5.14924e+32
     it=11: ==> |b2|=1.54257e+36
     logcf(*) used 12 iterations.
     it= 0: ==> |b2|=2450
     it= 1: ==> |b2|=372006
     it= 2: ==> |b2|=1.00485e+08
     it= 3: ==> |b2|=4.23633e+10
     it= 4: ==> |b2|=2.56713e+13
     it= 5: ==> |b2|=2.11343e+16
     it= 6: ==> |b2|=2.26869e+19
     it= 7: ==> |b2|=3.07772e+22
     it= 8: ==> |b2|=5.14811e+25
     it= 9: ==> |b2|=1.04082e+29
     it=10: ==> |b2|=2.50192e+32
     it=11: ==> |b2|=7.05238e+35
     it=12: ==> |b2|=2.30384e+39
     logcf(*) used 13 iterations.
     it= 0: ==> |b2|=2188.12
     it= 1: ==> |b2|=308271
     it= 2: ==> |b2|=7.72745e+07
     it= 3: ==> |b2|=3.02658e+10
     it= 4: ==> |b2|=1.70531e+13
     it= 5: ==> |b2|=1.30605e+16
     it= 6: ==> |b2|=1.30466e+19
     it= 7: ==> |b2|=1.64734e+22
     it= 8: ==> |b2|=2.56499e+25
     it= 9: ==> |b2|=4.82765e+28
     it=10: ==> |b2|=1.08039e+32
     it=11: ==> |b2|=2.83535e+35
     it=12: ==> |b2|=8.62389e+38
     it=13: ==> |b2|=3.00926e+42
     it=14: ==> |b2|=1.19409e+46
     it=15: ==> |b2|=5.34632e+49
     logcf(*) used 16 iterations.
     it= 0: ==> |b2|=1932.5
     it= 1: ==> |b2|=248832
     it= 2: ==> |b2|=5.68734e+07
     it= 3: ==> |b2|=2.03226e+10
     it= 4: ==> |b2|=1.04577e+13
     it= 5: ==> |b2|=7.32086e+15
     it= 6: ==> |b2|=6.68834e+18
     it= 7: ==> |b2|=7.72653e+21
     it= 8: ==> |b2|=1.10096e+25
     it= 9: ==> |b2|=1.89662e+28
     it=10: ==> |b2|=3.88536e+31
     it=11: ==> |b2|=9.33474e+34
     it=12: ==> |b2|=2.59938e+38
     it=13: ==> |b2|=8.30457e+41
     it=14: ==> |b2|=3.01718e+45
     it=15: ==> |b2|=1.23692e+49
     it=16: ==> |b2|=5.68258e+52
     it=17: ==> |b2|=2.90768e+56
     it=18: ==> |b2|=1.64796e+60
     logcf(*) used 19 iterations.
     it= 0: ==> |b2|=1683.12
     it= 1: ==> |b2|=193619
     it= 2: ==> |b2|=3.91439e+07
     it= 3: ==> |b2|=1.23338e+10
     it= 4: ==> |b2|=5.59551e+12
     it= 5: ==> |b2|=3.4562e+15
     it= 6: ==> |b2|=2.78868e+18
     it= 7: ==> |b2|=2.84748e+21
     it= 8: ==> |b2|=3.58854e+24
     it= 9: ==> |b2|=5.4701e+27
     it=10: ==> |b2|=9.91885e+30
     it=11: ==> |b2|=2.10987e+34
     it=12: ==> |b2|=5.20269e+37
     it=13: ==> |b2|=1.47211e+41
     it=14: ==> |b2|=4.73732e+44
     it=15: ==> |b2|=1.72036e+48
     it=16: ==> |b2|=7.00164e+51
     it=17: ==> |b2|=3.17394e+55
     it=18: ==> |b2|=1.59374e+59
     it=19: ==> |b2|=8.8205e+62
     it=20: ==> |b2|=5.35623e+66
     it=21: ==> |b2|=3.5541e+70
     it=22: ==> |b2|=2.56724e+74
     it=23: ==> |b2|=2.01172e+78 Lrg |b2|
     it=24: ==> |b2|=147221
     it=25: ==> |b2|=1.34508e+09
     it=26: ==> |b2|=1.32142e+13
     it=27: ==> |b2|=1.39232e+17
     logcf(*) used 28 iterations.
     > (lR <- logcfR(x, i=2, d=3, eps=1e-9))
     [1] 0.2225364 0.2273165 0.2323964 0.2378089 0.2435921 0.2497910 0.2564582
     [8] 0.2636564 0.2714610 0.2799634 0.2892758 0.2995378 0.3109259 0.3236668
     [15] 0.3380584 0.3545014 0.3735507 0.3960035 0.4230578 0.4566237 0.5000000
     [22] 0.5595850 0.6503112 0.8221318 0.8574109 0.8989257 0.9490572 1.0118259
     [29] 1.0948318 1.2152882 1.4286636
     > all.equal(lC, lR, tol = 0) # to see if ..
     [1] TRUE
     > stopifnot(all.equal(lC, lR, tol = 4e-16))
     > lRt <- logcfR(x, i=2, d=3, eps=1e-9, trace=TRUE) ; stopifnot(identical(lRt, lR))
     logcf(x[ 1]= -5, i=2, d=3, eps=1e-09) logcf(*) end: after 11 iterations.
     logcf(x[ 2]= -4.75, i=2, d=3, eps=1e-09) logcf(*) end: after 11 iterations.
     logcf(x[ 3]= -4.5, i=2, d=3, eps=1e-09) logcf(*) end: after 11 iterations.
     logcf(x[ 4]= -4.25, i=2, d=3, eps=1e-09) logcf(*) end: after 10 iterations.
     logcf(x[ 5]= -4, i=2, d=3, eps=1e-09) logcf(*) end: after 10 iterations.
     logcf(x[ 6]= -3.75, i=2, d=3, eps=1e-09) logcf(*) end: after 10 iterations.
     logcf(x[ 7]= -3.5, i=2, d=3, eps=1e-09) logcf(*) end: after 9 iterations.
     logcf(x[ 8]= -3.25, i=2, d=3, eps=1e-09) logcf(*) end: after 9 iterations.
     logcf(x[ 9]= -3, i=2, d=3, eps=1e-09) logcf(*) end: after 9 iterations.
     logcf(x[10]= -2.75, i=2, d=3, eps=1e-09) logcf(*) end: after 8 iterations.
     logcf(x[11]= -2.5, i=2, d=3, eps=1e-09) logcf(*) end: after 8 iterations.
     logcf(x[12]= -2.25, i=2, d=3, eps=1e-09) logcf(*) end: after 8 iterations.
     logcf(x[13]= -2, i=2, d=3, eps=1e-09) logcf(*) end: after 7 iterations.
     logcf(x[14]= -1.75, i=2, d=3, eps=1e-09) logcf(*) end: after 7 iterations.
     logcf(x[15]= -1.5, i=2, d=3, eps=1e-09) logcf(*) end: after 6 iterations.
     logcf(x[16]= -1.25, i=2, d=3, eps=1e-09) logcf(*) end: after 6 iterations.
     logcf(x[17]= -1, i=2, d=3, eps=1e-09) logcf(*) end: after 5 iterations.
     logcf(x[18]= -0.75, i=2, d=3, eps=1e-09) logcf(*) end: after 5 iterations.
     logcf(x[19]= -0.5, i=2, d=3, eps=1e-09) logcf(*) end: after 4 iterations.
     logcf(x[20]= -0.25, i=2, d=3, eps=1e-09) logcf(*) end: after 3 iterations.
     logcf(x[21]= 0, i=2, d=3, eps=1e-09) logcf(*) end: after 0 iterations.
     logcf(x[22]= 0.25, i=2, d=3, eps=1e-09) logcf(*) end: after 3 iterations.
     logcf(x[23]= 0.5, i=2, d=3, eps=1e-09) logcf(*) end: after 5 iterations.
     logcf(x[24]= 0.75, i=2, d=3, eps=1e-09) logcf(*) end: after 9 iterations.
     logcf(x[25]= 0.78125, i=2, d=3, eps=1e-09) logcf(*) end: after 10 iterations.
     logcf(x[26]= 0.8125, i=2, d=3, eps=1e-09) logcf(*) end: after 11 iterations.
     logcf(x[27]= 0.84375, i=2, d=3, eps=1e-09) logcf(*) end: after 12 iterations.
     logcf(x[28]= 0.875, i=2, d=3, eps=1e-09) logcf(*) end: after 13 iterations.
     logcf(x[29]= 0.90625, i=2, d=3, eps=1e-09) logcf(*) end: after 16 iterations.
     logcf(x[30]= 0.9375, i=2, d=3, eps=1e-09) logcf(*) end: after 19 iterations.
     logcf(x[31]= 0.96875, i=2, d=3, eps=1e-09) logcf(*) end: after 28 iterations.
     > lRt2 <- logcfR(x, i=2, d=3, eps=1e-9, trace= 2) ; stopifnot(identical(lRt2,lR))
     logcf(x[ 1]= -5, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 162720 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.0168627
     it= 2: ==> B2= 1.68458e+08 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00303811
     it= 3: ==> B2= 3.02689e+11 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000541327
     it= 4: ==> B2= 8.40216e+14 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 9.60626e-05
     it= 5: ==> B2= 3.33607e+18 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.70167e-05
     it= 6: ==> B2= 1.79478e+22 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.01154e-06
     it= 7: ==> B2= 1.25703e+26 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 5.32664e-07
     it= 8: ==> B2= 1.11146e+30 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 9.4179e-08
     it= 9: ==> B2= 1.21086e+34 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.66472e-08
     it=10: ==> B2= 1.5936e+38 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.942e-09
     it=11: ==> B2= 2.49268e+42 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 5.19854e-10
     logcf(*) end: after 11 iterations.
     logcf(x[ 2]= -4.75, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 151400 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.0157061
     it= 2: ==> B2= 1.519e+08 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00271234
     it= 3: ==> B2= 2.64707e+11 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000463242
     it= 4: ==> B2= 7.12814e+14 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 7.88006e-05
     it= 5: ==> B2= 2.74588e+18 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.33808e-05
     it= 6: ==> B2= 1.4333e+22 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.26999e-06
     it= 7: ==> B2= 9.73998e+25 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.84872e-07
     it= 8: ==> B2= 8.35605e+29 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 6.52292e-08
     it= 9: ==> B2= 8.83286e+33 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.10523e-08
     it=10: ==> B2= 1.12795e+38 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.8723e-09
     it=11: ==> B2= 1.71192e+42 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.1713e-10
     logcf(*) end: after 11 iterations.
     logcf(x[ 3]= -4.5, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 140480 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.014539
     it= 2: ==> B2= 1.36437e+08 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00239872
     it= 3: ==> B2= 2.30332e+11 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000391409
     it= 4: ==> B2= 6.0102e+14 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 6.36152e-05
     it= 5: ==> B2= 2.24367e+18 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.03211e-05
     it= 6: ==> B2= 1.135e+22 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.67293e-06
     it= 7: ==> B2= 7.47503e+25 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.71005e-07
     it= 8: ==> B2= 6.21522e+29 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 4.38843e-08
     it= 9: ==> B2= 6.3674e+33 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 7.10431e-09
     it=10: ==> B2= 7.88061e+37 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.14987e-09
     it=11: ==> B2= 1.15921e+42 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.86085e-10
     logcf(*) end: after 11 iterations.
     logcf(x[ 4]= -4.25, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 129960 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.0133641
     it= 2: ==> B2= 1.22034e+08 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00209863
     it= 3: ==> B2= 1.99336e+11 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000325959
     it= 4: ==> B2= 5.03394e+14 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 5.04302e-05
     it= 5: ==> B2= 1.81889e+18 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 7.78852e-06
     it= 6: ==> B2= 8.90621e+21 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.20172e-06
     it= 7: ==> B2= 5.67763e+25 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.85309e-07
     it= 8: ==> B2= 4.56957e+29 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.85641e-08
     it= 9: ==> B2= 4.53158e+33 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 4.40173e-09
     it=10: ==> B2= 5.429e+37 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 6.78171e-10
     logcf(*) end: after 10 iterations.
     logcf(x[ 5]= -4, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 119840 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.0121853
     it= 2: ==> B2= 1.08655e+08 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00181353
     it= 3: ==> B2= 1.71497e+11 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000266983
     it= 4: ==> B2= 4.18587e+14 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.91528e-05
     it= 5: ==> B2= 1.46194e+18 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 5.73167e-06
     it= 6: ==> B2= 6.91963e+21 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 8.38266e-07
     it= 7: ==> B2= 4.26415e+25 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.22525e-07
     it= 8: ==> B2= 3.31759e+29 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.79016e-08
     it= 9: ==> B2= 3.18042e+33 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.6148e-09
     it=10: ==> B2= 3.68336e+37 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.81854e-10
     logcf(*) end: after 10 iterations.
     logcf(x[ 6]= -3.75, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 110120 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.0110067
     it= 2: ==> B2= 9.62638e+07 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00154491
     it= 3: ==> B2= 1.46601e+11 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00021452
     it= 4: ==> B2= 3.45334e+14 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.96738e-05
     it= 5: ==> B2= 1.16411e+18 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 4.0975e-06
     it= 6: ==> B2= 5.31835e+21 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 5.65252e-07
     it= 7: ==> B2= 3.16349e+25 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 7.79297e-08
     it= 8: ==> B2= 2.37577e+29 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.07396e-08
     it= 9: ==> B2= 2.19845e+33 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.47962e-09
     it=10: ==> B2= 2.45771e+37 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.03808e-10
     logcf(*) end: after 10 iterations.
     logcf(x[ 7]= -3.5, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 100800 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00983372
     it= 2: ==> B2= 8.48232e+07 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00129431
     it= 3: ==> B2= 1.24442e+11 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000168553
     it= 4: ==> B2= 2.82452e+14 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.18673e-05
     it= 5: ==> B2= 9.17519e+17 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.83198e-06
     it= 6: ==> B2= 4.03952e+21 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.66405e-07
     it= 7: ==> B2= 2.3156e+25 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 4.73767e-08
     it= 8: ==> B2= 1.67591e+29 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 6.12337e-09
     it= 9: ==> B2= 1.49457e+33 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 7.91207e-10
     logcf(*) end: after 9 iterations.
     logcf(x[ 8]= -3.25, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 91880 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00867282
     it= 2: ==> B2= 7.42974e+07 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00106327
     it= 3: ==> B2= 1.04819e+11 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000128994
     it= 4: ==> B2= 2.28837e+14 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.55909e-05
     it= 5: ==> B2= 7.15064e+17 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.88109e-06
     it= 6: ==> B2= 3.02848e+21 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.26734e-07
     it= 7: ==> B2= 1.67007e+25 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.7312e-08
     it= 8: ==> B2= 1.1628e+29 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.28859e-09
     it= 9: ==> B2= 9.97611e+32 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.95855e-10
     logcf(*) end: after 9 iterations.
     logcf(x[ 9]= -3, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 83360 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00753175
     it= 2: ==> B2= 6.46501e+07 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000853254
     it= 3: ==> B2= 8.75389e+10 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 9.5671e-05
     it= 4: ==> B2= 1.83464e+14 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.06874e-05
     it= 5: ==> B2= 5.50387e+17 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.19178e-06
     it= 6: ==> B2= 2.23803e+21 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.32765e-07
     it= 7: ==> B2= 1.18496e+25 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.47808e-08
     it= 8: ==> B2= 7.92152e+28 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.64484e-09
     it= 9: ==> B2= 6.52535e+32 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.82987e-10
     logcf(*) end: after 9 iterations.
     logcf(x[10]= -2.75, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 75240 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00641978
     it= 2: ==> B2= 5.58449e+07 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000665641
     it= 3: ==> B2= 7.24171e+10 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 6.83222e-05
     it= 4: ==> B2= 1.45381e+14 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 6.98689e-06
     it= 5: ==> B2= 4.17809e+17 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 7.13234e-07
     it= 6: ==> B2= 1.6276e+21 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 7.27338e-08
     it= 7: ==> B2= 8.25594e+24 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 7.41242e-09
     it= 8: ==> B2= 5.28764e+28 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 7.55083e-10
     logcf(*) end: after 8 iterations.
     logcf(x[11]= -2.5, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 67520 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00534792
     it= 2: ==> B2= 4.78456e+07 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.0005016
     it= 3: ==> B2= 5.92745e+10 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 4.65818e-05
     it= 4: ==> B2= 1.13708e+14 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 4.31004e-06
     it= 5: ==> B2= 3.12287e+17 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.98074e-07
     it= 6: ==> B2= 1.16261e+21 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.67278e-08
     it= 7: ==> B2= 5.6361e+24 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.38642e-09
     it= 8: ==> B2= 3.44989e+28 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.12099e-10
     logcf(*) end: after 8 iterations.
     logcf(x[12]= -2.25, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 60200 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00432911
     it= 2: ==> B2= 4.06158e+07 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00036199
     it= 3: ==> B2= 4.79397e+10 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.99753e-05
     it= 4: ==> B2= 8.76351e+13 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.47309e-06
     it= 5: ==> B2= 2.2937e+17 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.03667e-07
     it= 6: ==> B2= 8.13827e+20 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.67549e-08
     it= 7: ==> B2= 3.76013e+24 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.37743e-09
     it= 8: ==> B2= 2.19363e+28 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.13188e-10
     logcf(*) end: after 8 iterations.
     logcf(x[13]= -2, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 53280 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00337838
     it= 2: ==> B2= 3.41194e+07 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00024722
     it= 3: ==> B2= 3.82483e+10 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.79187e-05
     it= 4: ==> B2= 6.64186e+13 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.29399e-06
     it= 5: ==> B2= 1.6515e+17 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 9.32713e-08
     it= 6: ==> B2= 5.56707e+20 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 6.71575e-09
     it= 7: ==> B2= 2.44378e+24 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 4.83216e-10
     logcf(*) end: after 7 iterations.
     logcf(x[14]= -1.75, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 46760 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00251277
     it= 2: ==> B2= 2.83198e+07 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000157067
     it= 3: ==> B2= 3.0043e+10 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 9.72602e-06
     it= 4: ==> B2= 4.93794e+13 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 6.00029e-07
     it= 5: ==> B2= 1.16224e+17 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.69475e-08
     it= 6: ==> B2= 3.70875e+20 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.27255e-09
     it= 7: ==> B2= 1.54119e+24 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.39681e-10
     logcf(*) end: after 7 iterations.
     logcf(x[15]= -1.5, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 40640 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.001751
     it= 2: ==> B2= 2.3181e+07 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 9.04805e-05
     it= 3: ==> B2= 2.31738e+10 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 4.63221e-06
     it= 4: ==> B2= 3.59e+13 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.36258e-07
     it= 5: ==> B2= 7.96488e+16 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.20265e-08
     it= 6: ==> B2= 2.39588e+20 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 6.11497e-10
     logcf(*) end: after 6 iterations.
     logcf(x[16]= -1.25, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 34920 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00111245
     it= 2: ==> B2= 1.86664e+07 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 4.53708e-05
     it= 3: ==> B2= 1.74976e+10 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.8334e-06
     it= 4: ==> B2= 2.5422e+13 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 7.38016e-08
     it= 5: ==> B2= 5.29017e+16 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.96486e-09
     it= 6: ==> B2= 1.49263e+20 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.18968e-10
     logcf(*) end: after 6 iterations.
     logcf(x[17]= -1, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 29600 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000614941
     it= 2: ==> B2= 1.474e+07 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.84647e-05
     it= 3: ==> B2= 1.28785e+10 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 5.49306e-07
     it= 4: ==> B2= 1.74436e+13 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.62767e-08
     it= 5: ==> B2= 3.38438e+16 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 4.81303e-10
     logcf(*) end: after 5 iterations.
     logcf(x[18]= -0.75, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 24680 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000270385
     it= 2: ==> B2= 1.13653e+07 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 5.33194e-06
     it= 3: ==> B2= 9.18785e+09 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.04153e-07
     it= 4: ==> B2= 1.1517e+13 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.02617e-09
     it= 5: ==> B2= 2.06815e+16 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.93312e-11
     logcf(*) end: after 5 iterations.
     logcf(x[19]= -0.5, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 20160 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 7.69704e-05
     it= 2: ==> B2= 8.50608e+06 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 8.02466e-07
     it= 3: ==> B2= 6.30386e+09 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 8.28445e-09
     it= 4: ==> B2= 7.24564e+12 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 8.51583e-11
     logcf(*) end: after 4 iterations.
     logcf(x[20]= -0.25, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 16040 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 7.26392e-06
     it= 2: ==> B2= 6.12601e+06 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.30773e-08
     it= 3: ==> B2= 4.11202e+09 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 7.25571e-11
     logcf(*) end: after 3 iterations.
     logcf(x[21]= 0, i=2, d=3, eps=1e-09) iterations:
     logcf(*) end: after 0 iterations.
     logcf(x[22]= 0.25, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 9000 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.04918e-05
     it= 2: ==> B2= 2.65815e+06 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.08623e-07
     it= 3: ==> B2= 1.38218e+09 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 5.68393e-10
     logcf(*) end: after 3 iterations.
     logcf(x[23]= 0.5, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 6080 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000659523
     it= 2: ==> B2= 1.49776e+06 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.00942e-05
     it= 3: ==> B2= 6.50656e+08 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 6.02264e-07
     it= 4: ==> B2= 4.39124e+11 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.79273e-08
     it= 5: ==> B2= 4.24985e+14 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 5.3174e-10
     logcf(*) end: after 5 iterations.
     logcf(x[24]= 0.75, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 3560 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00856402
     it= 2: ==> B2= 671330 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00100335
     it= 3: ==> B2= 2.24237e+08 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000114482
     it= 4: ==> B2= 1.16565e+11 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.2922e-05
     it= 5: ==> B2= 8.69636e+13 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.45089e-06
     it= 6: ==> B2= 8.80714e+16 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.62421e-07
     it= 7: ==> B2= 1.16246e+20 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.81486e-08
     it= 8: ==> B2= 1.93847e+23 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.02539e-09
     it= 9: ==> B2= 3.98491e+26 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.25837e-10
     logcf(*) end: after 9 iterations.
     logcf(x[25]= 0.78125, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 3273.12 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.0116683
     it= 2: ==> B2= 589700 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00162549
     it= 3: ==> B2= 1.88377e+08 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000220072
     it= 4: ==> B2= 9.36959e+10 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.94426e-05
     it= 5: ==> B2= 6.68994e+13 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.9163e-06
     it= 6: ==> B2= 6.48488e+16 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 5.19244e-07
     it= 7: ==> B2= 8.19327e+19 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 6.87066e-08
     it= 8: ==> B2= 1.30789e+23 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 9.07922e-09
     it= 9: ==> B2= 2.57381e+26 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.19866e-09
     it=10: ==> B2= 6.12129e+29 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.58143e-10
     logcf(*) end: after 10 iterations.
     logcf(x[26]= 0.8125, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 2992.5 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.0159401
     it= 2: ==> B2= 512650 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00265674
     it= 3: ==> B2= 1.55894e+08 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000429426
     it= 4: ==> B2= 7.3859e+10 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 6.84941e-05
     it= 5: ==> B2= 5.02475e+13 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.08546e-05
     it= 6: ==> B2= 4.64164e+16 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.71404e-06
     it= 7: ==> B2= 5.58911e+19 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.70071e-07
     it= 8: ==> B2= 8.50347e+22 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 4.2492e-08
     it= 9: ==> B2= 1.595e+26 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 6.6788e-09
     it=10: ==> B2= 3.61574e+29 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.04899e-09
     it=11: ==> B2= 9.74479e+32 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.64668e-10
     logcf(*) end: after 11 iterations.
     logcf(x[27]= 0.84375, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 2718.12 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.0218736
     it= 2: ==> B2= 440109 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00440022
     it= 3: ==> B2= 1.26644e+08 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000856838
     it= 4: ==> B2= 5.68225e+10 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000164362
     it= 5: ==> B2= 3.66244e+13 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.12964e-05
     it= 6: ==> B2= 3.20598e+16 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 5.93505e-06
     it= 7: ==> B2= 3.65864e+19 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.12276e-06
     it= 8: ==> B2= 5.27587e+22 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.12056e-07
     it= 9: ==> B2= 9.37997e+25 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 4.00067e-08
     it=10: ==> B2= 2.01557e+29 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 7.54162e-09
     it=11: ==> B2= 5.14924e+32 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.42081e-09
     it=12: ==> B2= 1.54257e+36 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.67552e-10
     logcf(*) end: after 12 iterations.
     logcf(x[28]= 0.875, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 2450 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.0302147
     it= 2: ==> B2= 372006 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00742718
     it= 3: ==> B2= 1.00485e+08 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00176572
     it= 4: ==> B2= 4.23633e+10 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000412688
     it= 5: ==> B2= 2.56713e+13 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 9.56191e-05
     it= 6: ==> B2= 2.11343e+16 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.20491e-05
     it= 7: ==> B2= 2.26869e+19 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 5.0698e-06
     it= 8: ==> B2= 3.07772e+22 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.16356e-06
     it= 9: ==> B2= 5.14811e+25 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.66706e-07
     it=10: ==> B2= 1.04082e+29 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 6.10778e-08
     it=11: ==> B2= 2.50192e+32 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.39778e-08
     it=12: ==> B2= 7.05238e+35 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.19721e-09
     it=13: ==> B2= 2.30384e+39 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 7.31016e-10
     logcf(*) end: after 13 iterations.
     logcf(x[29]= 0.90625, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 2188.12 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.0421192
     it= 2: ==> B2= 308271 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.0128742
     it= 3: ==> B2= 7.72745e+07 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00381265
     it= 4: ==> B2= 3.02658e+10 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00110791
     it= 5: ==> B2= 1.70531e+13 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000318587
     it= 6: ==> B2= 1.30605e+16 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 9.10705e-05
     it= 7: ==> B2= 1.30466e+19 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.5941e-05
     it= 8: ==> B2= 1.64734e+22 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 7.37247e-06
     it= 9: ==> B2= 2.56499e+25 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.09206e-06
     it=10: ==> B2= 4.82765e+28 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 5.93012e-07
     it=11: ==> B2= 1.08039e+32 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.6796e-07
     it=12: ==> B2= 2.83535e+35 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 4.75428e-08
     it=13: ==> B2= 8.62389e+38 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.34511e-08
     it=14: ==> B2= 3.00926e+42 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.80425e-09
     it=15: ==> B2= 1.19409e+46 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.07559e-09
     it=16: ==> B2= 5.34632e+49 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.04032e-10
     logcf(*) end: after 16 iterations.
     logcf(x[30]= 0.9375, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 1932.5 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.0594391
     it= 2: ==> B2= 248832 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.02317
     it= 3: ==> B2= 5.68734e+07 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00882488
     it= 4: ==> B2= 2.03226e+10 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00329775
     it= 5: ==> B2= 1.04577e+13 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00121712
     it= 6: ==> B2= 7.32086e+15 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000445765
     it= 7: ==> B2= 6.68834e+18 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00016248
     it= 8: ==> B2= 7.72653e+21 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 5.90414e-05
     it= 9: ==> B2= 1.10096e+25 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.14101e-05
     it=10: ==> B2= 1.89662e+28 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 7.7527e-06
     it=11: ==> B2= 3.88536e+31 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.80437e-06
     it=12: ==> B2= 9.33474e+34 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.01363e-06
     it=13: ==> B2= 2.59938e+38 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.6615e-07
     it=14: ==> B2= 8.30457e+41 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.32201e-07
     it=15: ==> B2= 3.01718e+45 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 4.77137e-08
     it=16: ==> B2= 1.23692e+49 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.72154e-08
     it=17: ==> B2= 5.68258e+52 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 6.20986e-09
     it=18: ==> B2= 2.90768e+56 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.23951e-09
     it=19: ==> B2= 1.64796e+60 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 8.07503e-10
     logcf(*) end: after 19 iterations.
     logcf(x[31]= 0.96875, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 1683.12 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.0852619
     it= 2: ==> B2= 193619 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.0440308
     it= 3: ==> B2= 3.91439e+07 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.0227933
     it= 4: ==> B2= 1.23338e+10 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.0116823
     it= 5: ==> B2= 5.59551e+12 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00592272
     it= 6: ==> B2= 3.4562e+15 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00297607
     it= 7: ==> B2= 2.78868e+18 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00148555
     it= 8: ==> B2= 2.84748e+21 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00073801
     it= 9: ==> B2= 3.58854e+24 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000365396
     it=10: ==> B2= 5.4701e+27 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000180472
     it=11: ==> B2= 9.91885e+30 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 8.89791e-05
     it=12: ==> B2= 2.10987e+34 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 4.38122e-05
     it=13: ==> B2= 5.20269e+37 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.15512e-05
     it=14: ==> B2= 1.47211e+41 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.05929e-05
     it=15: ==> B2= 4.73732e+44 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 5.20351e-06
     it=16: ==> B2= 1.72036e+48 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.55486e-06
     it=17: ==> B2= 7.00164e+51 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.25391e-06
     it=18: ==> B2= 3.17394e+55 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 6.15209e-07
     it=19: ==> B2= 1.59374e+59 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.0176e-07
     it=20: ==> B2= 8.8205e+62 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.47979e-07
     it=21: ==> B2= 5.35623e+66 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 7.25526e-08
     it=22: ==> B2= 3.5541e+70 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.55657e-08
     it=23: ==> B2= 2.56724e+74 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.7432e-08
     it=24: ==> B2= 2.01172e+78 Lrg m.B2
     --> crit. |A2*b1 - a1*B2|/|b1*B2| = 8.54292e-09
     it=25: ==> B2= 147221 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 4.18617e-09
     it=26: ==> B2= 1.34508e+09 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.05108e-09
     it=27: ==> B2= 1.32142e+13 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.00487e-09
     it=28: ==> B2= 1.39232e+17 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 4.92273e-10
     logcf(*) end: after 28 iterations.
     >
     > lR. <- logcfR.(x, i=2, d=3, eps=1e-9)
     > lR.t <- logcfR.(x, i=2, d=3, eps=1e-9, trace=TRUE) ; stopifnot(identical(lR.t, lR.))
     logcf(x[], i=2, d=3, eps=1e-09) iterations:
     it= 1: needIt: 30 TRUE, and 1 F.; length(x[<todo>])=30, m.B2= 23307.2
     it= 2: needIt: 30 TRUE; length(x[<todo>])=30, m.B2= 1.01159e+07
     it= 3: needIt: 30 TRUE; length(x[<todo>])=30, m.B2= 7.70605e+09
     it= 4: needIt: 28 TRUE, and 2 F.; length(x[<todo>])=28, m.B2= 1.00852e+13
     it= 5: needIt: 27 TRUE, and 1 F.; length(x[<todo>])=27, m.B2= 1.76419e+16
     it= 6: needIt: 24 TRUE, and 3 F.; length(x[<todo>])=24, m.B2= 4.75316e+19
     it= 7: needIt: 22 TRUE, and 2 F.; length(x[<todo>])=22, m.B2= 1.2798e+23
     it= 8: needIt: 20 TRUE, and 2 F.; length(x[<todo>])=20, m.B2= 3.63581e+26
     it= 9: needIt: 17 TRUE, and 3 F.; length(x[<todo>])=17, m.B2= 6.8674e+29
     it=10: needIt: 13 TRUE, and 4 F.; length(x[<todo>])=13, m.B2= 1.03776e+33
     it=11: needIt: 9 TRUE, and 4 F.; length(x[<todo>])= 9, m.B2= 3.09233e+35
     it=12: needIt: 5 TRUE, and 4 F.; length(x[<todo>])= 5, m.B2= 2.27357e+35
     it=13: needIt: 4 TRUE, and 1 F.; length(x[<todo>])= 4, m.B2= 4.04868e+38
     it=14: needIt: 3 TRUE, and 1 F.; length(x[<todo>])= 3, m.B2= 7.16537e+41
     it=15: needIt: 3 TRUE; length(x[<todo>])= 3, m.B2= 2.57468e+45
     it=16: needIt: 3 TRUE; length(x[<todo>])= 3, m.B2= 1.04393e+49
     it=17: needIt: 2 TRUE, and 1 F.; length(x[<todo>])= 2, m.B2= 1.99468e+52
     it=18: needIt: 2 TRUE; length(x[<todo>])= 2, m.B2= 9.60666e+55
     it=19: needIt: 2 TRUE; length(x[<todo>])= 2, m.B2= 5.12487e+59
     it=20: needIt: 1 TRUE, and 1 F.; length(x[<todo>])= 1, m.B2= 8.8205e+62
     it=21: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 5.35623e+66
     it=22: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 3.5541e+70
     it=23: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 2.56724e+74
     it=24: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 2.01172e+78 Lrg m.B2
     it=25: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 147221
     it=26: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 1.34508e+09
     it=27: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 1.32142e+13
     it=28: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 1.39232e+17
     logcf(*) end: after 28 iterations.
     >
     > all.equal(lC, lR., tol = 0) # TRUE !! (every where ?)
     [1] TRUE
     > all.equal(lR, lR., tol = 0) # TRUE !! " "
     [1] TRUE
     > stopifnot(all.equal(lC, lR., tol = 1e-15))
     > ## (even though they used eps=1e-9 .. i.e., are not *so* accurate)
     > showProc.time()
     Time (user system elapsed): 0.058 0.012 0.164
     >
     > ##--- now with improved logcfR.() {<< will become the new logcfR() at least for MPFR !}:
     >
     > ##require(Rmpfr) may be not, see if NS loading (via "::") is sufficient:
     > requireNamespace("Rmpfr") || quit("no")
     Loading required namespace: Rmpfr
     [1] TRUE
     > ## ----- ----------
     > xM <- Rmpfr::mpfr(x, 512)
     > (ct.14 <- system.time(lR.14 <- logcfR.(xM, i=2, d=3, eps=1e-20, trace=TRUE))) # 0.55 sec
     logcf(x[], i=2, d=3, eps=1e-20) iterations:
     it= 1: needIt: 30 TRUE, and 1 F.; length(x[<todo>])=30, m.B2= 23307.2
     it= 2: needIt: 30 TRUE; length(x[<todo>])=30, m.B2= 1.01159e+07
     it= 3: needIt: 30 TRUE; length(x[<todo>])=30, m.B2= 7.70605e+09
     it= 4: needIt: 30 TRUE; length(x[<todo>])=30, m.B2= 9.10781e+12
     it= 5: needIt: 30 TRUE; length(x[<todo>])=30, m.B2= 1.54287e+16
     it= 6: needIt: 30 TRUE; length(x[<todo>])=30, m.B2= 3.54543e+19
     it= 7: needIt: 30 TRUE; length(x[<todo>])=30, m.B2= 1.06137e+23
     it= 8: needIt: 29 TRUE, and 1 F.; length(x[<todo>])=29, m.B2= 4.19177e+26
     it= 9: needIt: 28 TRUE, and 1 F.; length(x[<todo>])=28, m.B2= 2.26761e+30
     it=10: needIt: 27 TRUE, and 1 F.; length(x[<todo>])=27, m.B2= 1.33011e+34
     it=11: needIt: 27 TRUE; length(x[<todo>])=27, m.B2= 9.0823e+37
     it=12: needIt: 26 TRUE, and 1 F.; length(x[<todo>])=26, m.B2= 7.15387e+41
     it=13: needIt: 25 TRUE, and 1 F.; length(x[<todo>])=25, m.B2= 6.21918e+45
     it=14: needIt: 24 TRUE, and 1 F.; length(x[<todo>])=24, m.B2= 9.51187e+49
     it=15: needIt: 23 TRUE, and 1 F.; length(x[<todo>])=23, m.B2= 1.04428e+54
     it=16: needIt: 22 TRUE, and 1 F.; length(x[<todo>])=22, m.B2= 1.19866e+58
     it=17: needIt: 21 TRUE, and 1 F.; length(x[<todo>])=21, m.B2= 1.40641e+62
     it=18: needIt: 20 TRUE, and 1 F.; length(x[<todo>])=20, m.B2= 1.64566e+66
     it=19: needIt: 19 TRUE, and 1 F.; length(x[<todo>])=19, m.B2= 1.86787e+70
     it=20: needIt: 17 TRUE, and 2 F.; length(x[<todo>])=17, m.B2= 9.5095e+73
     it=21: needIt: 15 TRUE, and 2 F.; length(x[<todo>])=15, m.B2= 2.07684e+78 Lrg m.B2
     it=22: needIt: 14 TRUE, and 1 F.; length(x[<todo>])=14, m.B2= 122830
     it=23: needIt: 11 TRUE, and 3 F.; length(x[<todo>])=11, m.B2= 3.76273e+08
     it=24: needIt: 10 TRUE, and 1 F.; length(x[<todo>])=10, m.B2= 7.77428e+11
     it=25: needIt: 7 TRUE, and 3 F.; length(x[<todo>])= 7, m.B2= 4.17254e+13
     it=26: needIt: 6 TRUE, and 1 F.; length(x[<todo>])= 6, m.B2= 1.55243e+15
     it=27: needIt: 5 TRUE, and 1 F.; length(x[<todo>])= 5, m.B2= 2.47748e+15
     it=28: needIt: 4 TRUE, and 1 F.; length(x[<todo>])= 4, m.B2= 1.06982e+19
     it=29: needIt: 4 TRUE; length(x[<todo>])= 4, m.B2= 1.40477e+23
     it=30: needIt: 4 TRUE; length(x[<todo>])= 4, m.B2= 1.9693e+27
     it=31: needIt: 3 TRUE, and 1 F.; length(x[<todo>])= 3, m.B2= 7.6538e+30
     it=32: needIt: 3 TRUE; length(x[<todo>])= 3, m.B2= 1.16488e+35
     it=33: needIt: 3 TRUE; length(x[<todo>])= 3, m.B2= 1.88175e+39
     it=34: needIt: 3 TRUE; length(x[<todo>])= 3, m.B2= 3.22081e+43
     it=35: needIt: 3 TRUE; length(x[<todo>])= 3, m.B2= 5.83159e+47
     it=36: needIt: 3 TRUE; length(x[<todo>])= 3, m.B2= 1.11521e+52
     it=37: needIt: 2 TRUE, and 1 F.; length(x[<todo>])= 2, m.B2= 3.51533e+55
     it=38: needIt: 2 TRUE; length(x[<todo>])= 2, m.B2= 7.10714e+59
     it=39: needIt: 2 TRUE; length(x[<todo>])= 2, m.B2= 1.51138e+64
     it=40: needIt: 2 TRUE; length(x[<todo>])= 2, m.B2= 3.37644e+68
     it=41: needIt: 2 TRUE; length(x[<todo>])= 2, m.B2= 7.91477e+72
     it=42: needIt: 2 TRUE; length(x[<todo>])= 2, m.B2= 1.94455e+77 Lrg m.B2
     it=43: needIt: 2 TRUE; length(x[<todo>])= 2, m.B2= 43197.4
     it=44: needIt: 2 TRUE; length(x[<todo>])= 2, m.B2= 1.16214e+09
     it=45: needIt: 1 TRUE, and 1 F.; length(x[<todo>])= 1, m.B2= 2.11103e+12
     it=46: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 5.83147e+16
     it=47: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 1.68004e+21
     it=48: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 5.04365e+25
     it=49: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 1.57649e+30
     it=50: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 5.12638e+34
     it=51: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 1.7329e+39
     it=52: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 6.08495e+43
     it=53: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 2.21796e+48
     it=54: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 8.38622e+52
     it=55: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 3.28706e+57
     it=56: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 1.33476e+62
     it=57: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 5.61156e+66
     it=58: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 2.44114e+71
     it=59: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 1.0982e+76
     it=60: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 5.10636e+80 Lrg m.B2
     it=61: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 2.1182e+08
     it=62: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 1.05047e+13
     it=63: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 5.37602e+17
     logcf(*) end: after 63 iterations.
     user system elapsed
     2.517 0.310 8.716
     > (ct14 <- system.time(lR14 <- logcfR (xM, i=2, d=3, eps=1e-20, trace=TRUE))) # 4 sec
     logcf(x[ 1]= -5, i=2, d=3, eps=1e-20) logcf(*) end: after 26 iterations.
     logcf(x[ 2]= -4.75, i=2, d=3, eps=1e-20) logcf(*) end: after 25 iterations.
     logcf(x[ 3]= -4.5, i=2, d=3, eps=1e-20) logcf(*) end: after 24 iterations.
     logcf(x[ 4]= -4.25, i=2, d=3, eps=1e-20) logcf(*) end: after 24 iterations.
     logcf(x[ 5]= -4, i=2, d=3, eps=1e-20) logcf(*) end: after 23 iterations.
     logcf(x[ 6]= -3.75, i=2, d=3, eps=1e-20) logcf(*) end: after 22 iterations.
     logcf(x[ 7]= -3.5, i=2, d=3, eps=1e-20) logcf(*) end: after 22 iterations.
     logcf(x[ 8]= -3.25, i=2, d=3, eps=1e-20) logcf(*) end: after 21 iterations.
     logcf(x[ 9]= -3, i=2, d=3, eps=1e-20) logcf(*) end: after 20 iterations.
     logcf(x[10]= -2.75, i=2, d=3, eps=1e-20) logcf(*) end: after 19 iterations.
     logcf(x[11]= -2.5, i=2, d=3, eps=1e-20) logcf(*) end: after 19 iterations.
     logcf(x[12]= -2.25, i=2, d=3, eps=1e-20) logcf(*) end: after 18 iterations.
     logcf(x[13]= -2, i=2, d=3, eps=1e-20) logcf(*) end: after 17 iterations.
     logcf(x[14]= -1.75, i=2, d=3, eps=1e-20) logcf(*) end: after 16 iterations.
     logcf(x[15]= -1.5, i=2, d=3, eps=1e-20) logcf(*) end: after 15 iterations.
     logcf(x[16]= -1.25, i=2, d=3, eps=1e-20) logcf(*) end: after 14 iterations.
     logcf(x[17]= -1, i=2, d=3, eps=1e-20) logcf(*) end: after 12 iterations.
     logcf(x[18]= -0.75, i=2, d=3, eps=1e-20) logcf(*) end: after 11 iterations.
     logcf(x[19]= -0.5, i=2, d=3, eps=1e-20) logcf(*) end: after 9 iterations.
     logcf(x[20]= -0.25, i=2, d=3, eps=1e-20) logcf(*) end: after 7 iterations.
     logcf(x[21]= 0, i=2, d=3, eps=1e-20) logcf(*) end: after 0 iterations.
     logcf(x[22]= 0.25, i=2, d=3, eps=1e-20) logcf(*) end: after 8 iterations.
     logcf(x[23]= 0.5, i=2, d=3, eps=1e-20) logcf(*) end: after 13 iterations.
     logcf(x[24]= 0.75, i=2, d=3, eps=1e-20) logcf(*) end: after 20 iterations.
     logcf(x[25]= 0.78125, i=2, d=3, eps=1e-20) logcf(*) end: after 22 iterations.
     logcf(x[26]= 0.8125, i=2, d=3, eps=1e-20) logcf(*) end: after 24 iterations.
     logcf(x[27]= 0.84375, i=2, d=3, eps=1e-20) logcf(*) end: after 27 iterations.
     logcf(x[28]= 0.875, i=2, d=3, eps=1e-20) logcf(*) end: after 30 iterations.
     logcf(x[29]= 0.90625, i=2, d=3, eps=1e-20) logcf(*) end: after 36 iterations.
     logcf(x[30]= 0.9375, i=2, d=3, eps=1e-20) logcf(*) end: after 44 iterations.
     logcf(x[31]= 0.96875, i=2, d=3, eps=1e-20) logcf(*) end: after 63 iterations.
     user system elapsed
     17.126 0.073 45.202
     >
     > all.equal(lR.14, lR14, tol=0) # TRUE
     [1] TRUE
     > identical(lR.14, lR14) # TRUE !! (not sure if on all platforms!)
     [1] TRUE
     >
     > SS <- function(ch, digits=7)
     + sub(paste0("([0-9]{1,",digits,"})[0-9]*e"), "\\1e", ch)
     > ## double prec <--> MPFR: vvvv (same eps)
     > lR.9 <- logcfR.(xM, 2,3, eps=1e-9)
     > ## show:
     > SS(Rmpfr::all.equal(Rmpfr::roundMpfr(lR.9, 64), lR, tol=0))# .. 5.1138e-16
     Error in target == current : comparison of these types is not implemented
     Calls: SS ... <Anonymous> -> <Anonymous> -> .local -> all.equal.numeric
     Execution halted
    Running the tests in ‘tests/qgamma-ex.R’ failed.
    Complete output:
     > library(DPQ)
     >
     > ###---> Automatically find places where qgamma() is not so precise (PR#2214) :
     > ### For PR#2214, had '1e-8' below and found quite a bit
     > ## see /u/maechler/R/MM/NUMERICS/dpq-functions/beta-gamma-etc/qgamma-ex.R ..
     >
     > ## FIXME: Timing ! --- partly these matplot() partly get quite slow ~?
     > source(system.file(package="Matrix", "test-tools-1.R", mustWork=TRUE))
     Loading required package: tools
     > ##--> showProc.time(), assertError(), relErrV(), ...
     > showProc.time()
     Time (user system elapsed): 1.938 0.067 3.292
     >
     > (doExtras <- DPQ:::doExtras())
     [1] FALSE
     > (sdir <- system.file("safe", package="DPQ")) ## save directory (to read from)
     [1] "/data/gannet/ripley/R/packages/tests-devel/DPQ.Rcheck/DPQ/safe"
     >
     > ### Nowadays finds cases in a special region for really small p and cutoff 1e-11 :
     > set.seed(47)
     > n <- if(doExtras) 100 else 32
     > res <- cbind(p=1,df=1,rE=1)[-1,]
     > for(M in 1:(if(doExtras) 20 else 10))
     + for(p in runif(n)) for(df in rlnorm(n)) {
     + r <- 1- pchisq(qchisq(p, df),df)/p
     + if(abs(r) > 1e-11) res <- rbind(res, c(p,df,r))
     + }
     >
     > ### use df in U[0,1]: finds two cases with bound 1e-11
     > for(p in runif(n)/2) for(df in runif(n)) {
     + qq <- qchisq(p, df)
     + if(qq > 0 && p > 0) {
     + r <- 1- pchisq(qq, df) / p
     + if(abs(r) > 1e-11) res <- rbind(res, c(p,df,r))
     + }
     + }
     >
     > ### now df very close to 0 : ==> finds more cases
     > for(p in sort(c(runif(64)/2, exp(-(1+rlnorm(256))))))
     + for(df in 2^-rlnorm(256, mean=2, sdlog=1.5)) {
     + qq <- qchisq(p, df)
     + if(qq > 0 && p > 0) {
     + r <- 1- pchisq(qq, df) / p
     + if(abs(r) > 1e-11) res <- rbind(res, c(p,df,r))
     + }
     + }
     > showProc.time()
     Time (user system elapsed): 1.134 0.118 3.239
     >
     > require(graphics)
     > if(!dev.interactive(orNone=TRUE)) pdf("qgamma-appr.pdf")
     > eaxis <- sfsmisc::eaxis
     >
     > showProc.time()
     Time (user system elapsed): 0.083 0.004 0.283
     > ## if(nrow(res) > 0) {
     > cat("Found inaccurate examples where pchisq(qchisq(p, df),df) != p\n")
     Found inaccurate examples where pchisq(qchisq(p, df),df) != p
     > ## sort in p, then df:
     > res <- res[order(res[,"p"], res[,"df"]), ]
     > rE <- res[,"rE"]
     > if(nrow(res) > 20) { hist(rE, breaks = 30); rug(rE) }
     > plot(res[,1:2])##--> quite interesting : all along one curve
     > ## p <= 1/2 and df <= 1 (about) !!
     > res <- cbind(res, nDig = round(-log10(abs(rE)), 1))
     > print(res, digits=12)
     p df rE nDig
     [1,] 0.000194375438651 0.02334079639198 -4.05718514340e-08 7.4
     [2,] 0.000605300028912 0.02041606754775 -1.99857908001e-11 10.7
     [3,] 0.001012316063255 0.01855615147677 -2.59145555106e-04 3.6
     [4,] 0.001248285290785 0.01838201076117 -2.84196000067e-10 9.5
     [5,] 0.001682388899865 0.01720736646288 5.53088974600e-04 3.3
     [6,] 0.001746787400790 0.01731189518997 -5.86897839217e-08 7.2
     [7,] 0.002664451237518 0.01599317398629 1.48421342013e-04 3.8
     [8,] 0.002664451237518 0.01618024201222 -3.82806282229e-08 7.4
     [9,] 0.003159421860255 0.01557612780310 -7.92117005632e-06 5.1
     [10,] 0.003159421860255 0.01568183691729 -4.52237520765e-08 7.3
     [11,] 0.004055462418244 0.01493858731306 4.15166391654e-06 5.4
     [12,] 0.004400694140827 0.01459101672970 9.07907026434e-04 3.0
     [13,] 0.004458811277768 0.01457506850867 9.03139988533e-05 4.0
     [14,] 0.004481882165743 0.01468883074316 -3.23309491179e-07 6.5
     [15,] 0.004939609905705 0.01440168350452 -2.81810098879e-06 5.6
     [16,] 0.008824465120182 0.01276352706510 1.21107345756e-04 3.9
     [17,] 0.009040265960535 0.01273711629661 1.38964402733e-05 4.9
     [18,] 0.010839089634828 0.01242499920422 2.63413624246e-10 9.6
     [19,] 0.011642124851282 0.01201471267173 1.44956234150e-04 3.8
     [20,] 0.014753716559535 0.01155624353203 1.52962087441e-10 9.8
     [21,] 0.015499213434879 0.01125420134457 -9.69695930770e-05 4.0
     [22,] 0.015499213434879 0.01135920381800 -9.55739012376e-08 7.0
     [23,] 0.018603016576955 0.01071716109330 1.63971046474e-03 2.8
     [24,] 0.018603016576955 0.01073655493589 2.14388784340e-04 3.7
     [25,] 0.022624242394389 0.01033379525113 -3.37865757594e-09 8.5
     [26,] 0.022624242394389 0.01034206121729 -2.76332994265e-08 7.6
     [27,] 0.023730217356634 0.01016252135853 -1.07732682708e-06 6.0
     [28,] 0.032427027472295 0.00942923095016 5.11205522358e-11 10.3
     [29,] 0.044753525441333 0.00839626444749 1.22224173549e-05 4.9
     [30,] 0.081818424963746 0.00686007746204 8.92777740624e-10 9.0
     [31,] 0.081818424963746 0.00689856335721 2.28502772259e-11 10.6
     [32,] 0.082800309102258 0.00681234719059 4.17997558788e-09 8.4
     [33,] 0.083507718914457 0.00680676700443 9.77167236016e-11 10.0
     [34,] 0.090821658072474 0.00655269761981 -7.16033632386e-09 8.1
     [35,] 0.102294760453517 0.00623563107239 3.69438657444e-09 8.4
     [36,] 0.110869751789691 0.00603268830251 -3.44006823028e-10 9.5
     [37,] 0.123950804624116 0.00571305309327 2.84683721041e-10 9.5
     [38,] 0.127405857731893 0.00562369059572 6.60541454867e-09 8.2
     [39,] 0.135229634154169 0.00540073357520 -2.34762594200e-05 4.6
     [40,] 0.137732279982451 0.00533092076413 2.99285844990e-04 3.5
     [41,] 0.138112917548194 0.00535138710974 -2.05335777981e-06 5.7
     [42,] 0.141100635980184 0.00527305771429 4.31593832968e-05 4.4
     [43,] 0.141100635980184 0.00537073537183 -3.00640179418e-10 9.5
     [44,] 0.142905299416015 0.00523680041306 3.48180824883e-04 3.5
     [45,] 0.145624557210331 0.00526923971034 -1.94501770245e-09 8.7
     [46,] 0.154606872884529 0.00506806894407 -4.59924667240e-07 6.3
     [47,] 0.154606872884529 0.00507366168703 2.72301046933e-07 6.6
     [48,] 0.163535630067488 0.00497650928578 3.39664962823e-11 10.5
     [49,] 0.169741036539408 0.00484181845356 5.31400978776e-09 8.3
     [50,] 0.177327576288650 0.00465956102839 5.53404362603e-05 4.3
     [51,] 0.178169157856761 0.00471949961255 4.79807527043e-10 9.3
     [52,] 0.190094017358772 0.00450373552308 -1.29698447116e-06 5.9
     [53,] 0.190147641510530 0.00453468705710 5.66235636157e-09 8.2
     [54,] 0.200112534472267 0.00442273120514 7.20473680715e-11 10.1
     [55,] 0.201518808589718 0.00439936964342 1.58748569845e-11 10.8
     [56,] 0.201518808589718 0.00439976887947 -9.97182336704e-11 10.0
     [57,] 0.210803673024037 0.00427351441034 -1.70232938856e-10 9.8
     [58,] 0.213058614771766 0.00426179831847 1.10152997834e-11 11.0
     [59,] 0.214780951412088 0.00419869272965 9.79194836326e-09 8.0
     [60,] 0.232805106603566 0.00395399315002 -9.17581020055e-08 7.0
     [61,] 0.249102914025652 0.00380019404026 -1.15818465929e-10 9.9
     [62,] 0.249102914025652 0.00382493512126 -1.39670497390e-11 10.9
     [63,] 0.252076511947811 0.00374903834738 -8.83337205604e-08 7.1
     [64,] 0.253082914021191 0.00375259362798 3.65436092498e-09 8.4
     [65,] 0.253922058700076 0.00371237348323 3.28994798726e-06 5.5
     [66,] 0.254289278570932 0.00374343873151 -1.05664899053e-09 9.0
     [67,] 0.260017499519858 0.00366179605930 2.34859742765e-07 6.6
     [68,] 0.270323906831467 0.00351999192121 -1.56164756277e-04 3.8
     [69,] 0.271699356057456 0.00355068132680 5.13092990317e-09 8.3
     [70,] 0.275516196070002 0.00346804047756 -4.35171547588e-04 3.4
     [71,] 0.280722231049885 0.00348224101220 5.48759926389e-10 9.3
     [72,] 0.284601233201101 0.00344936339590 1.57145851887e-10 9.8
     [73,] 0.290188543054775 0.00336613521112 -5.64443074502e-08 7.2
     [74,] 0.290579022038283 0.00334423496113 1.02667567892e-07 7.0
     [75,] 0.290579022038283 0.00336764858994 2.26061565023e-08 7.6
     [76,] 0.291850198713803 0.00333552811650 -1.27338760580e-06 5.9
     [77,] 0.296521136452775 0.00330308865102 2.25309977453e-07 6.6
     [78,] 0.298034174946132 0.00330462333485 8.42470393447e-09 8.1
     [79,] 0.300556783277253 0.00323922530004 4.66003314391e-05 4.3
     [80,] 0.303182283998467 0.00328704590597 -1.46205270113e-11 10.8
     [81,] 0.322319846303892 0.00306134512927 -1.15130830540e-05 4.9
     [82,] 0.322319846303892 0.00310689001755 8.57751647487e-11 10.1
     [83,] 0.325071272052651 0.00302343293053 -2.47088704493e-04 3.6
     [84,] 0.325071272052651 0.00304146419577 3.18761056051e-06 5.5
     [85,] 0.331888412404218 0.00300837121343 -4.96098895297e-09 8.3
     [86,] 0.362278153188527 0.00278204202032 4.53939330569e-10 9.3
     [87,] 0.385389476781711 0.00260981704384 7.37274796769e-10 9.1
     [88,] 0.425333956955001 0.00232995789362 1.82823025607e-08 7.7
     [89,] 0.439503709203564 0.00222452690840 -4.53585193982e-06 5.3
     [90,] 0.439503709203564 0.00224964327069 -3.02331937263e-10 9.5
     [91,] 0.450804624124430 0.00216770324934 -4.59455036239e-08 7.3
     >
     > if(requireNamespace("scatterplot3d")) {
     + scatterplot3d::scatterplot3d(res[,1:3], type ='h') ## quite interesting:
     + ## the inaccurate (p,df) points are on nice monotone curve !!!
     + ## this is *less* revealing
     + scatterplot3d::scatterplot3d(res[,c("p","df","nDig")], type ='h')
     + }
     Loading required namespace: scatterplot3d
     > rL <- res[abs(res[,'rE']) > 1e-9,]
     > rL <- rL[order(rL[,1],rL[,2]),]
     > rL
     p df rE nDig
     [1,] 0.0001943754 0.023340796 -4.057185e-08 7.4
     [2,] 0.0010123161 0.018556151 -2.591456e-04 3.6
     [3,] 0.0016823889 0.017207366 5.530890e-04 3.3
     [4,] 0.0017467874 0.017311895 -5.868978e-08 7.2
     [5,] 0.0026644512 0.015993174 1.484213e-04 3.8
     [6,] 0.0026644512 0.016180242 -3.828063e-08 7.4
     [7,] 0.0031594219 0.015576128 -7.921170e-06 5.1
     [8,] 0.0031594219 0.015681837 -4.522375e-08 7.3
     [9,] 0.0040554624 0.014938587 4.151664e-06 5.4
     [10,] 0.0044006941 0.014591017 9.079070e-04 3.0
     [11,] 0.0044588113 0.014575069 9.031400e-05 4.0
     [12,] 0.0044818822 0.014688831 -3.233095e-07 6.5
     [13,] 0.0049396099 0.014401684 -2.818101e-06 5.6
     [14,] 0.0088244651 0.012763527 1.211073e-04 3.9
     [15,] 0.0090402660 0.012737116 1.389644e-05 4.9
     [16,] 0.0116421249 0.012014713 1.449562e-04 3.8
     [17,] 0.0154992134 0.011254201 -9.696959e-05 4.0
     [18,] 0.0154992134 0.011359204 -9.557390e-08 7.0
     [19,] 0.0186030166 0.010717161 1.639710e-03 2.8
     [20,] 0.0186030166 0.010736555 2.143888e-04 3.7
     [21,] 0.0226242424 0.010333795 -3.378658e-09 8.5
     [22,] 0.0226242424 0.010342061 -2.763330e-08 7.6
     [23,] 0.0237302174 0.010162521 -1.077327e-06 6.0
     [24,] 0.0447535254 0.008396264 1.222242e-05 4.9
     [25,] 0.0828003091 0.006812347 4.179976e-09 8.4
     [26,] 0.0908216581 0.006552698 -7.160336e-09 8.1
     [27,] 0.1022947605 0.006235631 3.694387e-09 8.4
     [28,] 0.1274058577 0.005623691 6.605415e-09 8.2
     [29,] 0.1352296342 0.005400734 -2.347626e-05 4.6
     [30,] 0.1377322800 0.005330921 2.992858e-04 3.5
     [31,] 0.1381129175 0.005351387 -2.053358e-06 5.7
     [32,] 0.1411006360 0.005273058 4.315938e-05 4.4
     [33,] 0.1429052994 0.005236800 3.481808e-04 3.5
     [34,] 0.1456245572 0.005269240 -1.945018e-09 8.7
     [35,] 0.1546068729 0.005068069 -4.599247e-07 6.3
     [36,] 0.1546068729 0.005073662 2.723010e-07 6.6
     [37,] 0.1697410365 0.004841818 5.314010e-09 8.3
     [38,] 0.1773275763 0.004659561 5.534044e-05 4.3
     [39,] 0.1900940174 0.004503736 -1.296984e-06 5.9
     [40,] 0.1901476415 0.004534687 5.662356e-09 8.2
     [41,] 0.2147809514 0.004198693 9.791948e-09 8.0
     [42,] 0.2328051066 0.003953993 -9.175810e-08 7.0
     [43,] 0.2520765119 0.003749038 -8.833372e-08 7.1
     [44,] 0.2530829140 0.003752594 3.654361e-09 8.4
     [45,] 0.2539220587 0.003712373 3.289948e-06 5.5
     [46,] 0.2542892786 0.003743439 -1.056649e-09 9.0
     [47,] 0.2600174995 0.003661796 2.348597e-07 6.6
     [48,] 0.2703239068 0.003519992 -1.561648e-04 3.8
     [49,] 0.2716993561 0.003550681 5.130930e-09 8.3
     [50,] 0.2755161961 0.003468040 -4.351715e-04 3.4
     [51,] 0.2901885431 0.003366135 -5.644431e-08 7.2
     [52,] 0.2905790220 0.003344235 1.026676e-07 7.0
     [53,] 0.2905790220 0.003367649 2.260616e-08 7.6
     [54,] 0.2918501987 0.003335528 -1.273388e-06 5.9
     [55,] 0.2965211365 0.003303089 2.253100e-07 6.6
     [56,] 0.2980341749 0.003304623 8.424704e-09 8.1
     [57,] 0.3005567833 0.003239225 4.660033e-05 4.3
     [58,] 0.3223198463 0.003061345 -1.151308e-05 4.9
     [59,] 0.3250712721 0.003023433 -2.470887e-04 3.6
     [60,] 0.3250712721 0.003041464 3.187611e-06 5.5
     [61,] 0.3318884124 0.003008371 -4.960989e-09 8.3
     [62,] 0.4253339570 0.002329958 1.828230e-08 7.7
     [63,] 0.4395037092 0.002224527 -4.535852e-06 5.3
     [64,] 0.4508046241 0.002167703 -4.594550e-08 7.3
     > plot(rL[,1:2], type = "b", main = "inaccurate pchisq/qchisq pairs")
     >
     > plot(rL[,1:2], type = "b", log = "x", ylim = range(0, rL[,"df"]),
     + xaxt = "n",
     + main = "inaccurate pchisq/qchisq pairs"); abline(h = 0, lty=2)
     > ## aha -- a perfect line !!
     > lines(res[,1:2], col = adjustcolor(1, 0.5))
     > eaxis(1); axis(1, at = 1/2)
     >
     > d <- as.data.frame(res)
     > plot (df ~ log(p), data = d, type = "b", cex=1/4, col="gray")
     > points(df ~ log(p), data = as.data.frame(rL), col=2, cex = 1/2)
     >
     > summary(fm <- lm (df ~ log(p), data = d, weights = -log(abs(rE))))
    
     Call:
     lm(formula = df ~ log(p), data = d, weights = -log(abs(rE)))
    
     Weighted Residuals:
     Min 1Q Median 3Q Max
     -6.924e-04 -1.443e-04 -2.096e-05 7.786e-05 1.079e-03
    
     Coefficients:
     Estimate Std. Error t value Pr(>|t|)
     (Intercept) 5.168e-06 1.149e-05 0.45 0.654
     log(p) -2.725e-03 3.683e-06 -739.99 <2e-16 ***
     ---
     Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
    
     Residual standard error: 0.0002575 on 89 degrees of freedom
     Multiple R-squared: 0.9998, Adjusted R-squared: 0.9998
     F-statistic: 5.476e+05 on 1 and 89 DF, p-value: < 2.2e-16
    
     > ## R^2 = 0.9998
     >
     > p0 <- 2^seq(-50,-1, by=1/8)
     > dN <- data.frame(p = p0,
     + df = predict(fm, newdata = data.frame(p = p0)))
     > rE <- with(dN, 1- pchisq(qchisq(p, df),df)/p)
     > dN <- cbind(dN, rE = rE, nDig = round(-log10(abs(rE)), 1))
     > print(dN, digits=10)
     p df rE nDig
     1 8.881784197e-16 0.094454797738 -6.103206185e-07 6.2
     2 9.685654347e-16 0.094218673664 -2.417772682e-07 6.6
     3 1.056228096e-15 0.093982549590 1.482101845e-07 6.8
     4 1.151824906e-15 0.093746425517 5.596452101e-07 6.3
     5 1.256073967e-15 0.093510301443 9.925312783e-07 6.0
     6 1.369758374e-15 0.093274177369 -5.456117484e-07 6.3
     7 1.493732098e-15 0.093038053295 -6.476825187e-08 7.2
     8 1.628926404e-15 0.092801929221 4.375367337e-07 6.4
     9 1.776356839e-15 0.092565805147 9.613066765e-07 6.0
     10 1.937130869e-15 0.092329681073 -4.656782389e-07 6.3
     11 2.112456192e-15 0.092093557000 1.060769096e-07 7.0
     12 2.303649813e-15 0.091857432926 6.993074607e-07 6.2
     13 2.512147934e-15 0.091621308852 -6.429918680e-07 6.2
     14 2.739516748e-15 0.091385184778 -1.755328105e-09 8.8
     15 2.987464197e-15 0.091149060704 6.609670441e-07 6.2
     16 3.257852808e-15 0.090912936630 -5.966163394e-07 6.2
     17 3.552713679e-15 0.090676812556 1.141328021e-07 6.9
     18 3.874261739e-15 0.090440688483 8.463782111e-07 6.1
     19 4.224912384e-15 0.090204564409 -3.264588970e-07 6.5
     20 4.607299625e-15 0.089968440335 4.538340435e-07 6.3
     21 5.024295868e-15 0.089732316261 -6.607811160e-07 6.2
     22 5.479033495e-15 0.089496192187 1.675731415e-07 6.8
     23 5.974928394e-15 0.089260068113 -8.888069507e-07 6.1
     24 6.515705616e-15 0.089023944039 -1.237758629e-08 7.9
     25 7.105427358e-15 0.088787819965 8.855723791e-07 6.1
     26 7.748523477e-15 0.088551695892 -8.599123835e-08 7.1
     27 8.449824769e-15 0.088315571818 8.600545690e-07 6.1
     28 9.214599250e-15 0.088079447744 -5.324092944e-08 7.3
     29 1.004859174e-14 0.087843323670 -9.348371277e-07 6.0
     30 1.095806699e-14 0.087607199596 8.590019329e-08 7.1
     31 1.194985679e-14 0.087371075522 -7.374085143e-07 6.1
     32 1.303141123e-14 0.087134951448 3.314589819e-07 6.5
     33 1.421085472e-14 0.086898827375 -4.335491499e-07 6.4
     34 1.549704695e-14 0.086662703301 6.834622716e-07 6.2
     35 1.689964954e-14 0.086426579227 -2.323222548e-08 7.6
     36 1.842919850e-14 0.086190455153 -6.982056668e-07 6.2
     37 2.009718347e-14 0.085954331079 4.935690683e-07 6.3
     38 2.191613398e-14 0.085718207005 -1.230714122e-07 6.9
     39 2.389971358e-14 0.085482082931 -7.079816475e-07 6.1
     40 2.606282246e-14 0.085245958858 5.585870368e-07 6.3
     41 2.842170943e-14 0.085009834784 3.202904830e-08 7.5
     42 3.099409391e-14 0.084773710710 -4.627895094e-07 6.3
     43 3.379929908e-14 0.084537586636 8.786037373e-07 6.1
     44 3.685839700e-14 0.084301462562 4.421566921e-07 6.4
     45 4.019436694e-14 0.084065338488 3.745824395e-08 7.4
     46 4.383226796e-14 0.083829214414 -3.354886757e-07 6.5
     47 4.779942715e-14 0.083593090340 -6.766811509e-07 6.2
     48 5.212564492e-14 0.083356966267 7.928489877e-07 6.1
     49 5.684341886e-14 0.083120842193 5.100597127e-07 6.3
     50 6.198818782e-14 0.082884718119 2.590340281e-07 6.6
     51 6.759859815e-14 0.082648594045 3.977485741e-08 7.4
     52 7.371679400e-14 0.082412469971 -1.477148934e-07 6.8
     53 8.038873388e-14 0.082176345897 -3.034323055e-07 6.5
     54 8.766453592e-14 0.081940221823 -4.273744663e-07 6.4
     55 9.559885430e-14 0.081704097750 -5.195384647e-07 6.3
     56 1.042512898e-13 0.081467973676 -5.799213925e-07 6.2
     57 1.136868377e-13 0.081231849602 -6.085203379e-07 6.2
     58 1.239763756e-13 0.080995725528 -6.053324058e-07 6.2
     59 1.351971963e-13 0.080759601454 -5.703546833e-07 6.2
     60 1.474335880e-13 0.080523477380 -5.035842794e-07 6.3
     61 1.607774678e-13 0.080287353306 -4.050182891e-07 6.4
     62 1.753290718e-13 0.080051229233 -2.746538159e-07 6.6
     63 1.911977086e-13 0.079815105159 -1.124879638e-07 6.9
     64 2.085025797e-13 0.079578981085 8.148216124e-08 7.1
     65 2.273736754e-13 0.079342857011 3.072594555e-07 6.5
     66 2.479527513e-13 0.079106732937 5.648467992e-07 6.2
     67 2.703943926e-13 0.078870608863 -8.276082903e-07 6.1
     68 2.948671760e-13 0.078634484789 -5.012849573e-07 6.3
     69 3.215549355e-13 0.078398360715 -1.431432968e-07 6.8
     70 3.506581437e-13 0.078162236642 2.468195778e-07 6.6
     71 3.823954172e-13 0.077926112568 6.686065366e-07 6.2
     72 4.170051594e-13 0.077689988494 -5.340348010e-07 6.3
     73 4.547473509e-13 0.077453864420 -4.348594862e-08 7.4
     74 4.959055026e-13 0.077217740346 4.788952396e-07 6.3
     75 5.407887852e-13 0.076981616272 -6.077622940e-07 6.2
     76 5.897343520e-13 0.076745492198 -1.660412186e-08 7.8
     77 6.431098711e-13 0.076509368125 6.063946030e-07 6.2
     78 7.013162874e-13 0.076273244051 -3.642605502e-07 6.4
     79 7.647908344e-13 0.076037119977 3.275302165e-07 6.5
     80 8.340103188e-13 0.075800995903 -5.640570402e-07 6.2
     81 9.094947018e-13 0.075564871829 1.965354515e-07 6.7
     82 9.918110051e-13 0.075328747755 -6.159766142e-07 6.2
     83 1.081577570e-12 0.075092623681 2.134272701e-07 6.7
     84 1.179468704e-12 0.074856499608 -5.200023394e-07 6.3
     85 1.286219742e-12 0.074620375534 3.782225700e-07 6.4
     86 1.402632575e-12 0.074384251460 -2.761173419e-07 6.6
     87 1.529581669e-12 0.074148127386 6.909382196e-07 6.2
     88 1.668020638e-12 0.073912003312 1.156952232e-07 6.9
     89 1.818989404e-12 0.073675879238 -4.174157540e-07 6.4
     90 1.983622010e-12 0.073439755164 6.554521572e-07 6.2
     91 2.163155141e-12 0.073203631090 2.014481738e-07 6.7
     92 2.358937408e-12 0.072967507017 -2.104198829e-07 6.7
     93 2.572439484e-12 0.072731382943 -5.801509786e-07 6.2
     94 2.805265149e-12 0.072495258869 6.355293285e-07 6.2
     95 3.059163338e-12 0.072259134795 3.449181070e-07 6.5
     96 3.336041275e-12 0.072023010721 9.644772814e-08 7.0
     97 3.637978807e-12 0.071786886647 -1.098808029e-07 7.0
     98 3.967244020e-12 0.071550762573 -2.740664691e-07 6.6
     99 4.326310282e-12 0.071314638500 -3.961082631e-07 6.4
     100 4.717874816e-12 0.071078514426 -4.760051799e-07 6.3
     101 5.144878969e-12 0.070842390352 -5.137562191e-07 6.3
     102 5.610530299e-12 0.070606266278 -5.093603785e-07 6.3
     103 6.118326675e-12 0.070370142204 -4.628166750e-07 6.3
     104 6.672082550e-12 0.070134018130 -3.741241081e-07 6.4
     105 7.275957614e-12 0.069897894056 -2.432817023e-07 6.6
     106 7.934488041e-12 0.069661769983 -7.028846949e-08 7.2
     107 8.652620563e-12 0.069425645909 1.448565687e-07 6.8
     108 9.435749632e-12 0.069189521835 4.021543881e-07 6.4
     109 1.028975794e-11 0.068953397761 7.016059603e-07 6.2
     110 1.122106060e-11 0.068717273687 -4.176447372e-07 6.4
     111 1.223665335e-11 0.068481149613 -2.873865945e-08 7.5
     112 1.334416510e-11 0.068245025539 4.023232466e-07 6.4
     113 1.455191523e-11 0.068008901466 -5.698258618e-07 6.2
     114 1.586897608e-11 0.067772777392 -4.930799502e-08 7.3
     115 1.730524113e-11 0.067536653318 5.133677458e-07 6.3
     116 1.887149926e-11 0.067300529244 -3.116849310e-07 6.5
     117 2.057951587e-11 0.067064405170 3.404481462e-07 6.5
     118 2.244212120e-11 0.066828281096 -3.848111123e-07 6.4
     119 2.447330670e-11 0.066592157022 3.567795747e-07 6.4
     120 2.668833020e-11 0.066356032948 -2.686901401e-07 6.6
     121 2.910383046e-11 0.066119908875 5.623583754e-07 6.2
     122 3.173795216e-11 0.065883784801 3.667429294e-08 7.4
     123 3.461048225e-11 0.065647660727 -4.365232460e-07 6.4
     124 3.774299853e-11 0.065411536653 5.312784427e-07 6.3
     125 4.115903175e-11 0.065175412579 1.578594558e-07 6.8
     126 4.488424239e-11 0.064939288505 -1.630783120e-07 6.8
     127 4.894661340e-11 0.064703164431 -4.315370583e-07 6.4
     128 5.337666040e-11 0.064467040358 -6.475189818e-07 6.2
     129 5.820766091e-11 0.064230916284 5.516171563e-07 6.3
     130 6.347590433e-11 0.063994792210 4.354002830e-07 6.4
     131 6.922096451e-11 0.063758668136 3.716548337e-07 6.4
     132 7.548599706e-11 0.063522544062 3.603785849e-07 6.4
     133 8.231806350e-11 0.063286419988 4.015693077e-07 6.4
     134 8.976848478e-11 0.063050295914 4.952247737e-07 6.3
     135 9.789322680e-11 0.062814171841 6.413427337e-07 6.2
     136 1.067533208e-10 0.062578047767 -4.864750427e-07 6.3
     137 1.164153218e-10 0.062341923693 -2.302656055e-07 6.6
     138 1.269518087e-10 0.062105799619 7.839833394e-08 7.1
     139 1.384419290e-10 0.061869675545 4.395145157e-07 6.4
     140 1.509719941e-10 0.061633551471 -4.525763320e-07 6.3
     141 1.646361270e-10 0.061397427397 1.860554089e-08 7.7
     142 1.795369696e-10 0.061161303323 5.422315954e-07 6.3
     143 1.957864536e-10 0.060925179250 -1.718042137e-07 6.8
     144 2.135066416e-10 0.060689055176 4.618673999e-07 6.3
     145 2.328306437e-10 0.060452931102 -1.317493019e-07 6.9
     146 2.539036173e-10 0.060216807028 6.119534922e-07 6.2
     147 2.768838580e-10 0.059980682954 1.387356391e-07 6.9
     148 3.019439882e-10 0.059744558880 -2.716851728e-07 6.6
     149 3.292722540e-10 0.059508434806 -6.193156503e-07 6.2
     150 3.590739391e-10 0.059272310733 3.495618646e-07 6.5
     151 3.915729072e-10 0.059036186659 1.222864269e-07 6.9
     152 4.270132832e-10 0.058800062585 -4.221716110e-08 7.4
     153 4.656612873e-10 0.058563938511 -1.439556299e-07 6.8
     154 5.078072346e-10 0.058327814437 -1.829357232e-07 6.7
     155 5.537677160e-10 0.058091690363 -1.591641836e-07 6.8
     156 6.038879765e-10 0.057855566289 -7.264775914e-08 7.1
     157 6.585445080e-10 0.057619442216 7.660678825e-08 7.1
     158 7.181478783e-10 0.057383318142 2.885926981e-07 6.5
     159 7.831458144e-10 0.057147194068 5.633032045e-07 6.2
     160 8.540265665e-10 0.056911069994 -3.010137886e-07 6.5
     161 9.313225746e-10 0.056674945920 1.043142313e-07 7.0
     162 1.015614469e-09 0.056438821846 5.723448235e-07 6.2
     163 1.107535432e-09 0.056202697772 -8.306453525e-08 7.1
     164 1.207775953e-09 0.055966573698 5.155342088e-07 6.3
     165 1.317089016e-09 0.055730449625 1.091064128e-09 9.0
     166 1.436295757e-09 0.055494325551 -4.402707552e-07 6.4
     167 1.566291629e-09 0.055258201477 3.567050869e-07 6.4
     168 1.708053133e-09 0.055022077403 5.624478649e-08 7.2
     169 1.862645149e-09 0.054785953329 -1.711696178e-07 6.8
     170 2.031228938e-09 0.054549829255 -3.255506589e-07 6.5
     171 2.215070864e-09 0.054313705181 -4.069108679e-07 6.4
     172 2.415551906e-09 0.054077581108 -4.152627815e-07 6.4
     173 2.634178032e-09 0.053841457034 -3.506189497e-07 6.5
     174 2.872591513e-09 0.053605332960 -2.129919214e-07 6.7
     175 3.132583258e-09 0.053369208886 -2.394249909e-09 8.6
     176 3.416106266e-09 0.053133084812 2.811614974e-07 6.6
     177 3.725290298e-09 0.052896960738 -4.754652609e-07 6.3
     178 4.062457877e-09 0.052660836664 -4.082860428e-08 7.4
     179 4.430141728e-09 0.052424712591 4.667262948e-07 6.3
     180 4.831103812e-09 0.052188588517 -5.029516048e-08 7.3
     181 5.268356064e-09 0.051952464443 -4.839870040e-07 6.3
     182 5.745183026e-09 0.051716340369 2.526344739e-07 6.6
     183 6.265166516e-09 0.051480216295 -1.969243901e-08 7.7
     184 6.832212532e-09 0.051244092221 -2.087459541e-07 6.7
     185 7.450580597e-09 0.051007968147 -3.145456657e-07 6.5
     186 8.124915754e-09 0.050771844073 -3.371111803e-07 6.5
     187 8.860283457e-09 0.050535720000 -2.764621003e-07 6.6
     188 9.662207623e-09 0.050299595926 -1.326180317e-07 6.9
     189 1.053671213e-08 0.050063471852 9.440140747e-08 7.0
     190 1.149036605e-08 0.049827347778 4.045766026e-07 6.4
     191 1.253033303e-08 0.049591223704 -2.420871066e-07 6.6
     192 1.366442506e-08 0.049355099630 2.395533774e-07 6.6
     193 1.490116119e-08 0.049118975556 -2.252220010e-07 6.6
     194 1.624983151e-08 0.048882851483 4.277948780e-07 6.4
     195 1.772056691e-08 0.048646727409 1.448078844e-07 6.8
     196 1.932441525e-08 0.048410603335 -4.470068493e-08 7.3
     197 2.107342426e-08 0.048174479261 -1.407587547e-07 6.9
     198 2.298073210e-08 0.047938355187 -1.433942325e-07 6.8
     199 2.506066606e-08 0.047702231113 -5.263503899e-08 7.3
     200 2.732885013e-08 0.047466107039 1.314909133e-07 6.9
     201 2.980232239e-08 0.047229982966 4.089557111e-07 6.4
     202 3.249966302e-08 0.046993858892 -2.026429911e-07 6.7
     203 3.544113383e-08 0.046757734818 2.666381839e-07 6.6
     204 3.864883049e-08 0.046521610744 -1.427213570e-07 6.8
     205 4.214684851e-08 0.046285486670 -4.483846767e-07 6.3
     206 4.596146421e-08 0.046049362596 3.109955485e-07 6.5
     207 5.012133212e-08 0.045813238522 2.073633435e-07 6.7
     208 5.465770025e-08 0.045577114448 2.073183537e-07 6.7
     209 5.960464478e-08 0.045340990375 3.108231207e-07 6.5
     210 6.499932603e-08 0.045104866301 -4.225693764e-07 6.4
     211 7.088226765e-08 0.044868742227 -1.068421438e-07 7.0
     212 7.729766099e-08 0.044632618153 3.123190967e-07 6.5
     213 8.429369702e-08 0.044396494079 -8.979926758e-08 7.0
     214 9.192292842e-08 0.044160370005 -3.780712474e-07 6.4
     215 1.002426642e-07 0.043924245931 3.616102325e-07 6.4
     216 1.093154005e-07 0.043688121858 2.956315699e-07 6.5
     217 1.192092896e-07 0.043451997784 3.433583751e-07 6.5
     218 1.299986521e-07 0.043215873710 -3.936604087e-07 6.4
     219 1.417645353e-07 0.042979749636 -1.134241445e-07 6.9
     220 1.545953220e-07 0.042743625562 2.803691416e-07 6.6
     221 1.685873940e-07 0.042507501488 -9.497759912e-08 7.0
     222 1.838458568e-07 0.042271377414 -3.463650020e-07 6.5
     223 2.004853285e-07 0.042035253341 3.982654885e-07 6.4
     224 2.186308010e-07 0.041799129267 3.893573700e-07 6.4
     225 2.384185791e-07 0.041563005193 -3.573736593e-07 6.4
     226 2.599973041e-07 0.041326881119 -1.135260472e-07 6.9
     227 2.835290706e-07 0.041090757045 2.539798655e-07 6.6
     228 3.091906439e-07 0.040854632971 -1.007498764e-07 7.0
     229 3.371747881e-07 0.040618508897 -3.214330988e-07 6.5
     230 3.676917137e-07 0.040382384823 -4.081432583e-07 6.4
     231 4.009706570e-07 0.040146260750 -3.609537691e-07 6.4
     232 4.372616020e-07 0.039910136676 -1.799379994e-07 6.7
     233 4.768371582e-07 0.039674012602 1.348307247e-07 6.9
     234 5.199946082e-07 0.039437888528 -2.309643412e-07 6.6
     235 5.670581412e-07 0.039201764454 3.563334678e-07 6.4
     236 6.183812879e-07 0.038965640380 2.734302492e-07 6.6
     237 6.743495762e-07 0.038729516306 3.344816584e-07 6.5
     238 7.353834273e-07 0.038493392233 -2.537711326e-07 6.6
     239 8.019413140e-07 0.038257268159 1.001817561e-07 7.0
     240 8.745232040e-07 0.038021144085 -1.848126916e-07 6.7
     241 9.536743164e-07 0.037785020011 -3.156868371e-07 6.5
     242 1.039989216e-06 0.037548895937 -2.925440488e-07 6.5
     243 1.134116282e-06 0.037312771863 -1.154876221e-07 6.9
     244 1.236762576e-06 0.037076647789 2.153792392e-07 6.7
     245 1.348699152e-06 0.036840523716 -5.632338751e-08 7.2
     246 1.470766855e-06 0.036604399642 -1.638943248e-07 6.8
     247 1.603882628e-06 0.036368275568 -1.074536702e-07 7.0
     248 1.749046408e-06 0.036132151494 1.128785959e-07 6.9
     249 1.907348633e-06 0.035896027420 -2.381848818e-07 6.6
     250 2.079978433e-06 0.035659903346 3.148260044e-07 6.5
     251 2.268232565e-06 0.035423779272 3.067288504e-07 6.5
     252 2.473525152e-06 0.035187655198 -2.467805615e-07 6.6
     253 2.697398305e-06 0.034951531125 9.809279378e-08 7.0
     254 2.941533709e-06 0.034715407051 -9.217531383e-08 7.0
     255 3.207765256e-06 0.034479282977 -9.840820714e-08 7.0
     256 3.498092816e-06 0.034243158903 7.923717715e-08 7.1
     257 3.814697266e-06 0.034007034829 -2.523280838e-07 6.6
     258 4.159956866e-06 0.033770910755 2.978638209e-07 6.5
     259 4.536465130e-06 0.033534786681 -3.332980778e-07 6.5
     260 4.947050303e-06 0.033298662608 -8.305711496e-08 7.1
     261 5.394796609e-06 0.033062538534 -3.110326594e-07 6.5
     262 5.883067419e-06 0.032826414460 3.314533775e-07 6.5
     263 6.415530512e-06 0.032590290386 -1.553354063e-07 6.8
     264 6.996185632e-06 0.032354166312 2.279407690e-07 6.6
     265 7.629394531e-06 0.032118042238 1.638951825e-07 6.8
     266 8.319913732e-06 0.031881918164 3.135417774e-07 6.5
     267 9.072930260e-06 0.031645794091 3.656237890e-08 7.4
     268 9.894100606e-06 0.031409670017 -1.661173443e-08 7.8
     269 1.078959322e-05 0.031173545943 1.537749293e-07 6.8
     270 1.176613484e-05 0.030937421869 -7.675076596e-08 7.1
     271 1.283106102e-05 0.030701297795 -7.365044086e-08 7.1
     272 1.399237126e-05 0.030465173721 1.628068312e-07 6.8
     273 1.525878906e-05 0.030229049647 2.398761112e-08 7.6
     274 1.663982746e-05 0.029992925573 1.285369383e-07 6.9
     275 1.814586052e-05 0.029756801500 -1.216179157e-07 6.9
     276 1.978820121e-05 0.029520677426 -1.184344594e-07 6.9
     277 2.157918644e-05 0.029284553352 1.377655910e-07 6.9
     278 2.353226967e-05 0.029048429278 6.475006253e-08 7.2
     279 2.566212205e-05 0.028812305204 2.546562226e-07 6.6
     280 2.798474253e-05 0.028576181130 1.358057637e-07 6.9
     281 3.051757812e-05 0.028340057056 -2.762893581e-07 6.6
     282 3.327965493e-05 0.028103932983 1.553088241e-07 6.8
     283 3.629172104e-05 0.027867808909 -2.519743056e-07 6.6
     284 3.957640242e-05 0.027631684835 1.836182478e-07 6.7
     285 4.315837288e-05 0.027395560761 -1.887623498e-07 6.7
     286 4.706453935e-05 0.027159436687 -2.586992773e-07 6.6
     287 5.132424410e-05 0.026923312613 -2.666510612e-08 7.6
     288 5.596948506e-05 0.026687188539 -2.210357075e-08 7.7
     289 6.103515625e-05 0.026451064466 -2.296375117e-07 6.6
     290 6.655930986e-05 0.026214940392 -1.155410669e-07 6.9
     291 7.258344208e-05 0.025978816318 -1.934344240e-07 6.7
     292 7.915280485e-05 0.025742692244 5.977598905e-08 7.2
     293 8.631674575e-05 0.025506568170 1.410127927e-07 6.9
     294 9.412907870e-05 0.025270444096 6.554035215e-08 7.2
     295 1.026484882e-04 0.025034320022 -1.513968153e-07 6.8
     296 1.119389701e-04 0.024798195948 -7.984736650e-09 8.1
     297 1.220703125e-04 0.024562071875 1.377293413e-08 7.9
     298 1.331186197e-04 0.024325947801 -7.096807053e-08 7.1
     299 1.451668842e-04 0.024089823727 2.236347612e-07 6.7
     300 1.583056097e-04 0.023853699653 -3.406038118e-08 7.5
     301 1.726334915e-04 0.023617575579 1.071465702e-07 7.0
     302 1.882581574e-04 0.023381451505 1.915567694e-07 6.7
     303 2.052969764e-04 0.023145327431 -2.153880145e-07 6.7
     304 2.238779402e-04 0.022909203358 -1.943811789e-07 6.7
     305 2.441406250e-04 0.022673079284 -1.853585263e-07 6.7
     306 2.662372394e-04 0.022436955210 -1.734718758e-07 6.8
     307 2.903337683e-04 0.022200831136 -1.439218900e-07 6.8
     308 3.166112194e-04 0.021964707062 -8.196069534e-08 7.1
     309 3.452669830e-04 0.021728582988 2.710548919e-08 7.6
     310 3.765163148e-04 0.021492458914 1.979137831e-07 6.7
     311 4.105939528e-04 0.021256334841 3.784053582e-08 7.4
     312 4.477558805e-04 0.021020210767 -2.082512274e-08 7.7
     313 4.882812500e-04 0.020784086693 3.635452028e-08 7.4
     314 5.324744788e-04 0.020547962619 -1.675743422e-07 6.8
     315 5.806675366e-04 0.020311838545 1.696075650e-07 6.8
     316 6.332224388e-04 0.020075714471 -9.599203343e-08 7.0
     317 6.905339660e-04 0.019839590397 -1.676104155e-07 6.8
     318 7.530326296e-04 0.019603466323 -3.122730075e-08 7.5
     319 8.211879055e-04 0.019367342250 -3.779823499e-08 7.4
     320 8.955117609e-04 0.019131218176 -1.576477535e-07 6.8
     321 9.765625000e-04 0.018895094102 -6.902106886e-09 8.2
     322 1.064948958e-03 0.018658970028 7.899459131e-08 7.1
     323 1.161335073e-03 0.018422845954 1.293463390e-07 6.9
     324 1.266444878e-03 0.018186721880 -1.651599366e-07 6.8
     325 1.381067932e-03 0.017950597806 -9.328012252e-08 7.0
     326 1.506065259e-03 0.017714473733 3.008480842e-08 7.5
     327 1.642375811e-03 0.017478349659 -8.903183435e-08 7.1
     328 1.791023522e-03 0.017242225585 -8.893825099e-08 7.1
     329 1.953125000e-03 0.017006101511 5.866337671e-08 7.2
     330 2.129897915e-03 0.016769977437 7.502565913e-08 7.1
     331 2.322670146e-03 0.016533853363 3.812254845e-09 8.4
     332 2.532889755e-03 0.016297729289 -1.115639656e-07 7.0
     333 2.762135864e-03 0.016061605216 6.307704159e-08 7.2
     334 3.012130518e-03 0.015825481142 -1.675984129e-08 7.8
     335 3.284751622e-03 0.015589357068 -1.223428781e-08 7.9
     336 3.582047044e-03 0.015353232994 1.188536293e-07 6.9
     337 3.906250000e-03 0.015117108920 -1.217006735e-07 6.9
     338 4.259795831e-03 0.014880984846 -1.314145703e-07 6.9
     339 4.645340293e-03 0.014644860772 -1.288061258e-07 6.9
     340 5.065779510e-03 0.014408736698 -5.759581589e-08 7.2
     341 5.524271728e-03 0.014172612625 -1.110605841e-07 7.0
     342 6.024261037e-03 0.013936488551 2.552848555e-08 7.6
     343 6.569503244e-03 0.013700364477 -7.038977068e-08 7.2
     344 7.164094088e-03 0.013464240403 -8.014783059e-08 7.1
     345 7.812500000e-03 0.013228116329 6.514376105e-08 7.2
     346 8.519591661e-03 0.012991992255 -1.226700341e-08 7.9
     347 9.290680586e-03 0.012755868181 3.829306428e-09 8.4
     348 1.013155902e-02 0.012519744108 -1.720011800e-08 7.8
     349 1.104854346e-02 0.012283620034 2.105846508e-08 7.7
     350 1.204852207e-02 0.012047495960 1.170782316e-08 7.9
     351 1.313900649e-02 0.011811371886 6.422403376e-08 7.2
     352 1.432818818e-02 0.011575247812 9.486125396e-08 7.0
     353 1.562500000e-02 0.011339123738 3.894984368e-08 7.4
     354 1.703918332e-02 0.011102999664 3.208461985e-08 7.5
     355 1.858136117e-02 0.010866875591 3.155170891e-08 7.5
     356 2.026311804e-02 0.010630751517 1.297326291e-08 7.9
     357 2.209708691e-02 0.010394627443 -3.001926907e-08 7.5
     358 2.409704415e-02 0.010158503369 7.470777241e-08 7.1
     359 2.627801298e-02 0.009922379295 2.881776751e-08 7.5
     360 2.865637635e-02 0.009686255221 4.355808669e-08 7.4
     361 3.125000000e-02 0.009450131147 3.486256517e-08 7.5
     362 3.407836665e-02 0.009214007073 -4.540193843e-08 7.3
     363 3.716272234e-02 0.008977883000 6.057274970e-08 7.2
     364 4.052623608e-02 0.008741758926 -4.748039517e-08 7.3
     365 4.419417382e-02 0.008505634852 -4.947995325e-08 7.3
     366 4.819408829e-02 0.008269510778 5.303493644e-09 8.3
     367 5.255602595e-02 0.008033386704 2.750297434e-09 8.6
     368 5.731275270e-02 0.007797262630 7.649789580e-09 8.1
     369 6.250000000e-02 0.007561138556 2.575244529e-08 7.6
     370 6.815673329e-02 0.007325014483 2.588700820e-08 7.6
     371 7.432544469e-02 0.007088890409 -3.873515997e-08 7.4
     372 8.105247217e-02 0.006852766335 -2.868914595e-08 7.5
     373 8.838834765e-02 0.006616642261 8.820984831e-09 8.1
     374 9.638817659e-02 0.006380518187 -1.249980452e-08 7.9
     375 1.051120519e-01 0.006144394113 -2.542298283e-08 7.6
     376 1.146255054e-01 0.005908270039 -3.411814475e-08 7.5
     377 1.250000000e-01 0.005672145966 1.097271574e-08 8.0
     378 1.363134666e-01 0.005436021892 -5.883856513e-09 8.2
     379 1.486508894e-01 0.005199897818 -1.778869496e-08 7.7
     380 1.621049443e-01 0.004963773744 -2.477514438e-08 7.6
     381 1.767766953e-01 0.004727649670 -2.316977810e-08 7.6
     382 1.927763532e-01 0.004491525596 -1.391213433e-08 7.9
     383 2.102241038e-01 0.004255401522 4.032074008e-09 8.4
     384 2.292510108e-01 0.004019277448 1.322844057e-10 9.9
     385 2.500000000e-01 0.003783153375 1.667496141e-09 8.8
     386 2.726269332e-01 0.003547029301 -6.843543954e-09 8.2
     387 2.973017788e-01 0.003310905227 -5.830278704e-09 8.2
     388 3.242098887e-01 0.003074781153 -4.568749379e-09 8.3
     389 3.535533906e-01 0.002838657079 -6.372713468e-09 8.2
     390 3.855527064e-01 0.002602533005 4.293532641e-09 8.4
     391 4.204482076e-01 0.002366408931 4.466793713e-09 8.4
     392 4.585020216e-01 0.002130284858 -3.886739153e-09 8.4
     393 5.000000000e-01 0.001894160784 1.495896629e-09 8.8
     >
     > ## } ## only when we find inaccurate regions
     > showProc.time()
     Time (user system elapsed): 0.137 0.026 0.495
     >
     >
     > ## Oops: another qgamma() / qchisq() problem: mostly NaN's == all solved now
     > curve(qgamma(x, 20), 1e-16, 1e-10, log='x')
     > curve(qgamma(x, 20), 1e-300, .99 , log='xy') # and add the critical region from above:
     > abline(v=c(1e-16, 1e-10), col="light blue")
     > curve(qgamma(x, 20), 1e-26, 1e-07, log='x')
     > ##-> now using log=TRUE in same region:
     > curve(qgamma(x, 20, log=TRUE), -38, -16)## no problem!!
     > curve(qgamma(exp(x), 20), add=TRUE, col="green3", n=2001)
     > ## had problem here, but no longer !
     >
     > ##--> Further fix for qgamma: when 'x' is very small: use "log=TRUE of log(x)"!
     >
     > ## had bug (gave NaN), but no longer:
     > (q_12 <- qgamma(1e-12, 20))
     [1] 2.330042
     > all.equal(1e-12, pgamma(q_12, 20), tol=0)# show rel.err (Lnx 64-bit: 4.04e-16)
     [1] "Mean relative difference: 4.038968e-16"
     > stopifnot(
     + all.equal(1e-12, pgamma(q_12, 20), tolerance = 1e-14)
     + )
     >
     >
     > ## --- Nice graphic : --- but amazingly *S..L..O..W*
     >
     > p.qgammaSml <- function(from= 1e-110, to = 1e-5, ylim = c(0.4, 1000),
     + n = 201, k.lab = 3,
     + a1 = c(10, seq(10.1,20, by=.2), 21:105),
     + a2 = seq(110,330, by=10),
     + a3 = seq(350,1600, by=50))
     + {
     + ## Purpose: nice qgamma() lines ``for small x'' aka p
     + ## ----------------------------------------------------------------------
     + ## Author: Martin Maechler, Date: 22 Mar 2004, 14:23
     + x <- exp(seq(log(from), log(to), length = n))
     +
     + op <- par(las=1, lab = c(10,10, 7), xaxs = "i", mex = 0.8)
     + on.exit(par(op))
     + plot(x, qgamma(x, a1[1]), log="xy", ylim=ylim, type='l', xaxt = "n",
     + main = paste("qgamma(x, a) for very small x, a in [",
     + formatC(a1[1]),", ",formatC(max(a1,a2,a3)),"] - log-log", sep=''),
     + sub = R.version.string)
     + lab.x <- pretty(log10(c(from,to)), 20)
     + axis(1, at=10^lab.x, lab = paste("10^",formatC(lab.x),sep=''))
     + if(is.nan(qgamma(1e-12, 20)))
     + text(1e-60, 20, "all NaN", cex = 2)
     + if(!is.finite(qgamma(1e-140, 155)))
     + text(1e-240, 5, "all +Inf", cex = 2)
     +
     + lines.txt <- function(a.s, col = par("col")) {
     + col <- rep(col, length=length(a.s))
     + for(i in seq(along=a.s)) {
     + qx <- qgamma(x, (a <- a.s[i]))
     + if(i %% k.lab == 0 &&
     + any(ifi <- is.finite(qx) & qx >= ylim[1])) {
     + ik <- (i%%(2*k.lab))/k.lab # = 0 or 1
     + j <- quantile(which(ifi), c(.02,(1:3)/4+ ik/10, .98))
     + ## "segments" around the labels :
     + i0 <- 1
     + for(jj in j) {
     + ii <- i0:(jj-1)
     + i2 <- jj + -1:1
     + lines(x[ii], qx[ii], col=col[i])
     + lines(x[i2], qx[i2], col=col[i], type = 'c')
     + i0 <- jj+1
     + }
     + text(x[j], qx[j], formatC(a), col= "gray40", cex = 0.8)
     + }
     + else
     + lines(x, qx, col=col[i])
     +
     + }
     + }
     + oo <- options(warn = -1)
     + lines.txt(a1[-1])
     + lines.txt(a2, col= 2)
     + lines.txt(a3, col= rainbow(length(a3), .8, .8,
     + start = (max(a3)-min(a3))/(1+max(a3))))
     + invisible(options(oo))
     + }
     >
     > showProc.time()
     Time (user system elapsed): 0.037 0.001 0.113
     >
     > p.qgammaSml()
     > p.qgammaSml(1e-300)
     > p.qgammaSml(1e-300,1e-50, a2= seq(100,360, by=4), a3=seq(350,1500, by=10))
     >
     > showProc.time()
     Time (user system elapsed): 1.939 0.023 6.268
     >
     > ## The "upper" problematic corner:
     > p.qgammaSml(1e-19, 1e-3, a2=NULL,a3=NULL, ylim=c(.1,20))
     > p.qgammaSml(1e-19, 1e-3, a2=seq(1,12, by=.04), ylim=c(.1,20),a3=NULL,k.lab=10)
     > ## now shows the problem (quite well):
     > ## could it be in pgamma()'s inaccuracy, leading to qgamma() bias ?
     > aa <- c(seq(9, 22, by=0.25),seq(22.3,40,by=0.4))
     > caa <- formatC(range(aa))
     > sfsmisc::mult.fig(2)
     > curve(pgamma(x, sh=aa[1]), 0.5, 20, log = 'xy', ylim = c(1e-60, .2),
     + main = sprintf("pgamma(x, a) for a in [%s,%s]", caa[1],caa[2]))
     > for(sh in aa) curve(pgamma(x, sh), add = TRUE, col=2)
     > abline(h=c(1e-15), col="light blue", lty=2)
     >
     > curve(pgamma(x, sh=aa[1]), 0.5, 20, log = 'xy', ylim = c(1e-15, .8),
     + main = sprintf("pgamma(x, a) for a in [%s,%s]", caa[1],caa[2]))
     > for(sh in aa) curve(pgamma(x, sh), add = TRUE, col=2)
     > ## the "border curve" between "Pearson" and "Continued fraction (upper tail)"
     > ## in pgamma.c :
     > curve(pgamma(max(1,x), x), add = TRUE, col=4)
     > ## ==> pgamma() is perfect here {series expansion up to eps_C accuracy}!
     >
     > aa <- c(seq(9, 22, by=0.25),seq(22.3,40.4,by=0.4))
     > p.qgammaSml(1e-24, 1e-5, a1=aa, a2=NULL,a3=NULL, ylim=c(.8,8))
     > ## -------- save the above?
     > aa1 <- c(aa,seq(40.5,90, by=0.5))
     > p.qgammaSml(1e-60, 1e-5, a1=aa1, a2=NULL,a3=NULL, ylim=c(.9, 16))
     > aa2 <- c(aa1, seq(91,150, by= 1))
     > p.qgammaSml(1e-90, 1e-5, a1=aa2, a2=NULL,a3=NULL, ylim=c(.9, 35))
     > aa3 <- c(aa2, seq(150,250, by= 2), seq(253, 400, by=5))
     > p.qgammaSml(1e-200, 1e-5, a1=aa3, a2=NULL,a3=NULL, ylim=c(.9, 100))
     > p.qgammaSml(1e-200, 1e-5, a1=aa3, a2=NULL,a3=NULL, ylim=c(.9, 200),k.lab=9e9)
     > p.qgammaSml(1e-60, 1e-5, a1=aa3, a2=NULL,a3=NULL, ylim=c(.9, 200),k.lab=9e9)
     >
     > showProc.time()
     Time (user system elapsed): 5.188 0.039 13.072
     >
     > ## lower a \> 10
     >
     > curve(qgamma(x, 19), 1e-14, 1e-9, log='x')
     > curve(qgamma(x, 18), 1e-14, 1e-9, log='x')
     > curve(qgamma(x, 15), 1e-11, 5e-9, log='x')
     > curve(qgamma(x, 13), 5e-10, 1e-8, log='x')
     > curve(qgamma(x, 11), 1e-8, 5e-8, log='x')
     > curve(qgamma(x, 10.5), 4.2e-8, 6e-8, log='x')
     > curve(qgamma(x, 10.3), 6e-8, 7e-8, log='x')
     > curve(qgamma(x, 10.2), 7.1e-8, 7.6e-8, log='x')
     > curve(qgamma(x, 10.15),7.7e-8, 7.9e-8, log='x')
     > curve(qgamma(x, 10.14),7.88e-8,7.92e-8, log='x',n=10001)
     >
     > ## no more problems for smaller a!! here:
     > curve(qgamma(x, 10.13), 1e-10, 5e-4, log='x',n=20001)
     > curve(qgamma(x, 10.12), 1e-10, 5e-4, log='x',n=20001)
     > curve(qgamma(x, 10.1), 1e-10, 5e-4, log='x',n=20001)
     >
     > showProc.time()
     Time (user system elapsed): 0.805 0.015 2.032
     >
     > ##--- the "+Inf" / premature "0" case:
     > curve(qgamma(x, 155, log=TRUE), -1500, 0, log='y', n=2001,col=2)
     > curve(qgamma(x, 1e3, log=TRUE), -1500, 0, log='y', n=2001,col=2)
     > ## now works, but slowly and with kink
     > curve(qgamma (x, 1e5, log=TRUE), -3e5, 0, log='y', n=2001,col=2,lwd=3)
     > curve(qgammaAppr(x, 1e5, log=TRUE), add = TRUE, n=2001, col="blue",lwd=.4)
     > ## --- curves are almost "identical"
     > ## ===> the kink *does* come from the initial approx... hmm
     >
     > ## still "identical"
     > curve(qgamma (x, 1e4, log=TRUE), -3e4, 0, log='y', n=2001,col=2)
     > curve(qgammaAppr(x, 1e4, log=TRUE), add = TRUE, n=2001, col="tomato3")
     >
     > ## now see some difference (approx. has kink at ~ -165)
     > curve(qgamma (x, 100, log=TRUE), -200, 0, log='y', n=2001,col=2)
     > curve(qgammaAppr(x, 100, log=TRUE), add = TRUE, n=2001, col="tomato3")
     > ##
     > (kk <- 100 * 2/1.24)# 161.29
     [1] 161.2903
     > curve(qgamma (x, 100, log=TRUE), -1.1*kk, -.95*kk, log='y', n=2001,col=2)
     > curve(qgammaAppr(x, 100, log=TRUE), add = TRUE, n=2001, col="tomato3")
     > abline(v = -kk, col='blue', lty=2)# exactly: kink is at a * 2 / 1.24 = a / .62
     > curve(qgammaAppr(x - 100/.62, 100,log=TRUE), -1e-3, +1e-3)
     >
     > showProc.time()
     Time (user system elapsed): 0.19 0.003 0.44
     >
     > p.qgammaLog <- function(alpha, xl.f = 1.5, xr.f = 0.4, n = 2001)
     + {
     + ## Purpose:
     + ## ----------------------------------------------------------------------
     + ## Arguments:
     + ## ----------------------------------------------------------------------
     + ## Author: Martin Maechler, Date: 30 Mar 2004, 18:44
     + kk <- -alpha / .62 # = (alpha * 2) / (-1.24)
     + curve(qgamma(x, alpha, log=TRUE), xl.f*kk, xr.f*kk, log='y',
     + n=n, col=2, lwd=3.6, lty = 4,
     + main= paste("qgamma(x, alpha=",formatC(alpha,digits=10),", log = TRUE)"))
     + lines(kk, qgamma(kk, alpha, log=TRUE), type = 'h', lty = 3)
     + curve(qgamma (exp(x), alpha), add = TRUE, col="orange", n=n, lwd= 2)
     + curve(qgammaAppr(x, alpha, log=TRUE), add = TRUE, col=3, n=n,lwd = .4)
     + }
     >
     > showProc.time()
     Time (user system elapsed): 0.001 0 0.015
     >
     > p.qgammaLog(25)
     > p.qgammaLog(16)# ~ [-25, -20]
     > p.qgammaLog(12, 1.2, 0.8)# small problem remaining
     > p.qgammaLog(11, 1.2, 0.8)# even smaller
     > p.qgammaLog(10.5, 1.1, 0.9)# even smaller
     > p.qgammaLog(10.25, 1.1, 0.9)# even smaller
     > ## 2019-08: __nothing__ visible from here on:
     > p.qgammaLog(10.18, 1.02, 0.98)# even smaller
     > p.qgammaLog(10.15, 1.02, 0.98)# even smaller
     > p.qgammaLog(10.14, 1.001, 0.999)# even smaller
     > p.qgammaLog(10.139, 1.0002, 0.9998)#
     > p.qgammaLog(10.138, 1.0002, 0.9998)#
     > p.qgammaLog(10.137, 1.00001, 0.99999)#
     > p.qgammaLog(10.13699, 1.0000001, 0.9999999)#
     > p.qgammaLog(10.1369899, 1.0000001, 0.9999999)#
     > p.qgammaLog(10.1369894, 1.0000001, 0.9999999)#
     > p.qgammaLog(10.1369893, 1.0000001, 0.9999999)# even smaller at -16.34998
     >
     > showProc.time()
     Time (user system elapsed): 0.684 0.021 1.924
     >
     > ##-- here is the boundary --- for 64-bit AMD Opteron ---
     > ## and for 32-bit AMD Athlon
     >
     > p.qgammaLog(10.1369892, 1.0000001, 0.9999999)# no more
     > p.qgammaLog(10.136989, 1.0000001, 0.9999999)#
     > p.qgammaLog(10.136988, 1.0000001, 0.9999999)#
     > p.qgammaLog(10.136985, 1.0000001, 0.9999999)#
     > p.qgammaLog(10.13698, 1.0000001, 0.9999999)#
     > p.qgammaLog(10.13697, 1.0000001, 0.9999999)#
     > p.qgammaLog(10.13695, 1.0000001, 0.9999999)#
     > p.qgammaLog(10.1368, 1.000001, 0.999999)#
     > p.qgammaLog(10.1365, 1.000001, 0.999999)#
     > p.qgammaLog(10.136, 1.000001, 0.999999)#
     > p.qgammaLog(10.125, 1.1, 0.9)# --- see it now
     > p.qgammaLog(10, 1.2, 0.8)
     > p.qgammaLog(9)
     >
     > showProc.time()
     Time (user system elapsed): 0.541 0.012 1.677
     >
     > ## For large alpha: show difference to see problem better
     > ## ---> for alpha >= 10, the x problem starts *roughly* at x = -0.8*alpha
     > ##
     >
     > sfsmisc::mult.fig(2)
     > curve(qgammaAppr(x, 5, log=TRUE), - 8.1, -8, n=2001)
     > curve(qgammaAppr(x- 5/.62, 5, log=TRUE), -1e-15, 0)
     >
     > ## is the kink from pgamma() ? : no: this looks fine,
     > curve(pgamma(x, 1e5, log=TRUE), 1, 2e5, log='x', n=2001,col=2)
     > ## and this does too:
     > curve( dgamma(x, 1e5), .5e5, 2e5); par(new=TRUE)
     > curve( dgamma(x, 1e5, log=TRUE), .5e5, 2e5, col=2, yaxt="n")
     > axis(4,col.axis=2); par(new=TRUE)
     > curve( pgamma(x, 1e5), .5e5, 2e5, n=2001, col=3); par(new=TRUE)
     > curve( pgamma(x, 1e5, log=TRUE), .5e5, 2e5, n=2001, col=4); par(new=TRUE)
     > curve(-pgamma(x, 1e5, log=TRUE,lower=FALSE), .5e5, 2e5, n=2001, col=4)
     > ## all looking nice
     >
     >
     > x <- 10^seq(2,6, length=4001)
     > qx <- qgamma(pgamma(x, 1e5, log=TRUE), 1e5, log=TRUE)
     > plot(x, qx, type ='l', col=2, asp = 1); abline(0,1, lty=3)
     >
     > showProc.time()
     Time (user system elapsed): 0.098 0.002 0.334
     > <0c>
     > ###------------- Approximations of qgamma() ------
     > ##
     >
     > ## source("/u/maechler/R/MM/NUMERICS/dpq-functions/qchisqAppr.R")
     > ##--> qchisqAppr()
     > ##--> qchisqWH [ = Wilson Hilferty ]
     > ##--> qchisqKG [ = Kennedy & Gentle's improvements "a la AS 91" ]
     > ## dyn.load("/u/maechler/R/MM/NUMERICS/dpq-functions/qchisq_appr.so")
     >
     > ## Consider the two different implementations of
     > ## lgamma1p(a) := lgamma(1+a) == log(gamma(1+a) == log(a*gamma(a)) "stable":
     >
     > if(!exists("lseq", mode="function"))
     + lseq <- if(requireNamespace("sfsmisc")) sfsmisc::lseq else
     + function(from, to, length) exp(seq(log(from), log(to), length.out = length))
     >
     > if(require("Rmpfr")) { ##---------------- MPFR numbers -------------------------
     +
     + .mpfr.all.eq <- Rmpfr::all.equal
     + AllEq <- function(target, current, ...)
     + .mpfr.all.eq(target, current, ...,
     + formatFUN = function(x, ...) Rmpfr::format(x, digits = 9))
     +
     + print(gammaE <- Const("gamma",200)); pi. <- Const("pi",200)
     + print(a0 <- (gammaE^2 + pi.^2/6)/2)
     + print(psi2.1 <- -2*zeta(mpfr(3,200)))# == psigamma(1,2) =~ -2.4041138
     + print(a1 <- (psi2.1 - gammaE*(pi.^2/2 + gammaE^2))/6)
     +
     + x <- lseq(1e-30, 0.8, length = if(doExtras) 1000 else 125)
     + x. <- mpfr(x, 200)
     + xct. <- log(x. * gamma(x.)) ## using MPFR arithmetic .. no overflow ...
     + xc2. <- log(x.) + lgamma(x.)## (ditto)
     + print(AllEq(xct., xc2., tol = 0)) # 3.15779......e-57
     + xct <- as.numeric(xct.)
     + stopifnot(exprs = {
     + AllEq(xct., xc2., tol = 1e-45)
     + AllEq(xct , xc2., tol = 1e-15)
     + ##
     + all.equal(lgamma1p(x), lgamma1p(x, tol= 1e-16), tol=0)
     + ## -> no difference; i.e., default tol = 1e-14 seems fine enough!
     + })
     + showProc.time()
     +
     + m.appr <- cbind(log(x*gamma(x)), lgamma(1+x), log(x) + lgamma(x),
     + lgamma1p.(x, k=1, cut=3e-6),
     + lgamma1p.(x, k=2, cut=1e-4),
     + lgamma1p.(x, k=3, cut=8e-4),
     + lgamma1p(x))#, tol= 1e-14), # = default
     +
     + eMat <- m.appr - xct # absolute error
     + ## Relative errors:
     + str(reMat. <- m.appr/xct. - 1)
     + str(reMat <- as(reMat., "array")) # as(., "matrix") fails in older versions
     +
     + matplot(x, eMat , log="x", type="l", lty=1) #-> problematic log(x) + lgamma(x) for "large"
     + matplot(x, abs( eMat), log="xy", type="l", lty=1) #-> but good for small; lgamma1p is much better
     + matplot(x, abs(reMat), log="xy", type="l", lty=1)
     + abline(v= 3.47548562941137e-08, col = "gray80", lwd=3)#<- the cutoff value of lgamma1p()
     + ##---> should use earlier cutoff!
     + ## zoom in:
     +
     + matplot(x, abs(reMat), log="xy", type="l", col=1:7, lty=1,
     + lwd=2, xlim=c(8e-9, 1e-3), ylim = c(1e-18, 1e-7), axes=FALSE, frame=TRUE,
     + main = expression(lgamma1p(x) == log(Gamma(x+1)) ~~~ "approximations"
     + ~~~ abs(rel.Err(.))))
     + eaxis(1); eaxis(2)
     + abline(v= 3.47548562941137e-08, col = "gray80", lwd=3)#<- the cutoff value of lgamma1p()
     + abline(h= c(1,2,4)*.Machine$double.eps, lty=3, col="skyblue")
     + legend("topright", col=1:7, lty=1,lwd=2,
     + c("log(x*gamma(x))", "lgamma(1+x)", "log(x) + lgamma(x)",
     + "lgamma1p.(x, k=1, c=3e-6)",
     + "lgamma1p.(x, k=2, c=1e-4)",
     + "lgamma1p.(x, k=3, c=8e-4)",
     + "lgamma1p(x)"), bty="n", ncol=2)
     + abline(v = c(3e-6, 1e-4, 8e-4), col=4:6, lty=2, lwd=1/2)
     +
     + ## FIXME: do the same for the lgaamma1p_series()
     +
     + ## rm(x., xct., xc2., reMat., eMat, AllEq)
     + detach("package:Rmpfr")
     + showProc.time()
     +
     + } ## if( MPFR ) ----------------------------------------------------------------
     Loading required package: Rmpfr
     Loading required package: gmp
    
     Attaching package: 'gmp'
    
     The following objects are masked from 'package:base':
    
     %*%, apply, crossprod, matrix, tcrossprod
    
     C code of R package 'Rmpfr': GMP using 64 bits per limb
    
    
     Attaching package: 'Rmpfr'
    
     The following object is masked from 'package:gmp':
    
     outer
    
     The following object is masked from 'package:DPQ':
    
     log1mexp
    
     The following objects are masked from 'package:stats':
    
     dbinom, dgamma, dnbinom, dnorm, dpois, dt, pnorm
    
     The following objects are masked from 'package:base':
    
     cbind, pmax, pmin, rbind
    
     1 'mpfr' number of precision 200 bits
     [1] 0.57721566490153286060651209008240243104215933593992359880576723
     1 'mpfr' number of precision 200 bits
     [1] 0.98905599532797255539539565150063470793918352072821409044319567
     1 'mpfr' number of precision 200 bits
     [1] -2.404113806319188570799476323022899981529972584680997763584544
     1 'mpfr' number of precision 200 bits
     [1] -0.90747907608088628901656016735627511492861144907256376094133062
     Error in target == current : comparison of these types is not implemented
     Calls: print ... .mpfr.all.eq -> .mpfr.all.eq -> .local -> all.equal.numeric
     Execution halted
    Running the tests in ‘tests/stirlerr-tst.R’ failed.
    Complete output:
     > #### Testing stirlerr(), bd0(), ebd0(), dpois_raw(), ...
     > #### ===============================================
     >
     > require(DPQ)
     Loading required package: DPQ
     > for(pkg in c("Rmpfr", "DPQmpfr"))
     + if(!requireNamespace(pkg)) {
     + cat("no CRAN package", sQuote(pkg), " ---> no tests here.\n")
     + q("no")
     + }
     Loading required namespace: Rmpfr
     Loading required namespace: DPQmpfr
     > require("Rmpfr")
     Loading required package: Rmpfr
     Loading required package: gmp
    
     Attaching package: 'gmp'
    
     The following objects are masked from 'package:base':
    
     %*%, apply, crossprod, matrix, tcrossprod
    
     C code of R package 'Rmpfr': GMP using 64 bits per limb
    
    
     Attaching package: 'Rmpfr'
    
     The following object is masked from 'package:gmp':
    
     outer
    
     The following object is masked from 'package:DPQ':
    
     log1mexp
    
     The following objects are masked from 'package:stats':
    
     dbinom, dgamma, dnbinom, dnorm, dpois, dt, pnorm
    
     The following objects are masked from 'package:base':
    
     cbind, pmax, pmin, rbind
    
     >
     > cutoffs <- c(15,35,80,500) # cut points, n=*, in the above "algorithm"
     > ##
     > n <- c(seq(1,15, by=1/4),seq(16, 25, by=1/2), 26:30, seq(32,50, by=2), seq(55,1000, by=5),
     + 20*c(51:99), 50*(40:80), 150*(27:48), 500*(15:20))
     > st.n <- stirlerr(n)# rather use.halves=TRUE, just here , use.halves=FALSE)
     > plot(st.n ~ n, log="xy", type="b") ## looks good now
     > nM <- mpfr(n, 2048)
     > st.nM <- stirlerr(nM, use.halves=FALSE) ## << on purpose
     > all.equal(asNumeric(st.nM), st.n)# TRUE
     [1] TRUE
     > all.equal(st.nM, as(st.n,"mpfr"))# .. difference: 1.05884..............................e-15
     Error in target == current : comparison of these types is not implemented
     Calls: all.equal -> all.equal -> .local -> all.equal.numeric
     Execution halted
Flavor: r-devel-linux-x86_64-fedora-gcc

Version: 0.5-0
Check: examples
Result: ERROR
    Running examples in 'DPQ-Ex.R' failed
    The error most likely occurred in:
    
    > ### Name: ppoisson
    > ### Title: Direct Computation of 'ppois()' Poisson Distribution
    > ### Probabilities
    > ### Aliases: ppoisErr ppoisD
    > ### Keywords: distribution
    >
    > ### ** Examples
    >
    > (lams <- outer(c(1,2,5), 10^(0:3)))# 10^4 is already slow!
     [,1] [,2] [,3] [,4]
    [1,] 1 10 100 1000
    [2,] 2 20 200 2000
    [3,] 5 50 500 5000
    > system.time(e1 <- sapply(lams, ppoisErr))
     user system elapsed
     0.01 0.00 0.02
    > e1 / .Machine$double.eps
     [1] 0.0 0.5 -1.0 1.0 5.5 1.5 -4.0 -3.0 1.0 -1.0 2.0 2.0
    >
    > ## Try another 'ppFUN' :---------------------------------
    > ## this relies on the fact that it's *only* used on an 'x' of the form 0:M :
    > ppD0 <- function(x, lambda, all.from.0=TRUE)
    + cumsum(dpois(if(all.from.0) 0:x else x, lambda=lambda))
    > ## and test it:
    > p0 <- ppD0 ( 1000, lambda=10)
    > p1 <- ppois(0:1000, lambda=10)
    > stopifnot(all.equal(p0,p1, tol=8*.Machine$double.eps))
    >
    > system.time(p0.slow <- ppoisD(0:1000, lambda=10, all.from.0=FALSE))# not very slow, here
     user system elapsed
     0.01 0.00 0.02
    > p0.1 <- ppoisD(1000, lambda=10)
    > if(requireNamespace("Rmpfr")) {
    + ppoisMpfr <- function(x, lambda) cumsum(Rmpfr::dpois(x, lambda=lambda))
    + p0.best <- ppoisMpfr(0:1000, lambda = Rmpfr::mpfr(10, precBits = 256))
    + AllEq. <- Rmpfr::all.equal
    + AllEq <- function(target, current, ...)
    + AllEq.(target, current, ...,
    + formatFUN = function(x, ...) Rmpfr::format(x, digits = 9))
    + print(AllEq(p0.best, p0, tol = 0)) # 2.06e-18
    + print(AllEq(p0.best, p0.slow, tol = 0)) # the "worst" (4.44e-17)
    + print(AllEq(p0.best, p0.1, tol = 0)) # 1.08e-18
    + }
    Error in target == current : comparison of these types is not implemented
    Calls: print ... AllEq -> AllEq. -> AllEq. -> .local -> all.equal.numeric
    Execution halted
Flavor: r-devel-windows-x86_64-new-TK

Version: 0.5-0
Check: tests
Result: ERROR
     Running 'chisq-nonc-ex.R'
     Running 'dnbinom-tst.R'
    Running the tests in 'tests/dnbinom-tst.R' failed.
    Complete output:
     > #### Testing 1) dbinom_raw(), dnbinomR() and dnbinom.mu()
     > #### 2) log1pmx(), logcf() etc
     > require(DPQ)
     Loading required package: DPQ
     > source(system.file(package="Matrix", "test-tools-1.R", mustWork=TRUE))
     Loading required package: tools
     > ## -> showProc.time(), assertError()
     >
     > (doExtras <- DPQ:::doExtras() && !grepl("valgrind", R.home()))
     [1] FALSE
     >
     > if(!dev.interactive(orNone=TRUE)) pdf("wienergerm-accuracy.pdf")
     >
     >
     > ### 1. Testing dbinom_raw(), dnbinomR() and dnbinom.mu() >>> ../R/dbinom-nbinom.R <<<
     > ### ---------- ../man/dbinom_raw.Rd & ../man/dnbinomR.Rd
     >
     > ## "FIXME:" use sfsmisc :: relErrV() already here
     >
     > ### dbinom() vs dbinom.raw() :
     >
     > for(n in 1:20) {
     + cat("n=",n," ")
     + for(x in 0:n)
     + cat(".")
     + for(p in c(0, .1, .5, .8, 1)) {
     + stopifnot(all.equal(dbinom_raw(x, n, p, q=1-p, log=FALSE),
     + dbinom (x, n, p, log=FALSE)),
     + all.equal(dbinom_raw(x, n, p, q=1-p, log =TRUE),
     + dbinom (x, n, p, log =TRUE)))
     + }
     + cat("\n")
     + }
     n= 1 ..
     n= 2 ...
     n= 3 ....
     n= 4 .....
     n= 5 ......
     n= 6 .......
     n= 7 ........
     n= 8 .........
     n= 9 ..........
     n= 10 ...........
     n= 11 ............
     n= 12 .............
     n= 13 ..............
     n= 14 ...............
     n= 15 ................
     n= 16 .................
     n= 17 ..................
     n= 18 ...................
     n= 19 ....................
     n= 20 .....................
     > showProc.time()
     Time (user system elapsed): 1.56 0.06 1.62
     >
     > ### dnbinom*() :
     > stopifnot(exprs = {
     + dnbinomR(0, 1, 1) == 1
     + })
     >
     > ### exploring 'eps' == "true" tests must be done with Rmpfr !!
     >
     > ### 2. Testing log1pmx(), logcf() etc
     > ### ----------
     >
     > ### 2a: logcf()
     > ## == =======
     > x <- c((-20:3)/4, (25:31)/32) # close (but not too close) to upper bound 1
     >
     > (lC <- logcf (x, i=2, d=3, eps=1e-9))
     [1] 0.2225364 0.2273165 0.2323964 0.2378089 0.2435921 0.2497910 0.2564582
     [8] 0.2636564 0.2714610 0.2799634 0.2892758 0.2995378 0.3109259 0.3236668
     [15] 0.3380584 0.3545014 0.3735507 0.3960035 0.4230578 0.4566237 0.5000000
     [22] 0.5595850 0.6503112 0.8221318 0.8574109 0.8989257 0.9490572 1.0118259
     [29] 1.0948318 1.2152882 1.4286636
     > lCt <- logcf (x, i=2, d=3, eps=1e-9, trace=TRUE) ; stopifnot(identical(lCt, lC))
     it= 0: ==> |b2|=162720
     it= 1: ==> |b2|=1.68458e+08
     it= 2: ==> |b2|=3.02689e+11
     it= 3: ==> |b2|=8.40216e+14
     it= 4: ==> |b2|=3.33607e+18
     it= 5: ==> |b2|=1.79478e+22
     it= 6: ==> |b2|=1.25703e+26
     it= 7: ==> |b2|=1.11146e+30
     it= 8: ==> |b2|=1.21086e+34
     it= 9: ==> |b2|=1.5936e+38
     it=10: ==> |b2|=2.49268e+42
     logcf(*) used 11 iterations.
     it= 0: ==> |b2|=151400
     it= 1: ==> |b2|=1.519e+08
     it= 2: ==> |b2|=2.64707e+11
     it= 3: ==> |b2|=7.12814e+14
     it= 4: ==> |b2|=2.74588e+18
     it= 5: ==> |b2|=1.4333e+22
     it= 6: ==> |b2|=9.73998e+25
     it= 7: ==> |b2|=8.35605e+29
     it= 8: ==> |b2|=8.83286e+33
     it= 9: ==> |b2|=1.12795e+38
     it=10: ==> |b2|=1.71192e+42
     logcf(*) used 11 iterations.
     it= 0: ==> |b2|=140480
     it= 1: ==> |b2|=1.36437e+08
     it= 2: ==> |b2|=2.30332e+11
     it= 3: ==> |b2|=6.0102e+14
     it= 4: ==> |b2|=2.24367e+18
     it= 5: ==> |b2|=1.135e+22
     it= 6: ==> |b2|=7.47503e+25
     it= 7: ==> |b2|=6.21522e+29
     it= 8: ==> |b2|=6.3674e+33
     it= 9: ==> |b2|=7.88061e+37
     it=10: ==> |b2|=1.15921e+42
     logcf(*) used 11 iterations.
     it= 0: ==> |b2|=129960
     it= 1: ==> |b2|=1.22034e+08
     it= 2: ==> |b2|=1.99336e+11
     it= 3: ==> |b2|=5.03394e+14
     it= 4: ==> |b2|=1.81889e+18
     it= 5: ==> |b2|=8.90621e+21
     it= 6: ==> |b2|=5.67763e+25
     it= 7: ==> |b2|=4.56957e+29
     it= 8: ==> |b2|=4.53158e+33
     it= 9: ==> |b2|=5.429e+37
     logcf(*) used 10 iterations.
     it= 0: ==> |b2|=119840
     it= 1: ==> |b2|=1.08655e+08
     it= 2: ==> |b2|=1.71497e+11
     it= 3: ==> |b2|=4.18587e+14
     it= 4: ==> |b2|=1.46194e+18
     it= 5: ==> |b2|=6.91963e+21
     it= 6: ==> |b2|=4.26415e+25
     it= 7: ==> |b2|=3.31759e+29
     it= 8: ==> |b2|=3.18042e+33
     it= 9: ==> |b2|=3.68336e+37
     logcf(*) used 10 iterations.
     it= 0: ==> |b2|=110120
     it= 1: ==> |b2|=9.62638e+07
     it= 2: ==> |b2|=1.46601e+11
     it= 3: ==> |b2|=3.45334e+14
     it= 4: ==> |b2|=1.16411e+18
     it= 5: ==> |b2|=5.31835e+21
     it= 6: ==> |b2|=3.16349e+25
     it= 7: ==> |b2|=2.37577e+29
     it= 8: ==> |b2|=2.19845e+33
     it= 9: ==> |b2|=2.45771e+37
     logcf(*) used 10 iterations.
     it= 0: ==> |b2|=100800
     it= 1: ==> |b2|=8.48232e+07
     it= 2: ==> |b2|=1.24442e+11
     it= 3: ==> |b2|=2.82452e+14
     it= 4: ==> |b2|=9.17519e+17
     it= 5: ==> |b2|=4.03952e+21
     it= 6: ==> |b2|=2.3156e+25
     it= 7: ==> |b2|=1.67591e+29
     it= 8: ==> |b2|=1.49457e+33
     logcf(*) used 9 iterations.
     it= 0: ==> |b2|=91880
     it= 1: ==> |b2|=7.42974e+07
     it= 2: ==> |b2|=1.04819e+11
     it= 3: ==> |b2|=2.28837e+14
     it= 4: ==> |b2|=7.15064e+17
     it= 5: ==> |b2|=3.02848e+21
     it= 6: ==> |b2|=1.67007e+25
     it= 7: ==> |b2|=1.1628e+29
     it= 8: ==> |b2|=9.97611e+32
     logcf(*) used 9 iterations.
     it= 0: ==> |b2|=83360
     it= 1: ==> |b2|=6.46501e+07
     it= 2: ==> |b2|=8.75389e+10
     it= 3: ==> |b2|=1.83464e+14
     it= 4: ==> |b2|=5.50387e+17
     it= 5: ==> |b2|=2.23803e+21
     it= 6: ==> |b2|=1.18496e+25
     it= 7: ==> |b2|=7.92152e+28
     it= 8: ==> |b2|=6.52535e+32
     logcf(*) used 9 iterations.
     it= 0: ==> |b2|=75240
     it= 1: ==> |b2|=5.58449e+07
     it= 2: ==> |b2|=7.24171e+10
     it= 3: ==> |b2|=1.45381e+14
     it= 4: ==> |b2|=4.17809e+17
     it= 5: ==> |b2|=1.6276e+21
     it= 6: ==> |b2|=8.25594e+24
     it= 7: ==> |b2|=5.28764e+28
     logcf(*) used 8 iterations.
     it= 0: ==> |b2|=67520
     it= 1: ==> |b2|=4.78456e+07
     it= 2: ==> |b2|=5.92745e+10
     it= 3: ==> |b2|=1.13708e+14
     it= 4: ==> |b2|=3.12287e+17
     it= 5: ==> |b2|=1.16261e+21
     it= 6: ==> |b2|=5.6361e+24
     it= 7: ==> |b2|=3.44989e+28
     logcf(*) used 8 iterations.
     it= 0: ==> |b2|=60200
     it= 1: ==> |b2|=4.06159e+07
     it= 2: ==> |b2|=4.79397e+10
     it= 3: ==> |b2|=8.76351e+13
     it= 4: ==> |b2|=2.2937e+17
     it= 5: ==> |b2|=8.13827e+20
     it= 6: ==> |b2|=3.76013e+24
     it= 7: ==> |b2|=2.19363e+28
     logcf(*) used 8 iterations.
     it= 0: ==> |b2|=53280
     it= 1: ==> |b2|=3.41194e+07
     it= 2: ==> |b2|=3.82483e+10
     it= 3: ==> |b2|=6.64186e+13
     it= 4: ==> |b2|=1.6515e+17
     it= 5: ==> |b2|=5.56707e+20
     it= 6: ==> |b2|=2.44378e+24
     logcf(*) used 7 iterations.
     it= 0: ==> |b2|=46760
     it= 1: ==> |b2|=2.83198e+07
     it= 2: ==> |b2|=3.0043e+10
     it= 3: ==> |b2|=4.93794e+13
     it= 4: ==> |b2|=1.16224e+17
     it= 5: ==> |b2|=3.70875e+20
     it= 6: ==> |b2|=1.54119e+24
     logcf(*) used 7 iterations.
     it= 0: ==> |b2|=40640
     it= 1: ==> |b2|=2.3181e+07
     it= 2: ==> |b2|=2.31738e+10
     it= 3: ==> |b2|=3.59e+13
     it= 4: ==> |b2|=7.96488e+16
     it= 5: ==> |b2|=2.39588e+20
     logcf(*) used 6 iterations.
     it= 0: ==> |b2|=34920
     it= 1: ==> |b2|=1.86665e+07
     it= 2: ==> |b2|=1.74976e+10
     it= 3: ==> |b2|=2.5422e+13
     it= 4: ==> |b2|=5.29017e+16
     it= 5: ==> |b2|=1.49263e+20
     logcf(*) used 6 iterations.
     it= 0: ==> |b2|=29600
     it= 1: ==> |b2|=1.474e+07
     it= 2: ==> |b2|=1.28785e+10
     it= 3: ==> |b2|=1.74436e+13
     it= 4: ==> |b2|=3.38438e+16
     logcf(*) used 5 iterations.
     it= 0: ==> |b2|=24680
     it= 1: ==> |b2|=1.13653e+07
     it= 2: ==> |b2|=9.18785e+09
     it= 3: ==> |b2|=1.1517e+13
     it= 4: ==> |b2|=2.06815e+16
     logcf(*) used 5 iterations.
     it= 0: ==> |b2|=20160
     it= 1: ==> |b2|=8.50608e+06
     it= 2: ==> |b2|=6.30386e+09
     it= 3: ==> |b2|=7.24564e+12
     logcf(*) used 4 iterations.
     it= 0: ==> |b2|=16040
     it= 1: ==> |b2|=6.12601e+06
     it= 2: ==> |b2|=4.11202e+09
     logcf(*) used 3 iterations.
     logcf(*) used 0 iterations.
     it= 0: ==> |b2|=9000
     it= 1: ==> |b2|=2.65815e+06
     it= 2: ==> |b2|=1.38218e+09
     logcf(*) used 3 iterations.
     it= 0: ==> |b2|=6080
     it= 1: ==> |b2|=1.49776e+06
     it= 2: ==> |b2|=6.50656e+08
     it= 3: ==> |b2|=4.39124e+11
     it= 4: ==> |b2|=4.24985e+14
     logcf(*) used 5 iterations.
     it= 0: ==> |b2|=3560
     it= 1: ==> |b2|=671330
     it= 2: ==> |b2|=2.24237e+08
     it= 3: ==> |b2|=1.16565e+11
     it= 4: ==> |b2|=8.69636e+13
     it= 5: ==> |b2|=8.80714e+16
     it= 6: ==> |b2|=1.16246e+20
     it= 7: ==> |b2|=1.93847e+23
     it= 8: ==> |b2|=3.98491e+26
     logcf(*) used 9 iterations.
     it= 0: ==> |b2|=3273.12
     it= 1: ==> |b2|=589700
     it= 2: ==> |b2|=1.88377e+08
     it= 3: ==> |b2|=9.36959e+10
     it= 4: ==> |b2|=6.68994e+13
     it= 5: ==> |b2|=6.48488e+16
     it= 6: ==> |b2|=8.19327e+19
     it= 7: ==> |b2|=1.30789e+23
     it= 8: ==> |b2|=2.57381e+26
     it= 9: ==> |b2|=6.12129e+29
     logcf(*) used 10 iterations.
     it= 0: ==> |b2|=2992.5
     it= 1: ==> |b2|=512650
     it= 2: ==> |b2|=1.55894e+08
     it= 3: ==> |b2|=7.3859e+10
     it= 4: ==> |b2|=5.02475e+13
     it= 5: ==> |b2|=4.64164e+16
     it= 6: ==> |b2|=5.58911e+19
     it= 7: ==> |b2|=8.50347e+22
     it= 8: ==> |b2|=1.595e+26
     it= 9: ==> |b2|=3.61574e+29
     it=10: ==> |b2|=9.74479e+32
     logcf(*) used 11 iterations.
     it= 0: ==> |b2|=2718.12
     it= 1: ==> |b2|=440109
     it= 2: ==> |b2|=1.26644e+08
     it= 3: ==> |b2|=5.68225e+10
     it= 4: ==> |b2|=3.66244e+13
     it= 5: ==> |b2|=3.20598e+16
     it= 6: ==> |b2|=3.65864e+19
     it= 7: ==> |b2|=5.27587e+22
     it= 8: ==> |b2|=9.37997e+25
     it= 9: ==> |b2|=2.01557e+29
     it=10: ==> |b2|=5.14924e+32
     it=11: ==> |b2|=1.54257e+36
     logcf(*) used 12 iterations.
     it= 0: ==> |b2|=2450
     it= 1: ==> |b2|=372006
     it= 2: ==> |b2|=1.00485e+08
     it= 3: ==> |b2|=4.23633e+10
     it= 4: ==> |b2|=2.56713e+13
     it= 5: ==> |b2|=2.11343e+16
     it= 6: ==> |b2|=2.26869e+19
     it= 7: ==> |b2|=3.07772e+22
     it= 8: ==> |b2|=5.14811e+25
     it= 9: ==> |b2|=1.04082e+29
     it=10: ==> |b2|=2.50192e+32
     it=11: ==> |b2|=7.05238e+35
     it=12: ==> |b2|=2.30384e+39
     logcf(*) used 13 iterations.
     it= 0: ==> |b2|=2188.12
     it= 1: ==> |b2|=308271
     it= 2: ==> |b2|=7.72745e+07
     it= 3: ==> |b2|=3.02658e+10
     it= 4: ==> |b2|=1.70531e+13
     it= 5: ==> |b2|=1.30605e+16
     it= 6: ==> |b2|=1.30466e+19
     it= 7: ==> |b2|=1.64734e+22
     it= 8: ==> |b2|=2.56499e+25
     it= 9: ==> |b2|=4.82765e+28
     it=10: ==> |b2|=1.08039e+32
     it=11: ==> |b2|=2.83535e+35
     it=12: ==> |b2|=8.62389e+38
     it=13: ==> |b2|=3.00926e+42
     it=14: ==> |b2|=1.19409e+46
     it=15: ==> |b2|=5.34632e+49
     logcf(*) used 16 iterations.
     it= 0: ==> |b2|=1932.5
     it= 1: ==> |b2|=248832
     it= 2: ==> |b2|=5.68734e+07
     it= 3: ==> |b2|=2.03226e+10
     it= 4: ==> |b2|=1.04577e+13
     it= 5: ==> |b2|=7.32086e+15
     it= 6: ==> |b2|=6.68834e+18
     it= 7: ==> |b2|=7.72653e+21
     it= 8: ==> |b2|=1.10096e+25
     it= 9: ==> |b2|=1.89662e+28
     it=10: ==> |b2|=3.88536e+31
     it=11: ==> |b2|=9.33474e+34
     it=12: ==> |b2|=2.59938e+38
     it=13: ==> |b2|=8.30457e+41
     it=14: ==> |b2|=3.01718e+45
     it=15: ==> |b2|=1.23692e+49
     it=16: ==> |b2|=5.68258e+52
     it=17: ==> |b2|=2.90768e+56
     it=18: ==> |b2|=1.64796e+60
     logcf(*) used 19 iterations.
     it= 0: ==> |b2|=1683.12
     it= 1: ==> |b2|=193619
     it= 2: ==> |b2|=3.91439e+07
     it= 3: ==> |b2|=1.23338e+10
     it= 4: ==> |b2|=5.59551e+12
     it= 5: ==> |b2|=3.4562e+15
     it= 6: ==> |b2|=2.78868e+18
     it= 7: ==> |b2|=2.84748e+21
     it= 8: ==> |b2|=3.58854e+24
     it= 9: ==> |b2|=5.4701e+27
     it=10: ==> |b2|=9.91885e+30
     it=11: ==> |b2|=2.10987e+34
     it=12: ==> |b2|=5.20269e+37
     it=13: ==> |b2|=1.47211e+41
     it=14: ==> |b2|=4.73732e+44
     it=15: ==> |b2|=1.72036e+48
     it=16: ==> |b2|=7.00164e+51
     it=17: ==> |b2|=3.17394e+55
     it=18: ==> |b2|=1.59374e+59
     it=19: ==> |b2|=8.8205e+62
     it=20: ==> |b2|=5.35623e+66
     it=21: ==> |b2|=3.5541e+70
     it=22: ==> |b2|=2.56724e+74
     it=23: ==> |b2|=2.01172e+78 Lrg |b2|
     it=24: ==> |b2|=147221
     it=25: ==> |b2|=1.34508e+09
     it=26: ==> |b2|=1.32142e+13
     it=27: ==> |b2|=1.39232e+17
     logcf(*) used 28 iterations.
     > (lR <- logcfR(x, i=2, d=3, eps=1e-9))
     [1] 0.2225364 0.2273165 0.2323964 0.2378089 0.2435921 0.2497910 0.2564582
     [8] 0.2636564 0.2714610 0.2799634 0.2892758 0.2995378 0.3109259 0.3236668
     [15] 0.3380584 0.3545014 0.3735507 0.3960035 0.4230578 0.4566237 0.5000000
     [22] 0.5595850 0.6503112 0.8221318 0.8574109 0.8989257 0.9490572 1.0118259
     [29] 1.0948318 1.2152882 1.4286636
     > all.equal(lC, lR, tol = 0) # to see if ..
     [1] TRUE
     > stopifnot(all.equal(lC, lR, tol = 4e-16))
     > lRt <- logcfR(x, i=2, d=3, eps=1e-9, trace=TRUE) ; stopifnot(identical(lRt, lR))
     logcf(x[ 1]= -5, i=2, d=3, eps=1e-09) logcf(*) end: after 11 iterations.
     logcf(x[ 2]= -4.75, i=2, d=3, eps=1e-09) logcf(*) end: after 11 iterations.
     logcf(x[ 3]= -4.5, i=2, d=3, eps=1e-09) logcf(*) end: after 11 iterations.
     logcf(x[ 4]= -4.25, i=2, d=3, eps=1e-09) logcf(*) end: after 10 iterations.
     logcf(x[ 5]= -4, i=2, d=3, eps=1e-09) logcf(*) end: after 10 iterations.
     logcf(x[ 6]= -3.75, i=2, d=3, eps=1e-09) logcf(*) end: after 10 iterations.
     logcf(x[ 7]= -3.5, i=2, d=3, eps=1e-09) logcf(*) end: after 9 iterations.
     logcf(x[ 8]= -3.25, i=2, d=3, eps=1e-09) logcf(*) end: after 9 iterations.
     logcf(x[ 9]= -3, i=2, d=3, eps=1e-09) logcf(*) end: after 9 iterations.
     logcf(x[10]= -2.75, i=2, d=3, eps=1e-09) logcf(*) end: after 8 iterations.
     logcf(x[11]= -2.5, i=2, d=3, eps=1e-09) logcf(*) end: after 8 iterations.
     logcf(x[12]= -2.25, i=2, d=3, eps=1e-09) logcf(*) end: after 8 iterations.
     logcf(x[13]= -2, i=2, d=3, eps=1e-09) logcf(*) end: after 7 iterations.
     logcf(x[14]= -1.75, i=2, d=3, eps=1e-09) logcf(*) end: after 7 iterations.
     logcf(x[15]= -1.5, i=2, d=3, eps=1e-09) logcf(*) end: after 6 iterations.
     logcf(x[16]= -1.25, i=2, d=3, eps=1e-09) logcf(*) end: after 6 iterations.
     logcf(x[17]= -1, i=2, d=3, eps=1e-09) logcf(*) end: after 5 iterations.
     logcf(x[18]= -0.75, i=2, d=3, eps=1e-09) logcf(*) end: after 5 iterations.
     logcf(x[19]= -0.5, i=2, d=3, eps=1e-09) logcf(*) end: after 4 iterations.
     logcf(x[20]= -0.25, i=2, d=3, eps=1e-09) logcf(*) end: after 3 iterations.
     logcf(x[21]= 0, i=2, d=3, eps=1e-09) logcf(*) end: after 0 iterations.
     logcf(x[22]= 0.25, i=2, d=3, eps=1e-09) logcf(*) end: after 3 iterations.
     logcf(x[23]= 0.5, i=2, d=3, eps=1e-09) logcf(*) end: after 5 iterations.
     logcf(x[24]= 0.75, i=2, d=3, eps=1e-09) logcf(*) end: after 9 iterations.
     logcf(x[25]= 0.78125, i=2, d=3, eps=1e-09) logcf(*) end: after 10 iterations.
     logcf(x[26]= 0.8125, i=2, d=3, eps=1e-09) logcf(*) end: after 11 iterations.
     logcf(x[27]= 0.84375, i=2, d=3, eps=1e-09) logcf(*) end: after 12 iterations.
     logcf(x[28]= 0.875, i=2, d=3, eps=1e-09) logcf(*) end: after 13 iterations.
     logcf(x[29]= 0.90625, i=2, d=3, eps=1e-09) logcf(*) end: after 16 iterations.
     logcf(x[30]= 0.9375, i=2, d=3, eps=1e-09) logcf(*) end: after 19 iterations.
     logcf(x[31]= 0.96875, i=2, d=3, eps=1e-09) logcf(*) end: after 28 iterations.
     > lRt2 <- logcfR(x, i=2, d=3, eps=1e-9, trace= 2) ; stopifnot(identical(lRt2,lR))
     logcf(x[ 1]= -5, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 162720 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.0168627
     it= 2: ==> B2= 1.68458e+08 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00303811
     it= 3: ==> B2= 3.02689e+11 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000541327
     it= 4: ==> B2= 8.40216e+14 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 9.60626e-05
     it= 5: ==> B2= 3.33607e+18 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.70167e-05
     it= 6: ==> B2= 1.79478e+22 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.01154e-06
     it= 7: ==> B2= 1.25703e+26 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 5.32664e-07
     it= 8: ==> B2= 1.11146e+30 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 9.4179e-08
     it= 9: ==> B2= 1.21086e+34 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.66472e-08
     it=10: ==> B2= 1.5936e+38 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.942e-09
     it=11: ==> B2= 2.49268e+42 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 5.19854e-10
     logcf(*) end: after 11 iterations.
     logcf(x[ 2]= -4.75, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 151400 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.0157061
     it= 2: ==> B2= 1.519e+08 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00271234
     it= 3: ==> B2= 2.64707e+11 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000463242
     it= 4: ==> B2= 7.12814e+14 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 7.88006e-05
     it= 5: ==> B2= 2.74588e+18 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.33808e-05
     it= 6: ==> B2= 1.4333e+22 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.26999e-06
     it= 7: ==> B2= 9.73998e+25 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.84872e-07
     it= 8: ==> B2= 8.35605e+29 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 6.52292e-08
     it= 9: ==> B2= 8.83286e+33 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.10523e-08
     it=10: ==> B2= 1.12795e+38 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.8723e-09
     it=11: ==> B2= 1.71192e+42 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.1713e-10
     logcf(*) end: after 11 iterations.
     logcf(x[ 3]= -4.5, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 140480 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.014539
     it= 2: ==> B2= 1.36437e+08 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00239872
     it= 3: ==> B2= 2.30332e+11 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000391409
     it= 4: ==> B2= 6.0102e+14 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 6.36152e-05
     it= 5: ==> B2= 2.24367e+18 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.03211e-05
     it= 6: ==> B2= 1.135e+22 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.67293e-06
     it= 7: ==> B2= 7.47503e+25 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.71005e-07
     it= 8: ==> B2= 6.21522e+29 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 4.38843e-08
     it= 9: ==> B2= 6.3674e+33 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 7.10431e-09
     it=10: ==> B2= 7.88061e+37 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.14987e-09
     it=11: ==> B2= 1.15921e+42 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.86085e-10
     logcf(*) end: after 11 iterations.
     logcf(x[ 4]= -4.25, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 129960 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.0133641
     it= 2: ==> B2= 1.22034e+08 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00209863
     it= 3: ==> B2= 1.99336e+11 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000325959
     it= 4: ==> B2= 5.03394e+14 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 5.04302e-05
     it= 5: ==> B2= 1.81889e+18 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 7.78852e-06
     it= 6: ==> B2= 8.90621e+21 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.20172e-06
     it= 7: ==> B2= 5.67763e+25 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.85309e-07
     it= 8: ==> B2= 4.56957e+29 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.85641e-08
     it= 9: ==> B2= 4.53158e+33 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 4.40173e-09
     it=10: ==> B2= 5.429e+37 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 6.78171e-10
     logcf(*) end: after 10 iterations.
     logcf(x[ 5]= -4, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 119840 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.0121853
     it= 2: ==> B2= 1.08655e+08 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00181353
     it= 3: ==> B2= 1.71497e+11 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000266983
     it= 4: ==> B2= 4.18587e+14 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.91528e-05
     it= 5: ==> B2= 1.46194e+18 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 5.73167e-06
     it= 6: ==> B2= 6.91963e+21 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 8.38266e-07
     it= 7: ==> B2= 4.26415e+25 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.22525e-07
     it= 8: ==> B2= 3.31759e+29 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.79016e-08
     it= 9: ==> B2= 3.18042e+33 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.6148e-09
     it=10: ==> B2= 3.68336e+37 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.81854e-10
     logcf(*) end: after 10 iterations.
     logcf(x[ 6]= -3.75, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 110120 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.0110067
     it= 2: ==> B2= 9.62638e+07 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00154491
     it= 3: ==> B2= 1.46601e+11 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00021452
     it= 4: ==> B2= 3.45334e+14 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.96738e-05
     it= 5: ==> B2= 1.16411e+18 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 4.0975e-06
     it= 6: ==> B2= 5.31835e+21 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 5.65252e-07
     it= 7: ==> B2= 3.16349e+25 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 7.79297e-08
     it= 8: ==> B2= 2.37577e+29 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.07396e-08
     it= 9: ==> B2= 2.19845e+33 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.47962e-09
     it=10: ==> B2= 2.45771e+37 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.03808e-10
     logcf(*) end: after 10 iterations.
     logcf(x[ 7]= -3.5, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 100800 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00983372
     it= 2: ==> B2= 8.48232e+07 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00129431
     it= 3: ==> B2= 1.24442e+11 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000168553
     it= 4: ==> B2= 2.82452e+14 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.18673e-05
     it= 5: ==> B2= 9.17519e+17 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.83198e-06
     it= 6: ==> B2= 4.03952e+21 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.66405e-07
     it= 7: ==> B2= 2.3156e+25 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 4.73767e-08
     it= 8: ==> B2= 1.67591e+29 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 6.12337e-09
     it= 9: ==> B2= 1.49457e+33 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 7.91207e-10
     logcf(*) end: after 9 iterations.
     logcf(x[ 8]= -3.25, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 91880 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00867282
     it= 2: ==> B2= 7.42974e+07 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00106327
     it= 3: ==> B2= 1.04819e+11 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000128994
     it= 4: ==> B2= 2.28837e+14 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.55909e-05
     it= 5: ==> B2= 7.15064e+17 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.88109e-06
     it= 6: ==> B2= 3.02848e+21 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.26734e-07
     it= 7: ==> B2= 1.67007e+25 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.7312e-08
     it= 8: ==> B2= 1.1628e+29 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.28859e-09
     it= 9: ==> B2= 9.97611e+32 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.95855e-10
     logcf(*) end: after 9 iterations.
     logcf(x[ 9]= -3, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 83360 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00753175
     it= 2: ==> B2= 6.46501e+07 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000853254
     it= 3: ==> B2= 8.75389e+10 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 9.5671e-05
     it= 4: ==> B2= 1.83464e+14 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.06874e-05
     it= 5: ==> B2= 5.50387e+17 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.19178e-06
     it= 6: ==> B2= 2.23803e+21 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.32765e-07
     it= 7: ==> B2= 1.18496e+25 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.47808e-08
     it= 8: ==> B2= 7.92152e+28 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.64484e-09
     it= 9: ==> B2= 6.52535e+32 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.82987e-10
     logcf(*) end: after 9 iterations.
     logcf(x[10]= -2.75, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 75240 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00641978
     it= 2: ==> B2= 5.58449e+07 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000665641
     it= 3: ==> B2= 7.24171e+10 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 6.83222e-05
     it= 4: ==> B2= 1.45381e+14 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 6.98689e-06
     it= 5: ==> B2= 4.17809e+17 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 7.13234e-07
     it= 6: ==> B2= 1.6276e+21 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 7.27338e-08
     it= 7: ==> B2= 8.25594e+24 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 7.41242e-09
     it= 8: ==> B2= 5.28764e+28 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 7.55083e-10
     logcf(*) end: after 8 iterations.
     logcf(x[11]= -2.5, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 67520 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00534792
     it= 2: ==> B2= 4.78456e+07 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.0005016
     it= 3: ==> B2= 5.92745e+10 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 4.65818e-05
     it= 4: ==> B2= 1.13708e+14 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 4.31004e-06
     it= 5: ==> B2= 3.12287e+17 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.98074e-07
     it= 6: ==> B2= 1.16261e+21 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.67278e-08
     it= 7: ==> B2= 5.6361e+24 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.38642e-09
     it= 8: ==> B2= 3.44989e+28 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.12099e-10
     logcf(*) end: after 8 iterations.
     logcf(x[12]= -2.25, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 60200 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00432911
     it= 2: ==> B2= 4.06159e+07 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00036199
     it= 3: ==> B2= 4.79397e+10 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.99753e-05
     it= 4: ==> B2= 8.76351e+13 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.47309e-06
     it= 5: ==> B2= 2.2937e+17 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.03667e-07
     it= 6: ==> B2= 8.13827e+20 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.67549e-08
     it= 7: ==> B2= 3.76013e+24 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.37743e-09
     it= 8: ==> B2= 2.19363e+28 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.13188e-10
     logcf(*) end: after 8 iterations.
     logcf(x[13]= -2, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 53280 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00337838
     it= 2: ==> B2= 3.41194e+07 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00024722
     it= 3: ==> B2= 3.82483e+10 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.79187e-05
     it= 4: ==> B2= 6.64186e+13 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.29399e-06
     it= 5: ==> B2= 1.6515e+17 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 9.32713e-08
     it= 6: ==> B2= 5.56707e+20 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 6.71575e-09
     it= 7: ==> B2= 2.44378e+24 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 4.83216e-10
     logcf(*) end: after 7 iterations.
     logcf(x[14]= -1.75, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 46760 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00251277
     it= 2: ==> B2= 2.83198e+07 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000157067
     it= 3: ==> B2= 3.0043e+10 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 9.72602e-06
     it= 4: ==> B2= 4.93794e+13 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 6.00029e-07
     it= 5: ==> B2= 1.16224e+17 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.69475e-08
     it= 6: ==> B2= 3.70875e+20 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.27255e-09
     it= 7: ==> B2= 1.54119e+24 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.39681e-10
     logcf(*) end: after 7 iterations.
     logcf(x[15]= -1.5, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 40640 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.001751
     it= 2: ==> B2= 2.3181e+07 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 9.04805e-05
     it= 3: ==> B2= 2.31738e+10 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 4.63221e-06
     it= 4: ==> B2= 3.59e+13 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.36258e-07
     it= 5: ==> B2= 7.96488e+16 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.20265e-08
     it= 6: ==> B2= 2.39588e+20 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 6.11497e-10
     logcf(*) end: after 6 iterations.
     logcf(x[16]= -1.25, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 34920 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00111245
     it= 2: ==> B2= 1.86665e+07 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 4.53708e-05
     it= 3: ==> B2= 1.74976e+10 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.8334e-06
     it= 4: ==> B2= 2.5422e+13 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 7.38016e-08
     it= 5: ==> B2= 5.29017e+16 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.96486e-09
     it= 6: ==> B2= 1.49263e+20 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.18968e-10
     logcf(*) end: after 6 iterations.
     logcf(x[17]= -1, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 29600 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000614941
     it= 2: ==> B2= 1.474e+07 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.84647e-05
     it= 3: ==> B2= 1.28785e+10 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 5.49306e-07
     it= 4: ==> B2= 1.74436e+13 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.62767e-08
     it= 5: ==> B2= 3.38438e+16 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 4.81303e-10
     logcf(*) end: after 5 iterations.
     logcf(x[18]= -0.75, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 24680 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000270385
     it= 2: ==> B2= 1.13653e+07 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 5.33194e-06
     it= 3: ==> B2= 9.18785e+09 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.04153e-07
     it= 4: ==> B2= 1.1517e+13 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.02617e-09
     it= 5: ==> B2= 2.06815e+16 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.93312e-11
     logcf(*) end: after 5 iterations.
     logcf(x[19]= -0.5, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 20160 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 7.69704e-05
     it= 2: ==> B2= 8.50608e+06 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 8.02466e-07
     it= 3: ==> B2= 6.30386e+09 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 8.28445e-09
     it= 4: ==> B2= 7.24564e+12 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 8.51583e-11
     logcf(*) end: after 4 iterations.
     logcf(x[20]= -0.25, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 16040 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 7.26392e-06
     it= 2: ==> B2= 6.12601e+06 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.30773e-08
     it= 3: ==> B2= 4.11202e+09 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 7.25571e-11
     logcf(*) end: after 3 iterations.
     logcf(x[21]= 0, i=2, d=3, eps=1e-09) iterations:
     logcf(*) end: after 0 iterations.
     logcf(x[22]= 0.25, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 9000 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.04918e-05
     it= 2: ==> B2= 2.65815e+06 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.08623e-07
     it= 3: ==> B2= 1.38218e+09 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 5.68393e-10
     logcf(*) end: after 3 iterations.
     logcf(x[23]= 0.5, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 6080 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000659523
     it= 2: ==> B2= 1.49776e+06 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.00942e-05
     it= 3: ==> B2= 6.50656e+08 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 6.02264e-07
     it= 4: ==> B2= 4.39124e+11 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.79273e-08
     it= 5: ==> B2= 4.24985e+14 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 5.3174e-10
     logcf(*) end: after 5 iterations.
     logcf(x[24]= 0.75, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 3560 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00856402
     it= 2: ==> B2= 671330 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00100335
     it= 3: ==> B2= 2.24237e+08 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000114482
     it= 4: ==> B2= 1.16565e+11 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.2922e-05
     it= 5: ==> B2= 8.69636e+13 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.45089e-06
     it= 6: ==> B2= 8.80714e+16 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.62421e-07
     it= 7: ==> B2= 1.16246e+20 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.81486e-08
     it= 8: ==> B2= 1.93847e+23 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.02539e-09
     it= 9: ==> B2= 3.98491e+26 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.25837e-10
     logcf(*) end: after 9 iterations.
     logcf(x[25]= 0.78125, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 3273.12 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.0116683
     it= 2: ==> B2= 589700 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00162549
     it= 3: ==> B2= 1.88377e+08 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000220072
     it= 4: ==> B2= 9.36959e+10 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.94426e-05
     it= 5: ==> B2= 6.68994e+13 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.9163e-06
     it= 6: ==> B2= 6.48488e+16 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 5.19244e-07
     it= 7: ==> B2= 8.19327e+19 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 6.87066e-08
     it= 8: ==> B2= 1.30789e+23 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 9.07922e-09
     it= 9: ==> B2= 2.57381e+26 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.19866e-09
     it=10: ==> B2= 6.12129e+29 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.58143e-10
     logcf(*) end: after 10 iterations.
     logcf(x[26]= 0.8125, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 2992.5 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.0159401
     it= 2: ==> B2= 512650 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00265674
     it= 3: ==> B2= 1.55894e+08 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000429426
     it= 4: ==> B2= 7.3859e+10 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 6.84941e-05
     it= 5: ==> B2= 5.02475e+13 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.08546e-05
     it= 6: ==> B2= 4.64164e+16 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.71404e-06
     it= 7: ==> B2= 5.58911e+19 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.70071e-07
     it= 8: ==> B2= 8.50347e+22 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 4.2492e-08
     it= 9: ==> B2= 1.595e+26 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 6.6788e-09
     it=10: ==> B2= 3.61574e+29 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.04899e-09
     it=11: ==> B2= 9.74479e+32 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.64668e-10
     logcf(*) end: after 11 iterations.
     logcf(x[27]= 0.84375, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 2718.12 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.0218736
     it= 2: ==> B2= 440109 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00440022
     it= 3: ==> B2= 1.26644e+08 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000856838
     it= 4: ==> B2= 5.68225e+10 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000164362
     it= 5: ==> B2= 3.66244e+13 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.12964e-05
     it= 6: ==> B2= 3.20598e+16 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 5.93505e-06
     it= 7: ==> B2= 3.65864e+19 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.12276e-06
     it= 8: ==> B2= 5.27587e+22 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.12056e-07
     it= 9: ==> B2= 9.37997e+25 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 4.00067e-08
     it=10: ==> B2= 2.01557e+29 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 7.54162e-09
     it=11: ==> B2= 5.14924e+32 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.42081e-09
     it=12: ==> B2= 1.54257e+36 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.67552e-10
     logcf(*) end: after 12 iterations.
     logcf(x[28]= 0.875, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 2450 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.0302147
     it= 2: ==> B2= 372006 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00742718
     it= 3: ==> B2= 1.00485e+08 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00176572
     it= 4: ==> B2= 4.23633e+10 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000412688
     it= 5: ==> B2= 2.56713e+13 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 9.56191e-05
     it= 6: ==> B2= 2.11343e+16 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.20491e-05
     it= 7: ==> B2= 2.26869e+19 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 5.0698e-06
     it= 8: ==> B2= 3.07772e+22 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.16356e-06
     it= 9: ==> B2= 5.14811e+25 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.66706e-07
     it=10: ==> B2= 1.04082e+29 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 6.10778e-08
     it=11: ==> B2= 2.50192e+32 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.39778e-08
     it=12: ==> B2= 7.05238e+35 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.19721e-09
     it=13: ==> B2= 2.30384e+39 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 7.31016e-10
     logcf(*) end: after 13 iterations.
     logcf(x[29]= 0.90625, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 2188.12 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.0421192
     it= 2: ==> B2= 308271 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.0128742
     it= 3: ==> B2= 7.72745e+07 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00381265
     it= 4: ==> B2= 3.02658e+10 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00110791
     it= 5: ==> B2= 1.70531e+13 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000318587
     it= 6: ==> B2= 1.30605e+16 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 9.10705e-05
     it= 7: ==> B2= 1.30466e+19 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.5941e-05
     it= 8: ==> B2= 1.64734e+22 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 7.37247e-06
     it= 9: ==> B2= 2.56499e+25 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.09206e-06
     it=10: ==> B2= 4.82765e+28 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 5.93012e-07
     it=11: ==> B2= 1.08039e+32 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.6796e-07
     it=12: ==> B2= 2.83535e+35 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 4.75428e-08
     it=13: ==> B2= 8.62389e+38 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.34511e-08
     it=14: ==> B2= 3.00926e+42 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.80425e-09
     it=15: ==> B2= 1.19409e+46 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.07559e-09
     it=16: ==> B2= 5.34632e+49 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.04032e-10
     logcf(*) end: after 16 iterations.
     logcf(x[30]= 0.9375, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 1932.5 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.0594391
     it= 2: ==> B2= 248832 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.02317
     it= 3: ==> B2= 5.68734e+07 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00882488
     it= 4: ==> B2= 2.03226e+10 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00329775
     it= 5: ==> B2= 1.04577e+13 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00121712
     it= 6: ==> B2= 7.32086e+15 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000445765
     it= 7: ==> B2= 6.68834e+18 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00016248
     it= 8: ==> B2= 7.72653e+21 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 5.90414e-05
     it= 9: ==> B2= 1.10096e+25 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.14101e-05
     it=10: ==> B2= 1.89662e+28 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 7.7527e-06
     it=11: ==> B2= 3.88536e+31 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.80437e-06
     it=12: ==> B2= 9.33474e+34 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.01363e-06
     it=13: ==> B2= 2.59938e+38 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.6615e-07
     it=14: ==> B2= 8.30457e+41 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.32201e-07
     it=15: ==> B2= 3.01718e+45 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 4.77137e-08
     it=16: ==> B2= 1.23692e+49 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.72154e-08
     it=17: ==> B2= 5.68258e+52 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 6.20986e-09
     it=18: ==> B2= 2.90768e+56 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.23951e-09
     it=19: ==> B2= 1.64796e+60 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 8.07503e-10
     logcf(*) end: after 19 iterations.
     logcf(x[31]= 0.96875, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 1683.12 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.0852619
     it= 2: ==> B2= 193619 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.0440308
     it= 3: ==> B2= 3.91439e+07 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.0227933
     it= 4: ==> B2= 1.23338e+10 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.0116823
     it= 5: ==> B2= 5.59551e+12 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00592272
     it= 6: ==> B2= 3.4562e+15 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00297607
     it= 7: ==> B2= 2.78868e+18 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00148555
     it= 8: ==> B2= 2.84748e+21 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00073801
     it= 9: ==> B2= 3.58854e+24 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000365396
     it=10: ==> B2= 5.4701e+27 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000180472
     it=11: ==> B2= 9.91885e+30 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 8.89791e-05
     it=12: ==> B2= 2.10987e+34 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 4.38122e-05
     it=13: ==> B2= 5.20269e+37 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.15512e-05
     it=14: ==> B2= 1.47211e+41 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.05929e-05
     it=15: ==> B2= 4.73732e+44 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 5.20351e-06
     it=16: ==> B2= 1.72036e+48 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.55486e-06
     it=17: ==> B2= 7.00164e+51 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.25391e-06
     it=18: ==> B2= 3.17394e+55 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 6.15209e-07
     it=19: ==> B2= 1.59374e+59 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.0176e-07
     it=20: ==> B2= 8.8205e+62 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.47979e-07
     it=21: ==> B2= 5.35623e+66 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 7.25526e-08
     it=22: ==> B2= 3.5541e+70 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.55657e-08
     it=23: ==> B2= 2.56724e+74 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.7432e-08
     it=24: ==> B2= 2.01172e+78 Lrg m.B2
     --> crit. |A2*b1 - a1*B2|/|b1*B2| = 8.54292e-09
     it=25: ==> B2= 147221 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 4.18617e-09
     it=26: ==> B2= 1.34508e+09 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.05108e-09
     it=27: ==> B2= 1.32142e+13 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.00487e-09
     it=28: ==> B2= 1.39232e+17 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 4.92273e-10
     logcf(*) end: after 28 iterations.
     >
     > lR. <- logcfR.(x, i=2, d=3, eps=1e-9)
     > lR.t <- logcfR.(x, i=2, d=3, eps=1e-9, trace=TRUE) ; stopifnot(identical(lR.t, lR.))
     logcf(x[], i=2, d=3, eps=1e-09) iterations:
     it= 1: needIt: 30 TRUE, and 1 F.; length(x[<todo>])=30, m.B2= 23307.2
     it= 2: needIt: 30 TRUE; length(x[<todo>])=30, m.B2= 1.01159e+07
     it= 3: needIt: 30 TRUE; length(x[<todo>])=30, m.B2= 7.70605e+09
     it= 4: needIt: 28 TRUE, and 2 F.; length(x[<todo>])=28, m.B2= 1.00852e+13
     it= 5: needIt: 27 TRUE, and 1 F.; length(x[<todo>])=27, m.B2= 1.76419e+16
     it= 6: needIt: 24 TRUE, and 3 F.; length(x[<todo>])=24, m.B2= 4.75316e+19
     it= 7: needIt: 22 TRUE, and 2 F.; length(x[<todo>])=22, m.B2= 1.2798e+23
     it= 8: needIt: 20 TRUE, and 2 F.; length(x[<todo>])=20, m.B2= 3.63581e+26
     it= 9: needIt: 17 TRUE, and 3 F.; length(x[<todo>])=17, m.B2= 6.8674e+29
     it=10: needIt: 13 TRUE, and 4 F.; length(x[<todo>])=13, m.B2= 1.03776e+33
     it=11: needIt: 9 TRUE, and 4 F.; length(x[<todo>])= 9, m.B2= 3.09233e+35
     it=12: needIt: 5 TRUE, and 4 F.; length(x[<todo>])= 5, m.B2= 2.27357e+35
     it=13: needIt: 4 TRUE, and 1 F.; length(x[<todo>])= 4, m.B2= 4.04868e+38
     it=14: needIt: 3 TRUE, and 1 F.; length(x[<todo>])= 3, m.B2= 7.16537e+41
     it=15: needIt: 3 TRUE; length(x[<todo>])= 3, m.B2= 2.57468e+45
     it=16: needIt: 3 TRUE; length(x[<todo>])= 3, m.B2= 1.04393e+49
     it=17: needIt: 2 TRUE, and 1 F.; length(x[<todo>])= 2, m.B2= 1.99468e+52
     it=18: needIt: 2 TRUE; length(x[<todo>])= 2, m.B2= 9.60666e+55
     it=19: needIt: 2 TRUE; length(x[<todo>])= 2, m.B2= 5.12487e+59
     it=20: needIt: 1 TRUE, and 1 F.; length(x[<todo>])= 1, m.B2= 8.8205e+62
     it=21: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 5.35623e+66
     it=22: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 3.5541e+70
     it=23: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 2.56724e+74
     it=24: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 2.01172e+78 Lrg m.B2
     it=25: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 147221
     it=26: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 1.34508e+09
     it=27: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 1.32142e+13
     it=28: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 1.39232e+17
     logcf(*) end: after 28 iterations.
     >
     > all.equal(lC, lR., tol = 0) # TRUE !! (every where ?)
     [1] TRUE
     > all.equal(lR, lR., tol = 0) # TRUE !! " "
     [1] TRUE
     > stopifnot(all.equal(lC, lR., tol = 1e-15))
     > ## (even though they used eps=1e-9 .. i.e., are not *so* accurate)
     > showProc.time()
     Time (user system elapsed): 0.03 0 0.04
     >
     > ##--- now with improved logcfR.() {<< will become the new logcfR() at least for MPFR !}:
     >
     > ##require(Rmpfr) may be not, see if NS loading (via "::") is sufficient:
     > requireNamespace("Rmpfr") || quit("no")
     Loading required namespace: Rmpfr
     [1] TRUE
     > ## ----- ----------
     > xM <- Rmpfr::mpfr(x, 512)
     > (ct.14 <- system.time(lR.14 <- logcfR.(xM, i=2, d=3, eps=1e-20, trace=TRUE))) # 0.55 sec
     logcf(x[], i=2, d=3, eps=1e-20) iterations:
     it= 1: needIt: 30 TRUE, and 1 F.; length(x[<todo>])=30, m.B2= 23307.2
     it= 2: needIt: 30 TRUE; length(x[<todo>])=30, m.B2= 1.01159e+07
     it= 3: needIt: 30 TRUE; length(x[<todo>])=30, m.B2= 7.70605e+09
     it= 4: needIt: 30 TRUE; length(x[<todo>])=30, m.B2= 9.10781e+12
     it= 5: needIt: 30 TRUE; length(x[<todo>])=30, m.B2= 1.54287e+16
     it= 6: needIt: 30 TRUE; length(x[<todo>])=30, m.B2= 3.54543e+19
     it= 7: needIt: 30 TRUE; length(x[<todo>])=30, m.B2= 1.06137e+23
     it= 8: needIt: 29 TRUE, and 1 F.; length(x[<todo>])=29, m.B2= 4.19177e+26
     it= 9: needIt: 28 TRUE, and 1 F.; length(x[<todo>])=28, m.B2= 2.26761e+30
     it=10: needIt: 27 TRUE, and 1 F.; length(x[<todo>])=27, m.B2= 1.33011e+34
     it=11: needIt: 27 TRUE; length(x[<todo>])=27, m.B2= 9.0823e+37
     it=12: needIt: 26 TRUE, and 1 F.; length(x[<todo>])=26, m.B2= 7.15387e+41
     it=13: needIt: 25 TRUE, and 1 F.; length(x[<todo>])=25, m.B2= 6.21918e+45
     it=14: needIt: 24 TRUE, and 1 F.; length(x[<todo>])=24, m.B2= 9.51187e+49
     it=15: needIt: 23 TRUE, and 1 F.; length(x[<todo>])=23, m.B2= 1.04428e+54
     it=16: needIt: 22 TRUE, and 1 F.; length(x[<todo>])=22, m.B2= 1.19866e+58
     it=17: needIt: 21 TRUE, and 1 F.; length(x[<todo>])=21, m.B2= 1.40641e+62
     it=18: needIt: 20 TRUE, and 1 F.; length(x[<todo>])=20, m.B2= 1.64566e+66
     it=19: needIt: 19 TRUE, and 1 F.; length(x[<todo>])=19, m.B2= 1.86787e+70
     it=20: needIt: 17 TRUE, and 2 F.; length(x[<todo>])=17, m.B2= 9.5095e+73
     it=21: needIt: 15 TRUE, and 2 F.; length(x[<todo>])=15, m.B2= 2.07684e+78 Lrg m.B2
     it=22: needIt: 14 TRUE, and 1 F.; length(x[<todo>])=14, m.B2= 122830
     it=23: needIt: 11 TRUE, and 3 F.; length(x[<todo>])=11, m.B2= 3.76273e+08
     it=24: needIt: 10 TRUE, and 1 F.; length(x[<todo>])=10, m.B2= 7.77428e+11
     it=25: needIt: 7 TRUE, and 3 F.; length(x[<todo>])= 7, m.B2= 4.17254e+13
     it=26: needIt: 6 TRUE, and 1 F.; length(x[<todo>])= 6, m.B2= 1.55243e+15
     it=27: needIt: 5 TRUE, and 1 F.; length(x[<todo>])= 5, m.B2= 2.47748e+15
     it=28: needIt: 4 TRUE, and 1 F.; length(x[<todo>])= 4, m.B2= 1.06982e+19
     it=29: needIt: 4 TRUE; length(x[<todo>])= 4, m.B2= 1.40477e+23
     it=30: needIt: 4 TRUE; length(x[<todo>])= 4, m.B2= 1.9693e+27
     it=31: needIt: 3 TRUE, and 1 F.; length(x[<todo>])= 3, m.B2= 7.6538e+30
     it=32: needIt: 3 TRUE; length(x[<todo>])= 3, m.B2= 1.16488e+35
     it=33: needIt: 3 TRUE; length(x[<todo>])= 3, m.B2= 1.88175e+39
     it=34: needIt: 3 TRUE; length(x[<todo>])= 3, m.B2= 3.22081e+43
     it=35: needIt: 3 TRUE; length(x[<todo>])= 3, m.B2= 5.83159e+47
     it=36: needIt: 3 TRUE; length(x[<todo>])= 3, m.B2= 1.11521e+52
     it=37: needIt: 2 TRUE, and 1 F.; length(x[<todo>])= 2, m.B2= 3.51533e+55
     it=38: needIt: 2 TRUE; length(x[<todo>])= 2, m.B2= 7.10714e+59
     it=39: needIt: 2 TRUE; length(x[<todo>])= 2, m.B2= 1.51138e+64
     it=40: needIt: 2 TRUE; length(x[<todo>])= 2, m.B2= 3.37644e+68
     it=41: needIt: 2 TRUE; length(x[<todo>])= 2, m.B2= 7.91477e+72
     it=42: needIt: 2 TRUE; length(x[<todo>])= 2, m.B2= 1.94455e+77 Lrg m.B2
     it=43: needIt: 2 TRUE; length(x[<todo>])= 2, m.B2= 43197.4
     it=44: needIt: 2 TRUE; length(x[<todo>])= 2, m.B2= 1.16214e+09
     it=45: needIt: 1 TRUE, and 1 F.; length(x[<todo>])= 1, m.B2= 2.11103e+12
     it=46: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 5.83147e+16
     it=47: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 1.68004e+21
     it=48: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 5.04365e+25
     it=49: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 1.57649e+30
     it=50: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 5.12638e+34
     it=51: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 1.7329e+39
     it=52: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 6.08495e+43
     it=53: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 2.21796e+48
     it=54: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 8.38622e+52
     it=55: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 3.28706e+57
     it=56: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 1.33476e+62
     it=57: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 5.61156e+66
     it=58: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 2.44114e+71
     it=59: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 1.0982e+76
     it=60: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 5.10636e+80 Lrg m.B2
     it=61: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 2.1182e+08
     it=62: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 1.05047e+13
     it=63: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 5.37602e+17
     logcf(*) end: after 63 iterations.
     user system elapsed
     1.33 0.01 1.35
     > (ct14 <- system.time(lR14 <- logcfR (xM, i=2, d=3, eps=1e-20, trace=TRUE))) # 4 sec
     logcf(x[ 1]= -5, i=2, d=3, eps=1e-20) logcf(*) end: after 26 iterations.
     logcf(x[ 2]= -4.75, i=2, d=3, eps=1e-20) logcf(*) end: after 25 iterations.
     logcf(x[ 3]= -4.5, i=2, d=3, eps=1e-20) logcf(*) end: after 24 iterations.
     logcf(x[ 4]= -4.25, i=2, d=3, eps=1e-20) logcf(*) end: after 24 iterations.
     logcf(x[ 5]= -4, i=2, d=3, eps=1e-20) logcf(*) end: after 23 iterations.
     logcf(x[ 6]= -3.75, i=2, d=3, eps=1e-20) logcf(*) end: after 22 iterations.
     logcf(x[ 7]= -3.5, i=2, d=3, eps=1e-20) logcf(*) end: after 22 iterations.
     logcf(x[ 8]= -3.25, i=2, d=3, eps=1e-20) logcf(*) end: after 21 iterations.
     logcf(x[ 9]= -3, i=2, d=3, eps=1e-20) logcf(*) end: after 20 iterations.
     logcf(x[10]= -2.75, i=2, d=3, eps=1e-20) logcf(*) end: after 19 iterations.
     logcf(x[11]= -2.5, i=2, d=3, eps=1e-20) logcf(*) end: after 19 iterations.
     logcf(x[12]= -2.25, i=2, d=3, eps=1e-20) logcf(*) end: after 18 iterations.
     logcf(x[13]= -2, i=2, d=3, eps=1e-20) logcf(*) end: after 17 iterations.
     logcf(x[14]= -1.75, i=2, d=3, eps=1e-20) logcf(*) end: after 16 iterations.
     logcf(x[15]= -1.5, i=2, d=3, eps=1e-20) logcf(*) end: after 15 iterations.
     logcf(x[16]= -1.25, i=2, d=3, eps=1e-20) logcf(*) end: after 14 iterations.
     logcf(x[17]= -1, i=2, d=3, eps=1e-20) logcf(*) end: after 12 iterations.
     logcf(x[18]= -0.75, i=2, d=3, eps=1e-20) logcf(*) end: after 11 iterations.
     logcf(x[19]= -0.5, i=2, d=3, eps=1e-20) logcf(*) end: after 9 iterations.
     logcf(x[20]= -0.25, i=2, d=3, eps=1e-20) logcf(*) end: after 7 iterations.
     logcf(x[21]= 0, i=2, d=3, eps=1e-20) logcf(*) end: after 0 iterations.
     logcf(x[22]= 0.25, i=2, d=3, eps=1e-20) logcf(*) end: after 8 iterations.
     logcf(x[23]= 0.5, i=2, d=3, eps=1e-20) logcf(*) end: after 13 iterations.
     logcf(x[24]= 0.75, i=2, d=3, eps=1e-20) logcf(*) end: after 20 iterations.
     logcf(x[25]= 0.78125, i=2, d=3, eps=1e-20) logcf(*) end: after 22 iterations.
     logcf(x[26]= 0.8125, i=2, d=3, eps=1e-20) logcf(*) end: after 24 iterations.
     logcf(x[27]= 0.84375, i=2, d=3, eps=1e-20) logcf(*) end: after 27 iterations.
     logcf(x[28]= 0.875, i=2, d=3, eps=1e-20) logcf(*) end: after 30 iterations.
     logcf(x[29]= 0.90625, i=2, d=3, eps=1e-20) logcf(*) end: after 36 iterations.
     logcf(x[30]= 0.9375, i=2, d=3, eps=1e-20) logcf(*) end: after 44 iterations.
     logcf(x[31]= 0.96875, i=2, d=3, eps=1e-20) logcf(*) end: after 63 iterations.
     user system elapsed
     8.97 0.08 9.06
     >
     > all.equal(lR.14, lR14, tol=0) # TRUE
     [1] TRUE
     > identical(lR.14, lR14) # TRUE !! (not sure if on all platforms!)
     [1] TRUE
     >
     > SS <- function(ch, digits=7)
     + sub(paste0("([0-9]{1,",digits,"})[0-9]*e"), "\\1e", ch)
     > ## double prec <--> MPFR: vvvv (same eps)
     > lR.9 <- logcfR.(xM, 2,3, eps=1e-9)
     > ## show:
     > SS(Rmpfr::all.equal(Rmpfr::roundMpfr(lR.9, 64), lR, tol=0))# .. 5.1138e-16
     Error in target == current : comparison of these types is not implemented
     Calls: SS ... <Anonymous> -> <Anonymous> -> .local -> all.equal.numeric
     Execution halted
Flavor: r-devel-windows-x86_64-new-TK

Version: 0.5-0
Check: examples
Result: ERROR
    Running examples in 'DPQ-Ex.R' failed
    The error most likely occurred in:
    
    > ### Name: ppoisson
    > ### Title: Direct Computation of 'ppois()' Poisson Distribution
    > ### Probabilities
    > ### Aliases: ppoisErr ppoisD
    > ### Keywords: distribution
    >
    > ### ** Examples
    >
    > (lams <- outer(c(1,2,5), 10^(0:3)))# 10^4 is already slow!
     [,1] [,2] [,3] [,4]
    [1,] 1 10 100 1000
    [2,] 2 20 200 2000
    [3,] 5 50 500 5000
    > system.time(e1 <- sapply(lams, ppoisErr))
     user system elapsed
     0.01 0.00 0.02
    > e1 / .Machine$double.eps
     [1] 0.0 0.5 -1.0 1.0 5.5 1.5 -4.0 -3.0 1.0 -1.0 2.0 2.0
    >
    > ## Try another 'ppFUN' :---------------------------------
    > ## this relies on the fact that it's *only* used on an 'x' of the form 0:M :
    > ppD0 <- function(x, lambda, all.from.0=TRUE)
    + cumsum(dpois(if(all.from.0) 0:x else x, lambda=lambda))
    > ## and test it:
    > p0 <- ppD0 ( 1000, lambda=10)
    > p1 <- ppois(0:1000, lambda=10)
    > stopifnot(all.equal(p0,p1, tol=8*.Machine$double.eps))
    >
    > system.time(p0.slow <- ppoisD(0:1000, lambda=10, all.from.0=FALSE))# not very slow, here
     user system elapsed
     0 0 0
    > p0.1 <- ppoisD(1000, lambda=10)
    > if(requireNamespace("Rmpfr")) {
    + ppoisMpfr <- function(x, lambda) cumsum(Rmpfr::dpois(x, lambda=lambda))
    + p0.best <- ppoisMpfr(0:1000, lambda = Rmpfr::mpfr(10, precBits = 256))
    + AllEq. <- Rmpfr::all.equal
    + AllEq <- function(target, current, ...)
    + AllEq.(target, current, ...,
    + formatFUN = function(x, ...) Rmpfr::format(x, digits = 9))
    + print(AllEq(p0.best, p0, tol = 0)) # 2.06e-18
    + print(AllEq(p0.best, p0.slow, tol = 0)) # the "worst" (4.44e-17)
    + print(AllEq(p0.best, p0.1, tol = 0)) # 1.08e-18
    + }
    Error in target == current : comparison of these types is not implemented
    Calls: print ... AllEq -> AllEq. -> AllEq. -> .local -> all.equal.numeric
    Execution halted
Flavor: r-devel-windows-x86_64-old

Version: 0.5-0
Check: tests
Result: ERROR
     Running 'chisq-nonc-ex.R' [38s]
     Running 'dnbinom-tst.R' [15s]
     Running 'dnchisq-tst.R' [0s]
     Running 'hyper-dist-ex.R' [38s]
     Running 'pnbeta-tst.R' [0s]
     Running 'pnt-prec.R' [28s]
     Running 'ppois-ex.R' [2s]
     Running 'qPoisBinom-ex.R' [0s]
     Running 'qbeta-dist.R' [11s]
     Running 'qbeta-tst.R' [1s]
     Running 'qgamma-ex.R' [13s]
     Running 'stirlerr-tst.R' [4s]
     Running 't-nonc-tst.R' [6s]
     Running 'wienergerm-pchisq-tst.R' [0s]
     Running 'wienergerm_nchisq.R' [8s]
    Running the tests in 'tests/dnbinom-tst.R' failed.
    Complete output:
     > #### Testing 1) dbinom_raw(), dnbinomR() and dnbinom.mu()
     > #### 2) log1pmx(), logcf() etc
     > require(DPQ)
     Loading required package: DPQ
     > source(system.file(package="Matrix", "test-tools-1.R", mustWork=TRUE))
     Loading required package: tools
     > ## -> showProc.time(), assertError()
     >
     > (doExtras <- DPQ:::doExtras() && !grepl("valgrind", R.home()))
     [1] FALSE
     >
     > if(!dev.interactive(orNone=TRUE)) pdf("wienergerm-accuracy.pdf")
     >
     >
     > ### 1. Testing dbinom_raw(), dnbinomR() and dnbinom.mu() >>> ../R/dbinom-nbinom.R <<<
     > ### ---------- ../man/dbinom_raw.Rd & ../man/dnbinomR.Rd
     >
     > ## "FIXME:" use sfsmisc :: relErrV() already here
     >
     > ### dbinom() vs dbinom.raw() :
     >
     > for(n in 1:20) {
     + cat("n=",n," ")
     + for(x in 0:n)
     + cat(".")
     + for(p in c(0, .1, .5, .8, 1)) {
     + stopifnot(all.equal(dbinom_raw(x, n, p, q=1-p, log=FALSE),
     + dbinom (x, n, p, log=FALSE)),
     + all.equal(dbinom_raw(x, n, p, q=1-p, log =TRUE),
     + dbinom (x, n, p, log =TRUE)))
     + }
     + cat("\n")
     + }
     n= 1 ..
     n= 2 ...
     n= 3 ....
     n= 4 .....
     n= 5 ......
     n= 6 .......
     n= 7 ........
     n= 8 .........
     n= 9 ..........
     n= 10 ...........
     n= 11 ............
     n= 12 .............
     n= 13 ..............
     n= 14 ...............
     n= 15 ................
     n= 16 .................
     n= 17 ..................
     n= 18 ...................
     n= 19 ....................
     n= 20 .....................
     > showProc.time()
     Time (user system elapsed): 1.33 0.06 1.39
     >
     > ### dnbinom*() :
     > stopifnot(exprs = {
     + dnbinomR(0, 1, 1) == 1
     + })
     >
     > ### exploring 'eps' == "true" tests must be done with Rmpfr !!
     >
     > ### 2. Testing log1pmx(), logcf() etc
     > ### ----------
     >
     > ### 2a: logcf()
     > ## == =======
     > x <- c((-20:3)/4, (25:31)/32) # close (but not too close) to upper bound 1
     >
     > (lC <- logcf (x, i=2, d=3, eps=1e-9))
     [1] 0.2225364 0.2273165 0.2323964 0.2378089 0.2435921 0.2497910 0.2564582
     [8] 0.2636564 0.2714610 0.2799634 0.2892758 0.2995378 0.3109259 0.3236668
     [15] 0.3380584 0.3545014 0.3735507 0.3960035 0.4230578 0.4566237 0.5000000
     [22] 0.5595850 0.6503112 0.8221318 0.8574109 0.8989257 0.9490572 1.0118259
     [29] 1.0948318 1.2152882 1.4286636
     > lCt <- logcf (x, i=2, d=3, eps=1e-9, trace=TRUE) ; stopifnot(identical(lCt, lC))
     it= 0: ==> |b2|=162720
     it= 1: ==> |b2|=1.68458e+08
     it= 2: ==> |b2|=3.02689e+11
     it= 3: ==> |b2|=8.40216e+14
     it= 4: ==> |b2|=3.33607e+18
     it= 5: ==> |b2|=1.79478e+22
     it= 6: ==> |b2|=1.25703e+26
     it= 7: ==> |b2|=1.11146e+30
     it= 8: ==> |b2|=1.21086e+34
     it= 9: ==> |b2|=1.5936e+38
     it=10: ==> |b2|=2.49268e+42
     logcf(*) used 11 iterations.
     it= 0: ==> |b2|=151400
     it= 1: ==> |b2|=1.519e+08
     it= 2: ==> |b2|=2.64707e+11
     it= 3: ==> |b2|=7.12814e+14
     it= 4: ==> |b2|=2.74588e+18
     it= 5: ==> |b2|=1.4333e+22
     it= 6: ==> |b2|=9.73998e+25
     it= 7: ==> |b2|=8.35605e+29
     it= 8: ==> |b2|=8.83286e+33
     it= 9: ==> |b2|=1.12795e+38
     it=10: ==> |b2|=1.71192e+42
     logcf(*) used 11 iterations.
     it= 0: ==> |b2|=140480
     it= 1: ==> |b2|=1.36437e+08
     it= 2: ==> |b2|=2.30332e+11
     it= 3: ==> |b2|=6.0102e+14
     it= 4: ==> |b2|=2.24367e+18
     it= 5: ==> |b2|=1.135e+22
     it= 6: ==> |b2|=7.47503e+25
     it= 7: ==> |b2|=6.21522e+29
     it= 8: ==> |b2|=6.3674e+33
     it= 9: ==> |b2|=7.88061e+37
     it=10: ==> |b2|=1.15921e+42
     logcf(*) used 11 iterations.
     it= 0: ==> |b2|=129960
     it= 1: ==> |b2|=1.22034e+08
     it= 2: ==> |b2|=1.99336e+11
     it= 3: ==> |b2|=5.03394e+14
     it= 4: ==> |b2|=1.81889e+18
     it= 5: ==> |b2|=8.90621e+21
     it= 6: ==> |b2|=5.67763e+25
     it= 7: ==> |b2|=4.56957e+29
     it= 8: ==> |b2|=4.53158e+33
     it= 9: ==> |b2|=5.429e+37
     logcf(*) used 10 iterations.
     it= 0: ==> |b2|=119840
     it= 1: ==> |b2|=1.08655e+08
     it= 2: ==> |b2|=1.71497e+11
     it= 3: ==> |b2|=4.18587e+14
     it= 4: ==> |b2|=1.46194e+18
     it= 5: ==> |b2|=6.91963e+21
     it= 6: ==> |b2|=4.26415e+25
     it= 7: ==> |b2|=3.31759e+29
     it= 8: ==> |b2|=3.18042e+33
     it= 9: ==> |b2|=3.68336e+37
     logcf(*) used 10 iterations.
     it= 0: ==> |b2|=110120
     it= 1: ==> |b2|=9.62638e+07
     it= 2: ==> |b2|=1.46601e+11
     it= 3: ==> |b2|=3.45334e+14
     it= 4: ==> |b2|=1.16411e+18
     it= 5: ==> |b2|=5.31835e+21
     it= 6: ==> |b2|=3.16349e+25
     it= 7: ==> |b2|=2.37577e+29
     it= 8: ==> |b2|=2.19845e+33
     it= 9: ==> |b2|=2.45771e+37
     logcf(*) used 10 iterations.
     it= 0: ==> |b2|=100800
     it= 1: ==> |b2|=8.48232e+07
     it= 2: ==> |b2|=1.24442e+11
     it= 3: ==> |b2|=2.82452e+14
     it= 4: ==> |b2|=9.17519e+17
     it= 5: ==> |b2|=4.03952e+21
     it= 6: ==> |b2|=2.3156e+25
     it= 7: ==> |b2|=1.67591e+29
     it= 8: ==> |b2|=1.49457e+33
     logcf(*) used 9 iterations.
     it= 0: ==> |b2|=91880
     it= 1: ==> |b2|=7.42974e+07
     it= 2: ==> |b2|=1.04819e+11
     it= 3: ==> |b2|=2.28837e+14
     it= 4: ==> |b2|=7.15064e+17
     it= 5: ==> |b2|=3.02848e+21
     it= 6: ==> |b2|=1.67007e+25
     it= 7: ==> |b2|=1.1628e+29
     it= 8: ==> |b2|=9.97611e+32
     logcf(*) used 9 iterations.
     it= 0: ==> |b2|=83360
     it= 1: ==> |b2|=6.46501e+07
     it= 2: ==> |b2|=8.75389e+10
     it= 3: ==> |b2|=1.83464e+14
     it= 4: ==> |b2|=5.50387e+17
     it= 5: ==> |b2|=2.23803e+21
     it= 6: ==> |b2|=1.18496e+25
     it= 7: ==> |b2|=7.92152e+28
     it= 8: ==> |b2|=6.52535e+32
     logcf(*) used 9 iterations.
     it= 0: ==> |b2|=75240
     it= 1: ==> |b2|=5.58449e+07
     it= 2: ==> |b2|=7.24171e+10
     it= 3: ==> |b2|=1.45381e+14
     it= 4: ==> |b2|=4.17809e+17
     it= 5: ==> |b2|=1.6276e+21
     it= 6: ==> |b2|=8.25594e+24
     it= 7: ==> |b2|=5.28764e+28
     logcf(*) used 8 iterations.
     it= 0: ==> |b2|=67520
     it= 1: ==> |b2|=4.78456e+07
     it= 2: ==> |b2|=5.92745e+10
     it= 3: ==> |b2|=1.13708e+14
     it= 4: ==> |b2|=3.12287e+17
     it= 5: ==> |b2|=1.16261e+21
     it= 6: ==> |b2|=5.6361e+24
     it= 7: ==> |b2|=3.44989e+28
     logcf(*) used 8 iterations.
     it= 0: ==> |b2|=60200
     it= 1: ==> |b2|=4.06159e+07
     it= 2: ==> |b2|=4.79397e+10
     it= 3: ==> |b2|=8.76351e+13
     it= 4: ==> |b2|=2.2937e+17
     it= 5: ==> |b2|=8.13827e+20
     it= 6: ==> |b2|=3.76013e+24
     it= 7: ==> |b2|=2.19363e+28
     logcf(*) used 8 iterations.
     it= 0: ==> |b2|=53280
     it= 1: ==> |b2|=3.41194e+07
     it= 2: ==> |b2|=3.82483e+10
     it= 3: ==> |b2|=6.64186e+13
     it= 4: ==> |b2|=1.6515e+17
     it= 5: ==> |b2|=5.56707e+20
     it= 6: ==> |b2|=2.44378e+24
     logcf(*) used 7 iterations.
     it= 0: ==> |b2|=46760
     it= 1: ==> |b2|=2.83198e+07
     it= 2: ==> |b2|=3.0043e+10
     it= 3: ==> |b2|=4.93794e+13
     it= 4: ==> |b2|=1.16224e+17
     it= 5: ==> |b2|=3.70875e+20
     it= 6: ==> |b2|=1.54119e+24
     logcf(*) used 7 iterations.
     it= 0: ==> |b2|=40640
     it= 1: ==> |b2|=2.3181e+07
     it= 2: ==> |b2|=2.31738e+10
     it= 3: ==> |b2|=3.59e+13
     it= 4: ==> |b2|=7.96488e+16
     it= 5: ==> |b2|=2.39588e+20
     logcf(*) used 6 iterations.
     it= 0: ==> |b2|=34920
     it= 1: ==> |b2|=1.86665e+07
     it= 2: ==> |b2|=1.74976e+10
     it= 3: ==> |b2|=2.5422e+13
     it= 4: ==> |b2|=5.29017e+16
     it= 5: ==> |b2|=1.49263e+20
     logcf(*) used 6 iterations.
     it= 0: ==> |b2|=29600
     it= 1: ==> |b2|=1.474e+07
     it= 2: ==> |b2|=1.28785e+10
     it= 3: ==> |b2|=1.74436e+13
     it= 4: ==> |b2|=3.38438e+16
     logcf(*) used 5 iterations.
     it= 0: ==> |b2|=24680
     it= 1: ==> |b2|=1.13653e+07
     it= 2: ==> |b2|=9.18785e+09
     it= 3: ==> |b2|=1.1517e+13
     it= 4: ==> |b2|=2.06815e+16
     logcf(*) used 5 iterations.
     it= 0: ==> |b2|=20160
     it= 1: ==> |b2|=8.50608e+06
     it= 2: ==> |b2|=6.30386e+09
     it= 3: ==> |b2|=7.24564e+12
     logcf(*) used 4 iterations.
     it= 0: ==> |b2|=16040
     it= 1: ==> |b2|=6.12601e+06
     it= 2: ==> |b2|=4.11202e+09
     logcf(*) used 3 iterations.
     logcf(*) used 0 iterations.
     it= 0: ==> |b2|=9000
     it= 1: ==> |b2|=2.65815e+06
     it= 2: ==> |b2|=1.38218e+09
     logcf(*) used 3 iterations.
     it= 0: ==> |b2|=6080
     it= 1: ==> |b2|=1.49776e+06
     it= 2: ==> |b2|=6.50656e+08
     it= 3: ==> |b2|=4.39124e+11
     it= 4: ==> |b2|=4.24985e+14
     logcf(*) used 5 iterations.
     it= 0: ==> |b2|=3560
     it= 1: ==> |b2|=671330
     it= 2: ==> |b2|=2.24237e+08
     it= 3: ==> |b2|=1.16565e+11
     it= 4: ==> |b2|=8.69636e+13
     it= 5: ==> |b2|=8.80714e+16
     it= 6: ==> |b2|=1.16246e+20
     it= 7: ==> |b2|=1.93847e+23
     it= 8: ==> |b2|=3.98491e+26
     logcf(*) used 9 iterations.
     it= 0: ==> |b2|=3273.12
     it= 1: ==> |b2|=589700
     it= 2: ==> |b2|=1.88377e+08
     it= 3: ==> |b2|=9.36959e+10
     it= 4: ==> |b2|=6.68994e+13
     it= 5: ==> |b2|=6.48488e+16
     it= 6: ==> |b2|=8.19327e+19
     it= 7: ==> |b2|=1.30789e+23
     it= 8: ==> |b2|=2.57381e+26
     it= 9: ==> |b2|=6.12129e+29
     logcf(*) used 10 iterations.
     it= 0: ==> |b2|=2992.5
     it= 1: ==> |b2|=512650
     it= 2: ==> |b2|=1.55894e+08
     it= 3: ==> |b2|=7.3859e+10
     it= 4: ==> |b2|=5.02475e+13
     it= 5: ==> |b2|=4.64164e+16
     it= 6: ==> |b2|=5.58911e+19
     it= 7: ==> |b2|=8.50347e+22
     it= 8: ==> |b2|=1.595e+26
     it= 9: ==> |b2|=3.61574e+29
     it=10: ==> |b2|=9.74479e+32
     logcf(*) used 11 iterations.
     it= 0: ==> |b2|=2718.12
     it= 1: ==> |b2|=440109
     it= 2: ==> |b2|=1.26644e+08
     it= 3: ==> |b2|=5.68225e+10
     it= 4: ==> |b2|=3.66244e+13
     it= 5: ==> |b2|=3.20598e+16
     it= 6: ==> |b2|=3.65864e+19
     it= 7: ==> |b2|=5.27587e+22
     it= 8: ==> |b2|=9.37997e+25
     it= 9: ==> |b2|=2.01557e+29
     it=10: ==> |b2|=5.14924e+32
     it=11: ==> |b2|=1.54257e+36
     logcf(*) used 12 iterations.
     it= 0: ==> |b2|=2450
     it= 1: ==> |b2|=372006
     it= 2: ==> |b2|=1.00485e+08
     it= 3: ==> |b2|=4.23633e+10
     it= 4: ==> |b2|=2.56713e+13
     it= 5: ==> |b2|=2.11343e+16
     it= 6: ==> |b2|=2.26869e+19
     it= 7: ==> |b2|=3.07772e+22
     it= 8: ==> |b2|=5.14811e+25
     it= 9: ==> |b2|=1.04082e+29
     it=10: ==> |b2|=2.50192e+32
     it=11: ==> |b2|=7.05238e+35
     it=12: ==> |b2|=2.30384e+39
     logcf(*) used 13 iterations.
     it= 0: ==> |b2|=2188.12
     it= 1: ==> |b2|=308271
     it= 2: ==> |b2|=7.72745e+07
     it= 3: ==> |b2|=3.02658e+10
     it= 4: ==> |b2|=1.70531e+13
     it= 5: ==> |b2|=1.30605e+16
     it= 6: ==> |b2|=1.30466e+19
     it= 7: ==> |b2|=1.64734e+22
     it= 8: ==> |b2|=2.56499e+25
     it= 9: ==> |b2|=4.82765e+28
     it=10: ==> |b2|=1.08039e+32
     it=11: ==> |b2|=2.83535e+35
     it=12: ==> |b2|=8.62389e+38
     it=13: ==> |b2|=3.00926e+42
     it=14: ==> |b2|=1.19409e+46
     it=15: ==> |b2|=5.34632e+49
     logcf(*) used 16 iterations.
     it= 0: ==> |b2|=1932.5
     it= 1: ==> |b2|=248832
     it= 2: ==> |b2|=5.68734e+07
     it= 3: ==> |b2|=2.03226e+10
     it= 4: ==> |b2|=1.04577e+13
     it= 5: ==> |b2|=7.32086e+15
     it= 6: ==> |b2|=6.68834e+18
     it= 7: ==> |b2|=7.72653e+21
     it= 8: ==> |b2|=1.10096e+25
     it= 9: ==> |b2|=1.89662e+28
     it=10: ==> |b2|=3.88536e+31
     it=11: ==> |b2|=9.33474e+34
     it=12: ==> |b2|=2.59938e+38
     it=13: ==> |b2|=8.30457e+41
     it=14: ==> |b2|=3.01718e+45
     it=15: ==> |b2|=1.23692e+49
     it=16: ==> |b2|=5.68258e+52
     it=17: ==> |b2|=2.90768e+56
     it=18: ==> |b2|=1.64796e+60
     logcf(*) used 19 iterations.
     it= 0: ==> |b2|=1683.12
     it= 1: ==> |b2|=193619
     it= 2: ==> |b2|=3.91439e+07
     it= 3: ==> |b2|=1.23338e+10
     it= 4: ==> |b2|=5.59551e+12
     it= 5: ==> |b2|=3.4562e+15
     it= 6: ==> |b2|=2.78868e+18
     it= 7: ==> |b2|=2.84748e+21
     it= 8: ==> |b2|=3.58854e+24
     it= 9: ==> |b2|=5.4701e+27
     it=10: ==> |b2|=9.91885e+30
     it=11: ==> |b2|=2.10987e+34
     it=12: ==> |b2|=5.20269e+37
     it=13: ==> |b2|=1.47211e+41
     it=14: ==> |b2|=4.73732e+44
     it=15: ==> |b2|=1.72036e+48
     it=16: ==> |b2|=7.00164e+51
     it=17: ==> |b2|=3.17394e+55
     it=18: ==> |b2|=1.59374e+59
     it=19: ==> |b2|=8.8205e+62
     it=20: ==> |b2|=5.35623e+66
     it=21: ==> |b2|=3.5541e+70
     it=22: ==> |b2|=2.56724e+74
     it=23: ==> |b2|=2.01172e+78 Lrg |b2|
     it=24: ==> |b2|=147221
     it=25: ==> |b2|=1.34508e+09
     it=26: ==> |b2|=1.32142e+13
     it=27: ==> |b2|=1.39232e+17
     logcf(*) used 28 iterations.
     > (lR <- logcfR(x, i=2, d=3, eps=1e-9))
     [1] 0.2225364 0.2273165 0.2323964 0.2378089 0.2435921 0.2497910 0.2564582
     [8] 0.2636564 0.2714610 0.2799634 0.2892758 0.2995378 0.3109259 0.3236668
     [15] 0.3380584 0.3545014 0.3735507 0.3960035 0.4230578 0.4566237 0.5000000
     [22] 0.5595850 0.6503112 0.8221318 0.8574109 0.8989257 0.9490572 1.0118259
     [29] 1.0948318 1.2152882 1.4286636
     > all.equal(lC, lR, tol = 0) # to see if ..
     [1] TRUE
     > stopifnot(all.equal(lC, lR, tol = 4e-16))
     > lRt <- logcfR(x, i=2, d=3, eps=1e-9, trace=TRUE) ; stopifnot(identical(lRt, lR))
     logcf(x[ 1]= -5, i=2, d=3, eps=1e-09) logcf(*) end: after 11 iterations.
     logcf(x[ 2]= -4.75, i=2, d=3, eps=1e-09) logcf(*) end: after 11 iterations.
     logcf(x[ 3]= -4.5, i=2, d=3, eps=1e-09) logcf(*) end: after 11 iterations.
     logcf(x[ 4]= -4.25, i=2, d=3, eps=1e-09) logcf(*) end: after 10 iterations.
     logcf(x[ 5]= -4, i=2, d=3, eps=1e-09) logcf(*) end: after 10 iterations.
     logcf(x[ 6]= -3.75, i=2, d=3, eps=1e-09) logcf(*) end: after 10 iterations.
     logcf(x[ 7]= -3.5, i=2, d=3, eps=1e-09) logcf(*) end: after 9 iterations.
     logcf(x[ 8]= -3.25, i=2, d=3, eps=1e-09) logcf(*) end: after 9 iterations.
     logcf(x[ 9]= -3, i=2, d=3, eps=1e-09) logcf(*) end: after 9 iterations.
     logcf(x[10]= -2.75, i=2, d=3, eps=1e-09) logcf(*) end: after 8 iterations.
     logcf(x[11]= -2.5, i=2, d=3, eps=1e-09) logcf(*) end: after 8 iterations.
     logcf(x[12]= -2.25, i=2, d=3, eps=1e-09) logcf(*) end: after 8 iterations.
     logcf(x[13]= -2, i=2, d=3, eps=1e-09) logcf(*) end: after 7 iterations.
     logcf(x[14]= -1.75, i=2, d=3, eps=1e-09) logcf(*) end: after 7 iterations.
     logcf(x[15]= -1.5, i=2, d=3, eps=1e-09) logcf(*) end: after 6 iterations.
     logcf(x[16]= -1.25, i=2, d=3, eps=1e-09) logcf(*) end: after 6 iterations.
     logcf(x[17]= -1, i=2, d=3, eps=1e-09) logcf(*) end: after 5 iterations.
     logcf(x[18]= -0.75, i=2, d=3, eps=1e-09) logcf(*) end: after 5 iterations.
     logcf(x[19]= -0.5, i=2, d=3, eps=1e-09) logcf(*) end: after 4 iterations.
     logcf(x[20]= -0.25, i=2, d=3, eps=1e-09) logcf(*) end: after 3 iterations.
     logcf(x[21]= 0, i=2, d=3, eps=1e-09) logcf(*) end: after 0 iterations.
     logcf(x[22]= 0.25, i=2, d=3, eps=1e-09) logcf(*) end: after 3 iterations.
     logcf(x[23]= 0.5, i=2, d=3, eps=1e-09) logcf(*) end: after 5 iterations.
     logcf(x[24]= 0.75, i=2, d=3, eps=1e-09) logcf(*) end: after 9 iterations.
     logcf(x[25]= 0.78125, i=2, d=3, eps=1e-09) logcf(*) end: after 10 iterations.
     logcf(x[26]= 0.8125, i=2, d=3, eps=1e-09) logcf(*) end: after 11 iterations.
     logcf(x[27]= 0.84375, i=2, d=3, eps=1e-09) logcf(*) end: after 12 iterations.
     logcf(x[28]= 0.875, i=2, d=3, eps=1e-09) logcf(*) end: after 13 iterations.
     logcf(x[29]= 0.90625, i=2, d=3, eps=1e-09) logcf(*) end: after 16 iterations.
     logcf(x[30]= 0.9375, i=2, d=3, eps=1e-09) logcf(*) end: after 19 iterations.
     logcf(x[31]= 0.96875, i=2, d=3, eps=1e-09) logcf(*) end: after 28 iterations.
     > lRt2 <- logcfR(x, i=2, d=3, eps=1e-9, trace= 2) ; stopifnot(identical(lRt2,lR))
     logcf(x[ 1]= -5, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 162720 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.0168627
     it= 2: ==> B2= 1.68458e+08 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00303811
     it= 3: ==> B2= 3.02689e+11 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000541327
     it= 4: ==> B2= 8.40216e+14 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 9.60626e-05
     it= 5: ==> B2= 3.33607e+18 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.70167e-05
     it= 6: ==> B2= 1.79478e+22 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.01154e-06
     it= 7: ==> B2= 1.25703e+26 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 5.32664e-07
     it= 8: ==> B2= 1.11146e+30 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 9.4179e-08
     it= 9: ==> B2= 1.21086e+34 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.66472e-08
     it=10: ==> B2= 1.5936e+38 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.942e-09
     it=11: ==> B2= 2.49268e+42 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 5.19854e-10
     logcf(*) end: after 11 iterations.
     logcf(x[ 2]= -4.75, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 151400 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.0157061
     it= 2: ==> B2= 1.519e+08 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00271234
     it= 3: ==> B2= 2.64707e+11 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000463242
     it= 4: ==> B2= 7.12814e+14 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 7.88006e-05
     it= 5: ==> B2= 2.74588e+18 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.33808e-05
     it= 6: ==> B2= 1.4333e+22 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.26999e-06
     it= 7: ==> B2= 9.73998e+25 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.84872e-07
     it= 8: ==> B2= 8.35605e+29 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 6.52292e-08
     it= 9: ==> B2= 8.83286e+33 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.10523e-08
     it=10: ==> B2= 1.12795e+38 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.8723e-09
     it=11: ==> B2= 1.71192e+42 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.1713e-10
     logcf(*) end: after 11 iterations.
     logcf(x[ 3]= -4.5, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 140480 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.014539
     it= 2: ==> B2= 1.36437e+08 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00239872
     it= 3: ==> B2= 2.30332e+11 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000391409
     it= 4: ==> B2= 6.0102e+14 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 6.36152e-05
     it= 5: ==> B2= 2.24367e+18 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.03211e-05
     it= 6: ==> B2= 1.135e+22 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.67293e-06
     it= 7: ==> B2= 7.47503e+25 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.71005e-07
     it= 8: ==> B2= 6.21522e+29 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 4.38843e-08
     it= 9: ==> B2= 6.3674e+33 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 7.10431e-09
     it=10: ==> B2= 7.88061e+37 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.14987e-09
     it=11: ==> B2= 1.15921e+42 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.86085e-10
     logcf(*) end: after 11 iterations.
     logcf(x[ 4]= -4.25, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 129960 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.0133641
     it= 2: ==> B2= 1.22034e+08 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00209863
     it= 3: ==> B2= 1.99336e+11 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000325959
     it= 4: ==> B2= 5.03394e+14 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 5.04302e-05
     it= 5: ==> B2= 1.81889e+18 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 7.78852e-06
     it= 6: ==> B2= 8.90621e+21 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.20172e-06
     it= 7: ==> B2= 5.67763e+25 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.85309e-07
     it= 8: ==> B2= 4.56957e+29 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.85641e-08
     it= 9: ==> B2= 4.53158e+33 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 4.40173e-09
     it=10: ==> B2= 5.429e+37 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 6.78171e-10
     logcf(*) end: after 10 iterations.
     logcf(x[ 5]= -4, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 119840 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.0121853
     it= 2: ==> B2= 1.08655e+08 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00181353
     it= 3: ==> B2= 1.71497e+11 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000266983
     it= 4: ==> B2= 4.18587e+14 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.91528e-05
     it= 5: ==> B2= 1.46194e+18 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 5.73167e-06
     it= 6: ==> B2= 6.91963e+21 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 8.38266e-07
     it= 7: ==> B2= 4.26415e+25 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.22525e-07
     it= 8: ==> B2= 3.31759e+29 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.79016e-08
     it= 9: ==> B2= 3.18042e+33 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.6148e-09
     it=10: ==> B2= 3.68336e+37 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.81854e-10
     logcf(*) end: after 10 iterations.
     logcf(x[ 6]= -3.75, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 110120 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.0110067
     it= 2: ==> B2= 9.62638e+07 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00154491
     it= 3: ==> B2= 1.46601e+11 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00021452
     it= 4: ==> B2= 3.45334e+14 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.96738e-05
     it= 5: ==> B2= 1.16411e+18 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 4.0975e-06
     it= 6: ==> B2= 5.31835e+21 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 5.65252e-07
     it= 7: ==> B2= 3.16349e+25 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 7.79297e-08
     it= 8: ==> B2= 2.37577e+29 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.07396e-08
     it= 9: ==> B2= 2.19845e+33 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.47962e-09
     it=10: ==> B2= 2.45771e+37 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.03808e-10
     logcf(*) end: after 10 iterations.
     logcf(x[ 7]= -3.5, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 100800 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00983372
     it= 2: ==> B2= 8.48232e+07 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00129431
     it= 3: ==> B2= 1.24442e+11 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000168553
     it= 4: ==> B2= 2.82452e+14 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.18673e-05
     it= 5: ==> B2= 9.17519e+17 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.83198e-06
     it= 6: ==> B2= 4.03952e+21 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.66405e-07
     it= 7: ==> B2= 2.3156e+25 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 4.73767e-08
     it= 8: ==> B2= 1.67591e+29 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 6.12337e-09
     it= 9: ==> B2= 1.49457e+33 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 7.91207e-10
     logcf(*) end: after 9 iterations.
     logcf(x[ 8]= -3.25, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 91880 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00867282
     it= 2: ==> B2= 7.42974e+07 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00106327
     it= 3: ==> B2= 1.04819e+11 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000128994
     it= 4: ==> B2= 2.28837e+14 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.55909e-05
     it= 5: ==> B2= 7.15064e+17 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.88109e-06
     it= 6: ==> B2= 3.02848e+21 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.26734e-07
     it= 7: ==> B2= 1.67007e+25 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.7312e-08
     it= 8: ==> B2= 1.1628e+29 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.28859e-09
     it= 9: ==> B2= 9.97611e+32 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.95855e-10
     logcf(*) end: after 9 iterations.
     logcf(x[ 9]= -3, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 83360 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00753175
     it= 2: ==> B2= 6.46501e+07 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000853254
     it= 3: ==> B2= 8.75389e+10 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 9.5671e-05
     it= 4: ==> B2= 1.83464e+14 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.06874e-05
     it= 5: ==> B2= 5.50387e+17 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.19178e-06
     it= 6: ==> B2= 2.23803e+21 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.32765e-07
     it= 7: ==> B2= 1.18496e+25 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.47808e-08
     it= 8: ==> B2= 7.92152e+28 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.64484e-09
     it= 9: ==> B2= 6.52535e+32 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.82987e-10
     logcf(*) end: after 9 iterations.
     logcf(x[10]= -2.75, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 75240 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00641978
     it= 2: ==> B2= 5.58449e+07 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000665641
     it= 3: ==> B2= 7.24171e+10 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 6.83222e-05
     it= 4: ==> B2= 1.45381e+14 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 6.98689e-06
     it= 5: ==> B2= 4.17809e+17 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 7.13234e-07
     it= 6: ==> B2= 1.6276e+21 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 7.27338e-08
     it= 7: ==> B2= 8.25594e+24 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 7.41242e-09
     it= 8: ==> B2= 5.28764e+28 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 7.55083e-10
     logcf(*) end: after 8 iterations.
     logcf(x[11]= -2.5, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 67520 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00534792
     it= 2: ==> B2= 4.78456e+07 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.0005016
     it= 3: ==> B2= 5.92745e+10 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 4.65818e-05
     it= 4: ==> B2= 1.13708e+14 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 4.31004e-06
     it= 5: ==> B2= 3.12287e+17 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.98074e-07
     it= 6: ==> B2= 1.16261e+21 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.67278e-08
     it= 7: ==> B2= 5.6361e+24 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.38642e-09
     it= 8: ==> B2= 3.44989e+28 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.12099e-10
     logcf(*) end: after 8 iterations.
     logcf(x[12]= -2.25, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 60200 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00432911
     it= 2: ==> B2= 4.06159e+07 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00036199
     it= 3: ==> B2= 4.79397e+10 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.99753e-05
     it= 4: ==> B2= 8.76351e+13 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.47309e-06
     it= 5: ==> B2= 2.2937e+17 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.03667e-07
     it= 6: ==> B2= 8.13827e+20 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.67549e-08
     it= 7: ==> B2= 3.76013e+24 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.37743e-09
     it= 8: ==> B2= 2.19363e+28 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.13188e-10
     logcf(*) end: after 8 iterations.
     logcf(x[13]= -2, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 53280 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00337838
     it= 2: ==> B2= 3.41194e+07 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00024722
     it= 3: ==> B2= 3.82483e+10 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.79187e-05
     it= 4: ==> B2= 6.64186e+13 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.29399e-06
     it= 5: ==> B2= 1.6515e+17 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 9.32713e-08
     it= 6: ==> B2= 5.56707e+20 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 6.71575e-09
     it= 7: ==> B2= 2.44378e+24 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 4.83216e-10
     logcf(*) end: after 7 iterations.
     logcf(x[14]= -1.75, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 46760 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00251277
     it= 2: ==> B2= 2.83198e+07 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000157067
     it= 3: ==> B2= 3.0043e+10 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 9.72602e-06
     it= 4: ==> B2= 4.93794e+13 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 6.00029e-07
     it= 5: ==> B2= 1.16224e+17 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.69475e-08
     it= 6: ==> B2= 3.70875e+20 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.27255e-09
     it= 7: ==> B2= 1.54119e+24 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.39681e-10
     logcf(*) end: after 7 iterations.
     logcf(x[15]= -1.5, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 40640 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.001751
     it= 2: ==> B2= 2.3181e+07 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 9.04805e-05
     it= 3: ==> B2= 2.31738e+10 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 4.63221e-06
     it= 4: ==> B2= 3.59e+13 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.36258e-07
     it= 5: ==> B2= 7.96488e+16 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.20265e-08
     it= 6: ==> B2= 2.39588e+20 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 6.11497e-10
     logcf(*) end: after 6 iterations.
     logcf(x[16]= -1.25, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 34920 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00111245
     it= 2: ==> B2= 1.86665e+07 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 4.53708e-05
     it= 3: ==> B2= 1.74976e+10 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.8334e-06
     it= 4: ==> B2= 2.5422e+13 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 7.38016e-08
     it= 5: ==> B2= 5.29017e+16 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.96486e-09
     it= 6: ==> B2= 1.49263e+20 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.18968e-10
     logcf(*) end: after 6 iterations.
     logcf(x[17]= -1, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 29600 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000614941
     it= 2: ==> B2= 1.474e+07 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.84647e-05
     it= 3: ==> B2= 1.28785e+10 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 5.49306e-07
     it= 4: ==> B2= 1.74436e+13 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.62767e-08
     it= 5: ==> B2= 3.38438e+16 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 4.81303e-10
     logcf(*) end: after 5 iterations.
     logcf(x[18]= -0.75, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 24680 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000270385
     it= 2: ==> B2= 1.13653e+07 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 5.33194e-06
     it= 3: ==> B2= 9.18785e+09 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.04153e-07
     it= 4: ==> B2= 1.1517e+13 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.02617e-09
     it= 5: ==> B2= 2.06815e+16 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.93312e-11
     logcf(*) end: after 5 iterations.
     logcf(x[19]= -0.5, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 20160 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 7.69704e-05
     it= 2: ==> B2= 8.50608e+06 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 8.02466e-07
     it= 3: ==> B2= 6.30386e+09 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 8.28445e-09
     it= 4: ==> B2= 7.24564e+12 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 8.51583e-11
     logcf(*) end: after 4 iterations.
     logcf(x[20]= -0.25, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 16040 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 7.26392e-06
     it= 2: ==> B2= 6.12601e+06 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.30773e-08
     it= 3: ==> B2= 4.11202e+09 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 7.25571e-11
     logcf(*) end: after 3 iterations.
     logcf(x[21]= 0, i=2, d=3, eps=1e-09) iterations:
     logcf(*) end: after 0 iterations.
     logcf(x[22]= 0.25, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 9000 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.04918e-05
     it= 2: ==> B2= 2.65815e+06 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.08623e-07
     it= 3: ==> B2= 1.38218e+09 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 5.68393e-10
     logcf(*) end: after 3 iterations.
     logcf(x[23]= 0.5, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 6080 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000659523
     it= 2: ==> B2= 1.49776e+06 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.00942e-05
     it= 3: ==> B2= 6.50656e+08 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 6.02264e-07
     it= 4: ==> B2= 4.39124e+11 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.79273e-08
     it= 5: ==> B2= 4.24985e+14 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 5.3174e-10
     logcf(*) end: after 5 iterations.
     logcf(x[24]= 0.75, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 3560 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00856402
     it= 2: ==> B2= 671330 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00100335
     it= 3: ==> B2= 2.24237e+08 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000114482
     it= 4: ==> B2= 1.16565e+11 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.2922e-05
     it= 5: ==> B2= 8.69636e+13 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.45089e-06
     it= 6: ==> B2= 8.80714e+16 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.62421e-07
     it= 7: ==> B2= 1.16246e+20 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.81486e-08
     it= 8: ==> B2= 1.93847e+23 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.02539e-09
     it= 9: ==> B2= 3.98491e+26 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.25837e-10
     logcf(*) end: after 9 iterations.
     logcf(x[25]= 0.78125, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 3273.12 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.0116683
     it= 2: ==> B2= 589700 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00162549
     it= 3: ==> B2= 1.88377e+08 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000220072
     it= 4: ==> B2= 9.36959e+10 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.94426e-05
     it= 5: ==> B2= 6.68994e+13 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.9163e-06
     it= 6: ==> B2= 6.48488e+16 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 5.19244e-07
     it= 7: ==> B2= 8.19327e+19 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 6.87066e-08
     it= 8: ==> B2= 1.30789e+23 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 9.07922e-09
     it= 9: ==> B2= 2.57381e+26 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.19866e-09
     it=10: ==> B2= 6.12129e+29 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.58143e-10
     logcf(*) end: after 10 iterations.
     logcf(x[26]= 0.8125, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 2992.5 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.0159401
     it= 2: ==> B2= 512650 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00265674
     it= 3: ==> B2= 1.55894e+08 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000429426
     it= 4: ==> B2= 7.3859e+10 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 6.84941e-05
     it= 5: ==> B2= 5.02475e+13 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.08546e-05
     it= 6: ==> B2= 4.64164e+16 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.71404e-06
     it= 7: ==> B2= 5.58911e+19 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.70071e-07
     it= 8: ==> B2= 8.50347e+22 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 4.2492e-08
     it= 9: ==> B2= 1.595e+26 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 6.6788e-09
     it=10: ==> B2= 3.61574e+29 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.04899e-09
     it=11: ==> B2= 9.74479e+32 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.64668e-10
     logcf(*) end: after 11 iterations.
     logcf(x[27]= 0.84375, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 2718.12 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.0218736
     it= 2: ==> B2= 440109 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00440022
     it= 3: ==> B2= 1.26644e+08 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000856838
     it= 4: ==> B2= 5.68225e+10 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000164362
     it= 5: ==> B2= 3.66244e+13 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.12964e-05
     it= 6: ==> B2= 3.20598e+16 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 5.93505e-06
     it= 7: ==> B2= 3.65864e+19 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.12276e-06
     it= 8: ==> B2= 5.27587e+22 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.12056e-07
     it= 9: ==> B2= 9.37997e+25 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 4.00067e-08
     it=10: ==> B2= 2.01557e+29 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 7.54162e-09
     it=11: ==> B2= 5.14924e+32 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.42081e-09
     it=12: ==> B2= 1.54257e+36 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.67552e-10
     logcf(*) end: after 12 iterations.
     logcf(x[28]= 0.875, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 2450 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.0302147
     it= 2: ==> B2= 372006 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00742718
     it= 3: ==> B2= 1.00485e+08 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00176572
     it= 4: ==> B2= 4.23633e+10 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000412688
     it= 5: ==> B2= 2.56713e+13 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 9.56191e-05
     it= 6: ==> B2= 2.11343e+16 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.20491e-05
     it= 7: ==> B2= 2.26869e+19 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 5.0698e-06
     it= 8: ==> B2= 3.07772e+22 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.16356e-06
     it= 9: ==> B2= 5.14811e+25 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.66706e-07
     it=10: ==> B2= 1.04082e+29 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 6.10778e-08
     it=11: ==> B2= 2.50192e+32 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.39778e-08
     it=12: ==> B2= 7.05238e+35 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.19721e-09
     it=13: ==> B2= 2.30384e+39 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 7.31016e-10
     logcf(*) end: after 13 iterations.
     logcf(x[29]= 0.90625, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 2188.12 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.0421192
     it= 2: ==> B2= 308271 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.0128742
     it= 3: ==> B2= 7.72745e+07 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00381265
     it= 4: ==> B2= 3.02658e+10 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00110791
     it= 5: ==> B2= 1.70531e+13 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000318587
     it= 6: ==> B2= 1.30605e+16 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 9.10705e-05
     it= 7: ==> B2= 1.30466e+19 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.5941e-05
     it= 8: ==> B2= 1.64734e+22 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 7.37247e-06
     it= 9: ==> B2= 2.56499e+25 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.09206e-06
     it=10: ==> B2= 4.82765e+28 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 5.93012e-07
     it=11: ==> B2= 1.08039e+32 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.6796e-07
     it=12: ==> B2= 2.83535e+35 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 4.75428e-08
     it=13: ==> B2= 8.62389e+38 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.34511e-08
     it=14: ==> B2= 3.00926e+42 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.80425e-09
     it=15: ==> B2= 1.19409e+46 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.07559e-09
     it=16: ==> B2= 5.34632e+49 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.04032e-10
     logcf(*) end: after 16 iterations.
     logcf(x[30]= 0.9375, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 1932.5 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.0594391
     it= 2: ==> B2= 248832 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.02317
     it= 3: ==> B2= 5.68734e+07 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00882488
     it= 4: ==> B2= 2.03226e+10 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00329775
     it= 5: ==> B2= 1.04577e+13 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00121712
     it= 6: ==> B2= 7.32086e+15 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000445765
     it= 7: ==> B2= 6.68834e+18 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00016248
     it= 8: ==> B2= 7.72653e+21 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 5.90414e-05
     it= 9: ==> B2= 1.10096e+25 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.14101e-05
     it=10: ==> B2= 1.89662e+28 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 7.7527e-06
     it=11: ==> B2= 3.88536e+31 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.80437e-06
     it=12: ==> B2= 9.33474e+34 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.01363e-06
     it=13: ==> B2= 2.59938e+38 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.6615e-07
     it=14: ==> B2= 8.30457e+41 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.32201e-07
     it=15: ==> B2= 3.01718e+45 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 4.77137e-08
     it=16: ==> B2= 1.23692e+49 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.72154e-08
     it=17: ==> B2= 5.68258e+52 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 6.20986e-09
     it=18: ==> B2= 2.90768e+56 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.23951e-09
     it=19: ==> B2= 1.64796e+60 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 8.07503e-10
     logcf(*) end: after 19 iterations.
     logcf(x[31]= 0.96875, i=2, d=3, eps=1e-09) iterations:
     it= 1: ==> B2= 1683.12 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.0852619
     it= 2: ==> B2= 193619 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.0440308
     it= 3: ==> B2= 3.91439e+07 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.0227933
     it= 4: ==> B2= 1.23338e+10 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.0116823
     it= 5: ==> B2= 5.59551e+12 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00592272
     it= 6: ==> B2= 3.4562e+15 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00297607
     it= 7: ==> B2= 2.78868e+18 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00148555
     it= 8: ==> B2= 2.84748e+21 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.00073801
     it= 9: ==> B2= 3.58854e+24 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000365396
     it=10: ==> B2= 5.4701e+27 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 0.000180472
     it=11: ==> B2= 9.91885e+30 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 8.89791e-05
     it=12: ==> B2= 2.10987e+34 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 4.38122e-05
     it=13: ==> B2= 5.20269e+37 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.15512e-05
     it=14: ==> B2= 1.47211e+41 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.05929e-05
     it=15: ==> B2= 4.73732e+44 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 5.20351e-06
     it=16: ==> B2= 1.72036e+48 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.55486e-06
     it=17: ==> B2= 7.00164e+51 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.25391e-06
     it=18: ==> B2= 3.17394e+55 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 6.15209e-07
     it=19: ==> B2= 1.59374e+59 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.0176e-07
     it=20: ==> B2= 8.8205e+62 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.47979e-07
     it=21: ==> B2= 5.35623e+66 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 7.25526e-08
     it=22: ==> B2= 3.5541e+70 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 3.55657e-08
     it=23: ==> B2= 2.56724e+74 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.7432e-08
     it=24: ==> B2= 2.01172e+78 Lrg m.B2
     --> crit. |A2*b1 - a1*B2|/|b1*B2| = 8.54292e-09
     it=25: ==> B2= 147221 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 4.18617e-09
     it=26: ==> B2= 1.34508e+09 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 2.05108e-09
     it=27: ==> B2= 1.32142e+13 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 1.00487e-09
     it=28: ==> B2= 1.39232e+17 --> crit. |A2*b1 - a1*B2|/|b1*B2| = 4.92273e-10
     logcf(*) end: after 28 iterations.
     >
     > lR. <- logcfR.(x, i=2, d=3, eps=1e-9)
     > lR.t <- logcfR.(x, i=2, d=3, eps=1e-9, trace=TRUE) ; stopifnot(identical(lR.t, lR.))
     logcf(x[], i=2, d=3, eps=1e-09) iterations:
     it= 1: needIt: 30 TRUE, and 1 F.; length(x[<todo>])=30, m.B2= 23307.2
     it= 2: needIt: 30 TRUE; length(x[<todo>])=30, m.B2= 1.01159e+07
     it= 3: needIt: 30 TRUE; length(x[<todo>])=30, m.B2= 7.70605e+09
     it= 4: needIt: 28 TRUE, and 2 F.; length(x[<todo>])=28, m.B2= 1.00852e+13
     it= 5: needIt: 27 TRUE, and 1 F.; length(x[<todo>])=27, m.B2= 1.76419e+16
     it= 6: needIt: 24 TRUE, and 3 F.; length(x[<todo>])=24, m.B2= 4.75316e+19
     it= 7: needIt: 22 TRUE, and 2 F.; length(x[<todo>])=22, m.B2= 1.2798e+23
     it= 8: needIt: 20 TRUE, and 2 F.; length(x[<todo>])=20, m.B2= 3.63581e+26
     it= 9: needIt: 17 TRUE, and 3 F.; length(x[<todo>])=17, m.B2= 6.8674e+29
     it=10: needIt: 13 TRUE, and 4 F.; length(x[<todo>])=13, m.B2= 1.03776e+33
     it=11: needIt: 9 TRUE, and 4 F.; length(x[<todo>])= 9, m.B2= 3.09233e+35
     it=12: needIt: 5 TRUE, and 4 F.; length(x[<todo>])= 5, m.B2= 2.27357e+35
     it=13: needIt: 4 TRUE, and 1 F.; length(x[<todo>])= 4, m.B2= 4.04868e+38
     it=14: needIt: 3 TRUE, and 1 F.; length(x[<todo>])= 3, m.B2= 7.16537e+41
     it=15: needIt: 3 TRUE; length(x[<todo>])= 3, m.B2= 2.57468e+45
     it=16: needIt: 3 TRUE; length(x[<todo>])= 3, m.B2= 1.04393e+49
     it=17: needIt: 2 TRUE, and 1 F.; length(x[<todo>])= 2, m.B2= 1.99468e+52
     it=18: needIt: 2 TRUE; length(x[<todo>])= 2, m.B2= 9.60666e+55
     it=19: needIt: 2 TRUE; length(x[<todo>])= 2, m.B2= 5.12487e+59
     it=20: needIt: 1 TRUE, and 1 F.; length(x[<todo>])= 1, m.B2= 8.8205e+62
     it=21: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 5.35623e+66
     it=22: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 3.5541e+70
     it=23: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 2.56724e+74
     it=24: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 2.01172e+78 Lrg m.B2
     it=25: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 147221
     it=26: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 1.34508e+09
     it=27: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 1.32142e+13
     it=28: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 1.39232e+17
     logcf(*) end: after 28 iterations.
     >
     > all.equal(lC, lR., tol = 0) # TRUE !! (every where ?)
     [1] TRUE
     > all.equal(lR, lR., tol = 0) # TRUE !! " "
     [1] TRUE
     > stopifnot(all.equal(lC, lR., tol = 1e-15))
     > ## (even though they used eps=1e-9 .. i.e., are not *so* accurate)
     > showProc.time()
     Time (user system elapsed): 0.03 0.02 0.05
     >
     > ##--- now with improved logcfR.() {<< will become the new logcfR() at least for MPFR !}:
     >
     > ##require(Rmpfr) may be not, see if NS loading (via "::") is sufficient:
     > requireNamespace("Rmpfr") || quit("no")
     Loading required namespace: Rmpfr
     [1] TRUE
     > ## ----- ----------
     > xM <- Rmpfr::mpfr(x, 512)
     > (ct.14 <- system.time(lR.14 <- logcfR.(xM, i=2, d=3, eps=1e-20, trace=TRUE))) # 0.55 sec
     logcf(x[], i=2, d=3, eps=1e-20) iterations:
     it= 1: needIt: 30 TRUE, and 1 F.; length(x[<todo>])=30, m.B2= 23307.2
     it= 2: needIt: 30 TRUE; length(x[<todo>])=30, m.B2= 1.01159e+07
     it= 3: needIt: 30 TRUE; length(x[<todo>])=30, m.B2= 7.70605e+09
     it= 4: needIt: 30 TRUE; length(x[<todo>])=30, m.B2= 9.10781e+12
     it= 5: needIt: 30 TRUE; length(x[<todo>])=30, m.B2= 1.54287e+16
     it= 6: needIt: 30 TRUE; length(x[<todo>])=30, m.B2= 3.54543e+19
     it= 7: needIt: 30 TRUE; length(x[<todo>])=30, m.B2= 1.06137e+23
     it= 8: needIt: 29 TRUE, and 1 F.; length(x[<todo>])=29, m.B2= 4.19177e+26
     it= 9: needIt: 28 TRUE, and 1 F.; length(x[<todo>])=28, m.B2= 2.26761e+30
     it=10: needIt: 27 TRUE, and 1 F.; length(x[<todo>])=27, m.B2= 1.33011e+34
     it=11: needIt: 27 TRUE; length(x[<todo>])=27, m.B2= 9.0823e+37
     it=12: needIt: 26 TRUE, and 1 F.; length(x[<todo>])=26, m.B2= 7.15387e+41
     it=13: needIt: 25 TRUE, and 1 F.; length(x[<todo>])=25, m.B2= 6.21918e+45
     it=14: needIt: 24 TRUE, and 1 F.; length(x[<todo>])=24, m.B2= 9.51187e+49
     it=15: needIt: 23 TRUE, and 1 F.; length(x[<todo>])=23, m.B2= 1.04428e+54
     it=16: needIt: 22 TRUE, and 1 F.; length(x[<todo>])=22, m.B2= 1.19866e+58
     it=17: needIt: 21 TRUE, and 1 F.; length(x[<todo>])=21, m.B2= 1.40641e+62
     it=18: needIt: 20 TRUE, and 1 F.; length(x[<todo>])=20, m.B2= 1.64566e+66
     it=19: needIt: 19 TRUE, and 1 F.; length(x[<todo>])=19, m.B2= 1.86787e+70
     it=20: needIt: 17 TRUE, and 2 F.; length(x[<todo>])=17, m.B2= 9.5095e+73
     it=21: needIt: 15 TRUE, and 2 F.; length(x[<todo>])=15, m.B2= 2.07684e+78 Lrg m.B2
     it=22: needIt: 14 TRUE, and 1 F.; length(x[<todo>])=14, m.B2= 122830
     it=23: needIt: 11 TRUE, and 3 F.; length(x[<todo>])=11, m.B2= 3.76273e+08
     it=24: needIt: 10 TRUE, and 1 F.; length(x[<todo>])=10, m.B2= 7.77428e+11
     it=25: needIt: 7 TRUE, and 3 F.; length(x[<todo>])= 7, m.B2= 4.17254e+13
     it=26: needIt: 6 TRUE, and 1 F.; length(x[<todo>])= 6, m.B2= 1.55243e+15
     it=27: needIt: 5 TRUE, and 1 F.; length(x[<todo>])= 5, m.B2= 2.47748e+15
     it=28: needIt: 4 TRUE, and 1 F.; length(x[<todo>])= 4, m.B2= 1.06982e+19
     it=29: needIt: 4 TRUE; length(x[<todo>])= 4, m.B2= 1.40477e+23
     it=30: needIt: 4 TRUE; length(x[<todo>])= 4, m.B2= 1.9693e+27
     it=31: needIt: 3 TRUE, and 1 F.; length(x[<todo>])= 3, m.B2= 7.6538e+30
     it=32: needIt: 3 TRUE; length(x[<todo>])= 3, m.B2= 1.16488e+35
     it=33: needIt: 3 TRUE; length(x[<todo>])= 3, m.B2= 1.88175e+39
     it=34: needIt: 3 TRUE; length(x[<todo>])= 3, m.B2= 3.22081e+43
     it=35: needIt: 3 TRUE; length(x[<todo>])= 3, m.B2= 5.83159e+47
     it=36: needIt: 3 TRUE; length(x[<todo>])= 3, m.B2= 1.11521e+52
     it=37: needIt: 2 TRUE, and 1 F.; length(x[<todo>])= 2, m.B2= 3.51533e+55
     it=38: needIt: 2 TRUE; length(x[<todo>])= 2, m.B2= 7.10714e+59
     it=39: needIt: 2 TRUE; length(x[<todo>])= 2, m.B2= 1.51138e+64
     it=40: needIt: 2 TRUE; length(x[<todo>])= 2, m.B2= 3.37644e+68
     it=41: needIt: 2 TRUE; length(x[<todo>])= 2, m.B2= 7.91477e+72
     it=42: needIt: 2 TRUE; length(x[<todo>])= 2, m.B2= 1.94455e+77 Lrg m.B2
     it=43: needIt: 2 TRUE; length(x[<todo>])= 2, m.B2= 43197.4
     it=44: needIt: 2 TRUE; length(x[<todo>])= 2, m.B2= 1.16214e+09
     it=45: needIt: 1 TRUE, and 1 F.; length(x[<todo>])= 1, m.B2= 2.11103e+12
     it=46: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 5.83147e+16
     it=47: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 1.68004e+21
     it=48: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 5.04365e+25
     it=49: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 1.57649e+30
     it=50: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 5.12638e+34
     it=51: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 1.7329e+39
     it=52: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 6.08495e+43
     it=53: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 2.21796e+48
     it=54: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 8.38622e+52
     it=55: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 3.28706e+57
     it=56: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 1.33476e+62
     it=57: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 5.61156e+66
     it=58: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 2.44114e+71
     it=59: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 1.0982e+76
     it=60: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 5.10636e+80 Lrg m.B2
     it=61: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 2.1182e+08
     it=62: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 1.05047e+13
     it=63: needIt: 1 TRUE; length(x[<todo>])= 1, m.B2= 5.37602e+17
     logcf(*) end: after 63 iterations.
     user system elapsed
     1.06 0.00 1.06
     > (ct14 <- system.time(lR14 <- logcfR (xM, i=2, d=3, eps=1e-20, trace=TRUE))) # 4 sec
     logcf(x[ 1]= -5, i=2, d=3, eps=1e-20) logcf(*) end: after 26 iterations.
     logcf(x[ 2]= -4.75, i=2, d=3, eps=1e-20) logcf(*) end: after 25 iterations.
     logcf(x[ 3]= -4.5, i=2, d=3, eps=1e-20) logcf(*) end: after 24 iterations.
     logcf(x[ 4]= -4.25, i=2, d=3, eps=1e-20) logcf(*) end: after 24 iterations.
     logcf(x[ 5]= -4, i=2, d=3, eps=1e-20) logcf(*) end: after 23 iterations.
     logcf(x[ 6]= -3.75, i=2, d=3, eps=1e-20) logcf(*) end: after 22 iterations.
     logcf(x[ 7]= -3.5, i=2, d=3, eps=1e-20) logcf(*) end: after 22 iterations.
     logcf(x[ 8]= -3.25, i=2, d=3, eps=1e-20) logcf(*) end: after 21 iterations.
     logcf(x[ 9]= -3, i=2, d=3, eps=1e-20) logcf(*) end: after 20 iterations.
     logcf(x[10]= -2.75, i=2, d=3, eps=1e-20) logcf(*) end: after 19 iterations.
     logcf(x[11]= -2.5, i=2, d=3, eps=1e-20) logcf(*) end: after 19 iterations.
     logcf(x[12]= -2.25, i=2, d=3, eps=1e-20) logcf(*) end: after 18 iterations.
     logcf(x[13]= -2, i=2, d=3, eps=1e-20) logcf(*) end: after 17 iterations.
     logcf(x[14]= -1.75, i=2, d=3, eps=1e-20) logcf(*) end: after 16 iterations.
     logcf(x[15]= -1.5, i=2, d=3, eps=1e-20) logcf(*) end: after 15 iterations.
     logcf(x[16]= -1.25, i=2, d=3, eps=1e-20) logcf(*) end: after 14 iterations.
     logcf(x[17]= -1, i=2, d=3, eps=1e-20) logcf(*) end: after 12 iterations.
     logcf(x[18]= -0.75, i=2, d=3, eps=1e-20) logcf(*) end: after 11 iterations.
     logcf(x[19]= -0.5, i=2, d=3, eps=1e-20) logcf(*) end: after 9 iterations.
     logcf(x[20]= -0.25, i=2, d=3, eps=1e-20) logcf(*) end: after 7 iterations.
     logcf(x[21]= 0, i=2, d=3, eps=1e-20) logcf(*) end: after 0 iterations.
     logcf(x[22]= 0.25, i=2, d=3, eps=1e-20) logcf(*) end: after 8 iterations.
     logcf(x[23]= 0.5, i=2, d=3, eps=1e-20) logcf(*) end: after 13 iterations.
     logcf(x[24]= 0.75, i=2, d=3, eps=1e-20) logcf(*) end: after 20 iterations.
     logcf(x[25]= 0.78125, i=2, d=3, eps=1e-20) logcf(*) end: after 22 iterations.
     logcf(x[26]= 0.8125, i=2, d=3, eps=1e-20) logcf(*) end: after 24 iterations.
     logcf(x[27]= 0.84375, i=2, d=3, eps=1e-20) logcf(*) end: after 27 iterations.
     logcf(x[28]= 0.875, i=2, d=3, eps=1e-20) logcf(*) end: after 30 iterations.
     logcf(x[29]= 0.90625, i=2, d=3, eps=1e-20) logcf(*) end: after 36 iterations.
     logcf(x[30]= 0.9375, i=2, d=3, eps=1e-20) logcf(*) end: after 44 iterations.
     logcf(x[31]= 0.96875, i=2, d=3, eps=1e-20) logcf(*) end: after 63 iterations.
     user system elapsed
     11.03 0.00 11.01
     >
     > all.equal(lR.14, lR14, tol=0) # TRUE
     [1] TRUE
     > identical(lR.14, lR14) # TRUE !! (not sure if on all platforms!)
     [1] TRUE
     >
     > SS <- function(ch, digits=7)
     + sub(paste0("([0-9]{1,",digits,"})[0-9]*e"), "\\1e", ch)
     > ## double prec <--> MPFR: vvvv (same eps)
     > lR.9 <- logcfR.(xM, 2,3, eps=1e-9)
     > ## show:
     > SS(Rmpfr::all.equal(Rmpfr::roundMpfr(lR.9, 64), lR, tol=0))# .. 5.1138e-16
     Error in target == current : comparison of these types is not implemented
     Calls: SS ... <Anonymous> -> <Anonymous> -> .local -> all.equal.numeric
     Execution halted
    Running the tests in 'tests/qgamma-ex.R' failed.
    Complete output:
     > library(DPQ)
     >
     > ###---> Automatically find places where qgamma() is not so precise (PR#2214) :
     > ### For PR#2214, had '1e-8' below and found quite a bit
     > ## see /u/maechler/R/MM/NUMERICS/dpq-functions/beta-gamma-etc/qgamma-ex.R ..
     >
     > ## FIXME: Timing ! --- partly these matplot() partly get quite slow ~?
     > source(system.file(package="Matrix", "test-tools-1.R", mustWork=TRUE))
     Loading required package: tools
     > ##--> showProc.time(), assertError(), relErrV(), ...
     > showProc.time()
     Time (user system elapsed): 1.15 0.01 1.19
     >
     > (doExtras <- DPQ:::doExtras())
     [1] FALSE
     > (sdir <- system.file("safe", package="DPQ")) ## save directory (to read from)
     [1] "D:/temp/RtmpSktvIp/RLIBS_127bc1605436a/DPQ/safe"
     >
     > ### Nowadays finds cases in a special region for really small p and cutoff 1e-11 :
     > set.seed(47)
     > n <- if(doExtras) 100 else 32
     > res <- cbind(p=1,df=1,rE=1)[-1,]
     > for(M in 1:(if(doExtras) 20 else 10))
     + for(p in runif(n)) for(df in rlnorm(n)) {
     + r <- 1- pchisq(qchisq(p, df),df)/p
     + if(abs(r) > 1e-11) res <- rbind(res, c(p,df,r))
     + }
     >
     > ### use df in U[0,1]: finds two cases with bound 1e-11
     > for(p in runif(n)/2) for(df in runif(n)) {
     + qq <- qchisq(p, df)
     + if(qq > 0 && p > 0) {
     + r <- 1- pchisq(qq, df) / p
     + if(abs(r) > 1e-11) res <- rbind(res, c(p,df,r))
     + }
     + }
     >
     > ### now df very close to 0 : ==> finds more cases
     > for(p in sort(c(runif(64)/2, exp(-(1+rlnorm(256))))))
     + for(df in 2^-rlnorm(256, mean=2, sdlog=1.5)) {
     + qq <- qchisq(p, df)
     + if(qq > 0 && p > 0) {
     + r <- 1- pchisq(qq, df) / p
     + if(abs(r) > 1e-11) res <- rbind(res, c(p,df,r))
     + }
     + }
     > showProc.time()
     Time (user system elapsed): 0.6 0.02 0.6
     >
     > require(graphics)
     > if(!dev.interactive(orNone=TRUE)) pdf("qgamma-appr.pdf")
     > eaxis <- sfsmisc::eaxis
     >
     > showProc.time()
     Time (user system elapsed): 0.04 0 0.05
     > ## if(nrow(res) > 0) {
     > cat("Found inaccurate examples where pchisq(qchisq(p, df),df) != p\n")
     Found inaccurate examples where pchisq(qchisq(p, df),df) != p
     > ## sort in p, then df:
     > res <- res[order(res[,"p"], res[,"df"]), ]
     > rE <- res[,"rE"]
     > if(nrow(res) > 20) { hist(rE, breaks = 30); rug(rE) }
     > plot(res[,1:2])##--> quite interesting : all along one curve
     > ## p <= 1/2 and df <= 1 (about) !!
     > res <- cbind(res, nDig = round(-log10(abs(rE)), 1))
     > print(res, digits=12)
     p df rE nDig
     [1,] 0.000194375438651 0.02334079639198 -4.05718514340e-08 7.4
     [2,] 0.000605300028912 0.02041606754775 -1.99857908001e-11 10.7
     [3,] 0.001012316063255 0.01855615147677 -2.59145555106e-04 3.6
     [4,] 0.001248285290785 0.01838201076117 -2.84196000067e-10 9.5
     [5,] 0.001682388899865 0.01720736646288 5.53088974600e-04 3.3
     [6,] 0.001746787400790 0.01731189518997 -5.86897839217e-08 7.2
     [7,] 0.002664451237518 0.01599317398629 1.48421342013e-04 3.8
     [8,] 0.002664451237518 0.01618024201222 -3.82806282229e-08 7.4
     [9,] 0.003159421860255 0.01557612780310 -7.92117005632e-06 5.1
     [10,] 0.003159421860255 0.01568183691729 -4.52237520765e-08 7.3
     [11,] 0.004055462418244 0.01493858731306 4.15166391654e-06 5.4
     [12,] 0.004400694140827 0.01459101672970 9.07907026434e-04 3.0
     [13,] 0.004458811277768 0.01457506850867 9.03139988533e-05 4.0
     [14,] 0.004481882165743 0.01468883074316 -3.23309491179e-07 6.5
     [15,] 0.004939609905705 0.01440168350452 -2.81810098879e-06 5.6
     [16,] 0.008824465120182 0.01276352706510 1.21107345756e-04 3.9
     [17,] 0.009040265960535 0.01273711629661 1.38964402733e-05 4.9
     [18,] 0.010839089634828 0.01242499920422 2.63413624246e-10 9.6
     [19,] 0.011642124851282 0.01201471267173 1.44956234150e-04 3.8
     [20,] 0.014753716559535 0.01155624353203 1.52962087441e-10 9.8
     [21,] 0.015499213434879 0.01125420134457 -9.69695930770e-05 4.0
     [22,] 0.015499213434879 0.01135920381800 -9.55739012376e-08 7.0
     [23,] 0.018603016576955 0.01071716109330 1.63971046474e-03 2.8
     [24,] 0.018603016576955 0.01073655493589 2.14388784340e-04 3.7
     [25,] 0.022624242394389 0.01033379525113 -3.37865757594e-09 8.5
     [26,] 0.022624242394389 0.01034206121729 -2.76332994265e-08 7.6
     [27,] 0.023730217356634 0.01016252135853 -1.07732682708e-06 6.0
     [28,] 0.032427027472295 0.00942923095016 5.11205522358e-11 10.3
     [29,] 0.044753525441333 0.00839626444749 1.22224173549e-05 4.9
     [30,] 0.081818424963746 0.00686007746204 8.92777740624e-10 9.0
     [31,] 0.081818424963746 0.00689856335721 2.28502772259e-11 10.6
     [32,] 0.082800309102258 0.00681234719059 4.17997558788e-09 8.4
     [33,] 0.083507718914457 0.00680676700443 9.77167236016e-11 10.0
     [34,] 0.090821658072474 0.00655269761981 -7.16033632386e-09 8.1
     [35,] 0.102294760453517 0.00623563107239 3.69438657444e-09 8.4
     [36,] 0.110869751789691 0.00603268830251 -3.44006823028e-10 9.5
     [37,] 0.123950804624116 0.00571305309327 2.84683721041e-10 9.5
     [38,] 0.127405857731893 0.00562369059572 6.60541454867e-09 8.2
     [39,] 0.135229634154169 0.00540073357520 -2.34762594200e-05 4.6
     [40,] 0.137732279982451 0.00533092076413 2.99285844990e-04 3.5
     [41,] 0.138112917548194 0.00535138710974 -2.05335777981e-06 5.7
     [42,] 0.141100635980184 0.00527305771429 4.31593832968e-05 4.4
     [43,] 0.141100635980184 0.00537073537183 -3.00640179418e-10 9.5
     [44,] 0.142905299416015 0.00523680041306 3.48180824883e-04 3.5
     [45,] 0.145624557210331 0.00526923971034 -1.94501770245e-09 8.7
     [46,] 0.154606872884529 0.00506806894407 -4.59924667240e-07 6.3
     [47,] 0.154606872884529 0.00507366168703 2.72301046933e-07 6.6
     [48,] 0.163535630067488 0.00497650928578 3.39664962823e-11 10.5
     [49,] 0.169741036539408 0.00484181845356 5.31400978776e-09 8.3
     [50,] 0.177327576288650 0.00465956102839 5.53404362603e-05 4.3
     [51,] 0.178169157856761 0.00471949961255 4.79807527043e-10 9.3
     [52,] 0.190094017358772 0.00450373552308 -1.29698447116e-06 5.9
     [53,] 0.190147641510530 0.00453468705710 5.66235636157e-09 8.2
     [54,] 0.200112534472267 0.00442273120514 7.20473680715e-11 10.1
     [55,] 0.201518808589718 0.00439936964342 1.58748569845e-11 10.8
     [56,] 0.201518808589718 0.00439976887947 -9.97182336704e-11 10.0
     [57,] 0.210803673024037 0.00427351441034 -1.70232938856e-10 9.8
     [58,] 0.213058614771766 0.00426179831847 1.10152997834e-11 11.0
     [59,] 0.214780951412088 0.00419869272965 9.79194836326e-09 8.0
     [60,] 0.232805106603566 0.00395399315002 -9.17581020055e-08 7.0
     [61,] 0.249102914025652 0.00380019404026 -1.15818465929e-10 9.9
     [62,] 0.249102914025652 0.00382493512126 -1.39670497390e-11 10.9
     [63,] 0.252076511947811 0.00374903834738 -8.83337205604e-08 7.1
     [64,] 0.253082914021191 0.00375259362798 3.65436092498e-09 8.4
     [65,] 0.253922058700076 0.00371237348323 3.28994798726e-06 5.5
     [66,] 0.254289278570932 0.00374343873151 -1.05664899053e-09 9.0
     [67,] 0.260017499519858 0.00366179605930 2.34859742765e-07 6.6
     [68,] 0.270323906831467 0.00351999192121 -1.56164756277e-04 3.8
     [69,] 0.271699356057456 0.00355068132680 5.13092990317e-09 8.3
     [70,] 0.275516196070002 0.00346804047756 -4.35171547588e-04 3.4
     [71,] 0.280722231049885 0.00348224101220 5.48759926389e-10 9.3
     [72,] 0.284601233201101 0.00344936339590 1.57145851887e-10 9.8
     [73,] 0.290188543054775 0.00336613521112 -5.64443074502e-08 7.2
     [74,] 0.290579022038283 0.00334423496113 1.02667567781e-07 7.0
     [75,] 0.290579022038283 0.00336764858994 2.26061565023e-08 7.6
     [76,] 0.291850198713803 0.00333552811650 -1.27338760580e-06 5.9
     [77,] 0.296521136452775 0.00330308865102 2.25309977453e-07 6.6
     [78,] 0.298034174946132 0.00330462333485 8.42470393447e-09 8.1
     [79,] 0.300556783277253 0.00323922530004 4.66003314391e-05 4.3
     [80,] 0.303182283998467 0.00328704590597 -1.46205270113e-11 10.8
     [81,] 0.322319846303892 0.00306134512927 -1.15130830540e-05 4.9
     [82,] 0.322319846303892 0.00310689001755 8.57751647487e-11 10.1
     [83,] 0.325071272052651 0.00302343293053 -2.47088704493e-04 3.6
     [84,] 0.325071272052651 0.00304146419577 3.18761056051e-06 5.5
     [85,] 0.331888412404218 0.00300837121343 -4.96098895297e-09 8.3
     [86,] 0.362278153188527 0.00278204202032 4.53939330569e-10 9.3
     [87,] 0.385389476781711 0.00260981704384 7.37274796769e-10 9.1
     [88,] 0.425333956955001 0.00232995789362 1.82823025607e-08 7.7
     [89,] 0.439503709203564 0.00222452690840 -4.53585193982e-06 5.3
     [90,] 0.439503709203564 0.00224964327069 -3.02331937263e-10 9.5
     [91,] 0.450804624124430 0.00216770324934 -4.59455036239e-08 7.3
     >
     > if(requireNamespace("scatterplot3d")) {
     + scatterplot3d::scatterplot3d(res[,1:3], type ='h') ## quite interesting:
     + ## the inaccurate (p,df) points are on nice monotone curve !!!
     + ## this is *less* revealing
     + scatterplot3d::scatterplot3d(res[,c("p","df","nDig")], type ='h')
     + }
     Loading required namespace: scatterplot3d
     > rL <- res[abs(res[,'rE']) > 1e-9,]
     > rL <- rL[order(rL[,1],rL[,2]),]
     > rL
     p df rE nDig
     [1,] 0.0001943754 0.023340796 -4.057185e-08 7.4
     [2,] 0.0010123161 0.018556151 -2.591456e-04 3.6
     [3,] 0.0016823889 0.017207366 5.530890e-04 3.3
     [4,] 0.0017467874 0.017311895 -5.868978e-08 7.2
     [5,] 0.0026644512 0.015993174 1.484213e-04 3.8
     [6,] 0.0026644512 0.016180242 -3.828063e-08 7.4
     [7,] 0.0031594219 0.015576128 -7.921170e-06 5.1
     [8,] 0.0031594219 0.015681837 -4.522375e-08 7.3
     [9,] 0.0040554624 0.014938587 4.151664e-06 5.4
     [10,] 0.0044006941 0.014591017 9.079070e-04 3.0
     [11,] 0.0044588113 0.014575069 9.031400e-05 4.0
     [12,] 0.0044818822 0.014688831 -3.233095e-07 6.5
     [13,] 0.0049396099 0.014401684 -2.818101e-06 5.6
     [14,] 0.0088244651 0.012763527 1.211073e-04 3.9
     [15,] 0.0090402660 0.012737116 1.389644e-05 4.9
     [16,] 0.0116421249 0.012014713 1.449562e-04 3.8
     [17,] 0.0154992134 0.011254201 -9.696959e-05 4.0
     [18,] 0.0154992134 0.011359204 -9.557390e-08 7.0
     [19,] 0.0186030166 0.010717161 1.639710e-03 2.8
     [20,] 0.0186030166 0.010736555 2.143888e-04 3.7
     [21,] 0.0226242424 0.010333795 -3.378658e-09 8.5
     [22,] 0.0226242424 0.010342061 -2.763330e-08 7.6
     [23,] 0.0237302174 0.010162521 -1.077327e-06 6.0
     [24,] 0.0447535254 0.008396264 1.222242e-05 4.9
     [25,] 0.0828003091 0.006812347 4.179976e-09 8.4
     [26,] 0.0908216581 0.006552698 -7.160336e-09 8.1
     [27,] 0.1022947605 0.006235631 3.694387e-09 8.4
     [28,] 0.1274058577 0.005623691 6.605415e-09 8.2
     [29,] 0.1352296342 0.005400734 -2.347626e-05 4.6
     [30,] 0.1377322800 0.005330921 2.992858e-04 3.5
     [31,] 0.1381129175 0.005351387 -2.053358e-06 5.7
     [32,] 0.1411006360 0.005273058 4.315938e-05 4.4
     [33,] 0.1429052994 0.005236800 3.481808e-04 3.5
     [34,] 0.1456245572 0.005269240 -1.945018e-09 8.7
     [35,] 0.1546068729 0.005068069 -4.599247e-07 6.3
     [36,] 0.1546068729 0.005073662 2.723010e-07 6.6
     [37,] 0.1697410365 0.004841818 5.314010e-09 8.3
     [38,] 0.1773275763 0.004659561 5.534044e-05 4.3
     [39,] 0.1900940174 0.004503736 -1.296984e-06 5.9
     [40,] 0.1901476415 0.004534687 5.662356e-09 8.2
     [41,] 0.2147809514 0.004198693 9.791948e-09 8.0
     [42,] 0.2328051066 0.003953993 -9.175810e-08 7.0
     [43,] 0.2520765119 0.003749038 -8.833372e-08 7.1
     [44,] 0.2530829140 0.003752594 3.654361e-09 8.4
     [45,] 0.2539220587 0.003712373 3.289948e-06 5.5
     [46,] 0.2542892786 0.003743439 -1.056649e-09 9.0
     [47,] 0.2600174995 0.003661796 2.348597e-07 6.6
     [48,] 0.2703239068 0.003519992 -1.561648e-04 3.8
     [49,] 0.2716993561 0.003550681 5.130930e-09 8.3
     [50,] 0.2755161961 0.003468040 -4.351715e-04 3.4
     [51,] 0.2901885431 0.003366135 -5.644431e-08 7.2
     [52,] 0.2905790220 0.003344235 1.026676e-07 7.0
     [53,] 0.2905790220 0.003367649 2.260616e-08 7.6
     [54,] 0.2918501987 0.003335528 -1.273388e-06 5.9
     [55,] 0.2965211365 0.003303089 2.253100e-07 6.6
     [56,] 0.2980341749 0.003304623 8.424704e-09 8.1
     [57,] 0.3005567833 0.003239225 4.660033e-05 4.3
     [58,] 0.3223198463 0.003061345 -1.151308e-05 4.9
     [59,] 0.3250712721 0.003023433 -2.470887e-04 3.6
     [60,] 0.3250712721 0.003041464 3.187611e-06 5.5
     [61,] 0.3318884124 0.003008371 -4.960989e-09 8.3
     [62,] 0.4253339570 0.002329958 1.828230e-08 7.7
     [63,] 0.4395037092 0.002224527 -4.535852e-06 5.3
     [64,] 0.4508046241 0.002167703 -4.594550e-08 7.3
     > plot(rL[,1:2], type = "b", main = "inaccurate pchisq/qchisq pairs")
     >
     > plot(rL[,1:2], type = "b", log = "x", ylim = range(0, rL[,"df"]),
     + xaxt = "n",
     + main = "inaccurate pchisq/qchisq pairs"); abline(h = 0, lty=2)
     > ## aha -- a perfect line !!
     > lines(res[,1:2], col = adjustcolor(1, 0.5))
     > eaxis(1); axis(1, at = 1/2)
     >
     > d <- as.data.frame(res)
     > plot (df ~ log(p), data = d, type = "b", cex=1/4, col="gray")
     > points(df ~ log(p), data = as.data.frame(rL), col=2, cex = 1/2)
     >
     > summary(fm <- lm (df ~ log(p), data = d, weights = -log(abs(rE))))
    
     Call:
     lm(formula = df ~ log(p), data = d, weights = -log(abs(rE)))
    
     Weighted Residuals:
     Min 1Q Median 3Q Max
     -6.924e-04 -1.443e-04 -2.096e-05 7.786e-05 1.079e-03
    
     Coefficients:
     Estimate Std. Error t value Pr(>|t|)
     (Intercept) 5.168e-06 1.149e-05 0.45 0.654
     log(p) -2.725e-03 3.683e-06 -739.99 <2e-16 ***
     ---
     Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
    
     Residual standard error: 0.0002575 on 89 degrees of freedom
     Multiple R-squared: 0.9998, Adjusted R-squared: 0.9998
     F-statistic: 5.476e+05 on 1 and 89 DF, p-value: < 2.2e-16
    
     > ## R^2 = 0.9998
     >
     > p0 <- 2^seq(-50,-1, by=1/8)
     > dN <- data.frame(p = p0,
     + df = predict(fm, newdata = data.frame(p = p0)))
     > rE <- with(dN, 1- pchisq(qchisq(p, df),df)/p)
     > dN <- cbind(dN, rE = rE, nDig = round(-log10(abs(rE)), 1))
     > print(dN, digits=10)
     p df rE nDig
     1 8.881784197e-16 0.094454797738 -6.103205523e-07 6.2
     2 9.685654347e-16 0.094218673664 -2.417772020e-07 6.6
     3 1.056228096e-15 0.093982549590 1.482102507e-07 6.8
     4 1.151824906e-15 0.093746425517 5.596452761e-07 6.3
     5 1.256073967e-15 0.093510301443 9.925313446e-07 6.0
     6 1.369758374e-15 0.093274177369 -5.456116821e-07 6.3
     7 1.493732098e-15 0.093038053295 -6.476818548e-08 7.2
     8 1.628926404e-15 0.092801929221 4.375367997e-07 6.4
     9 1.776356839e-15 0.092565805147 9.613067427e-07 6.0
     10 1.937130869e-15 0.092329681073 -4.656781729e-07 6.3
     11 2.112456192e-15 0.092093557000 1.060769756e-07 7.0
     12 2.303649813e-15 0.091857432926 6.993075217e-07 6.2
     13 2.512147934e-15 0.091621308852 -6.429918018e-07 6.2
     14 2.739516748e-15 0.091385184778 -1.755261936e-09 8.8
     15 2.987464197e-15 0.091149060704 6.609671102e-07 6.2
     16 3.257852808e-15 0.090912936630 -5.966162733e-07 6.2
     17 3.552713679e-15 0.090676812556 1.141328682e-07 6.9
     18 3.874261739e-15 0.090440688483 8.463782721e-07 6.1
     19 4.224912384e-15 0.090204564409 -3.264588360e-07 6.5
     20 4.607299625e-15 0.089968440335 4.538341097e-07 6.3
     21 5.024295868e-15 0.089732316261 -6.607810550e-07 6.2
     22 5.479033495e-15 0.089496192187 1.675732028e-07 6.8
     23 5.974928394e-15 0.089260068113 -8.888068896e-07 6.1
     24 6.515705616e-15 0.089023944039 -1.237752500e-08 7.9
     25 7.105427358e-15 0.088787819965 8.855724453e-07 6.1
     26 7.748523477e-15 0.088551695892 -8.599117218e-08 7.1
     27 8.449824769e-15 0.088315571818 8.600546298e-07 6.1
     28 9.214599250e-15 0.088079447744 -5.324086838e-08 7.3
     29 1.004859174e-14 0.087843323670 -9.348370664e-07 6.0
     30 1.095806699e-14 0.087607199596 8.590025435e-08 7.1
     31 1.194985679e-14 0.087371075522 -7.374084532e-07 6.1
     32 1.303141123e-14 0.087134951448 3.314590431e-07 6.5
     33 1.421085472e-14 0.086898827375 -4.335490891e-07 6.4
     34 1.549704695e-14 0.086662703301 6.834623328e-07 6.2
     35 1.689964954e-14 0.086426579227 -2.323216441e-08 7.6
     36 1.842919850e-14 0.086190455153 -6.982056056e-07 6.2
     37 2.009718347e-14 0.085954331079 4.935691293e-07 6.3
     38 2.191613398e-14 0.085718207005 -1.230713562e-07 6.9
     39 2.389971358e-14 0.085482082931 -7.079815914e-07 6.1
     40 2.606282246e-14 0.085245958858 5.585870981e-07 6.3
     41 2.842170943e-14 0.085009834784 3.202910948e-08 7.5
     42 3.099409391e-14 0.084773710710 -4.627894483e-07 6.3
     43 3.379929908e-14 0.084537586636 8.786037984e-07 6.1
     44 3.685839700e-14 0.084301462562 4.421567533e-07 6.4
     45 4.019436694e-14 0.084065338488 3.745830501e-08 7.4
     46 4.383226796e-14 0.083829214414 -3.354886196e-07 6.5
     47 4.779942715e-14 0.083593090340 -6.766810949e-07 6.2
     48 5.212564492e-14 0.083356966267 7.928490436e-07 6.1
     49 5.684341886e-14 0.083120842193 5.100597688e-07 6.3
     50 6.198818782e-14 0.082884718119 2.590340894e-07 6.6
     51 6.759859815e-14 0.082648594045 3.977491347e-08 7.4
     52 7.371679400e-14 0.082412469971 -1.477148375e-07 6.8
     53 8.038873388e-14 0.082176345897 -3.034322495e-07 6.5
     54 8.766453592e-14 0.081940221823 -4.273744101e-07 6.4
     55 9.559885430e-14 0.081704097750 -5.195384087e-07 6.3
     56 1.042512898e-13 0.081467973676 -5.799213367e-07 6.2
     57 1.136868377e-13 0.081231849602 -6.085202819e-07 6.2
     58 1.239763756e-13 0.080995725528 -6.053323498e-07 6.2
     59 1.351971963e-13 0.080759601454 -5.703546275e-07 6.2
     60 1.474335880e-13 0.080523477380 -5.035842232e-07 6.3
     61 1.607774678e-13 0.080287353306 -4.050182332e-07 6.4
     62 1.753290718e-13 0.080051229233 -2.746537600e-07 6.6
     63 1.911977086e-13 0.079815105159 -1.124879079e-07 6.9
     64 2.085025797e-13 0.079578981085 8.148221731e-08 7.1
     65 2.273736754e-13 0.079342857011 3.072595063e-07 6.5
     66 2.479527513e-13 0.079106732937 5.648468501e-07 6.2
     67 2.703943926e-13 0.078870608863 -8.276082344e-07 6.1
     68 2.948671760e-13 0.078634484789 -5.012849011e-07 6.3
     69 3.215549355e-13 0.078398360715 -1.431432406e-07 6.8
     70 3.506581437e-13 0.078162236642 2.468196286e-07 6.6
     71 3.823954172e-13 0.077926112568 6.686065874e-07 6.2
     72 4.170051594e-13 0.077689988494 -5.340347500e-07 6.3
     73 4.547473509e-13 0.077453864420 -4.348589755e-08 7.4
     74 4.959055026e-13 0.077217740346 4.788952904e-07 6.3
     75 5.407887852e-13 0.076981616272 -6.077622430e-07 6.2
     76 5.897343520e-13 0.076745492198 -1.660406568e-08 7.8
     77 6.431098711e-13 0.076509368125 6.063946588e-07 6.2
     78 7.013162874e-13 0.076273244051 -3.642604993e-07 6.4
     79 7.647908344e-13 0.076037119977 3.275302674e-07 6.5
     80 8.340103188e-13 0.075800995903 -5.640569893e-07 6.2
     81 9.094947018e-13 0.075564871829 1.965355025e-07 6.7
     82 9.918110051e-13 0.075328747755 -6.159765633e-07 6.2
     83 1.081577570e-12 0.075092623681 2.134273159e-07 6.7
     84 1.179468704e-12 0.074856499608 -5.200022886e-07 6.3
     85 1.286219742e-12 0.074620375534 3.782226209e-07 6.4
     86 1.402632575e-12 0.074384251460 -2.761172913e-07 6.6
     87 1.529581669e-12 0.074148127386 6.909382704e-07 6.2
     88 1.668020638e-12 0.073912003312 1.156952742e-07 6.9
     89 1.818989404e-12 0.073675879238 -4.174157031e-07 6.4
     90 1.983622010e-12 0.073439755164 6.554522081e-07 6.2
     91 2.163155141e-12 0.073203631090 2.014482196e-07 6.7
     92 2.358937408e-12 0.072967507017 -2.104198369e-07 6.7
     93 2.572439484e-12 0.072731382943 -5.801509328e-07 6.2
     94 2.805265149e-12 0.072495258869 6.355293793e-07 6.2
     95 3.059163338e-12 0.072259134795 3.449181578e-07 6.5
     96 3.336041275e-12 0.072023010721 9.644777399e-08 7.0
     97 3.637978807e-12 0.071786886647 -1.098807569e-07 7.0
     98 3.967244020e-12 0.071550762573 -2.740664233e-07 6.6
     99 4.326310282e-12 0.071314638500 -3.961082171e-07 6.4
     100 4.717874816e-12 0.071078514426 -4.760051342e-07 6.3
     101 5.144878969e-12 0.070842390352 -5.137561683e-07 6.3
     102 5.610530299e-12 0.070606266278 -5.093603328e-07 6.3
     103 6.118326675e-12 0.070370142204 -4.628166292e-07 6.3
     104 6.672082550e-12 0.070134018130 -3.741240624e-07 6.4
     105 7.275957614e-12 0.069897894056 -2.432816564e-07 6.6
     106 7.934488041e-12 0.069661769983 -7.028842375e-08 7.2
     107 8.652620563e-12 0.069425645909 1.448566145e-07 6.8
     108 9.435749632e-12 0.069189521835 4.021544341e-07 6.4
     109 1.028975794e-11 0.068953397761 7.016060063e-07 6.2
     110 1.122106060e-11 0.068717273687 -4.176446966e-07 6.4
     111 1.223665335e-11 0.068481149613 -2.873861349e-08 7.5
     112 1.334416510e-11 0.068245025539 4.023232923e-07 6.4
     113 1.455191523e-11 0.068008901466 -5.698258159e-07 6.2
     114 1.586897608e-11 0.067772777392 -4.930794928e-08 7.3
     115 1.730524113e-11 0.067536653318 5.133677865e-07 6.3
     116 1.887149926e-11 0.067300529244 -3.116848903e-07 6.5
     117 2.057951587e-11 0.067064405170 3.404481869e-07 6.5
     118 2.244212120e-11 0.066828281096 -3.848110663e-07 6.4
     119 2.447330670e-11 0.066592157022 3.567796153e-07 6.4
     120 2.668833020e-11 0.066356032948 -2.686900942e-07 6.6
     121 2.910383046e-11 0.066119908875 5.623584213e-07 6.2
     122 3.173795216e-11 0.065883784801 3.667433890e-08 7.4
     123 3.461048225e-11 0.065647660727 -4.365232051e-07 6.4
     124 3.774299853e-11 0.065411536653 5.312784832e-07 6.3
     125 4.115903175e-11 0.065175412579 1.578594965e-07 6.8
     126 4.488424239e-11 0.064939288505 -1.630782713e-07 6.8
     127 4.894661340e-11 0.064703164431 -4.315370177e-07 6.4
     128 5.337666040e-11 0.064467040358 -6.475189409e-07 6.2
     129 5.820766091e-11 0.064230916284 5.516171970e-07 6.3
     130 6.347590433e-11 0.063994792210 4.354003237e-07 6.4
     131 6.922096451e-11 0.063758668136 3.716548743e-07 6.4
     132 7.548599706e-11 0.063522544062 3.603786256e-07 6.4
     133 8.231806350e-11 0.063286419988 4.015693486e-07 6.4
     134 8.976848478e-11 0.063050295914 4.952248145e-07 6.3
     135 9.789322680e-11 0.062814171841 6.413427747e-07 6.2
     136 1.067533208e-10 0.062578047767 -4.864750069e-07 6.3
     137 1.164153218e-10 0.062341923693 -2.302655673e-07 6.6
     138 1.269518087e-10 0.062105799619 7.839837224e-08 7.1
     139 1.384419290e-10 0.061869675545 4.395145565e-07 6.4
     140 1.509719941e-10 0.061633551471 -4.525762940e-07 6.3
     141 1.646361270e-10 0.061397427397 1.860557897e-08 7.7
     142 1.795369696e-10 0.061161303323 5.422316337e-07 6.3
     143 1.957864536e-10 0.060925179250 -1.718041758e-07 6.8
     144 2.135066416e-10 0.060689055176 4.618674380e-07 6.3
     145 2.328306437e-10 0.060452931102 -1.317492637e-07 6.9
     146 2.539036173e-10 0.060216807028 6.119535279e-07 6.2
     147 2.768838580e-10 0.059980682954 1.387356774e-07 6.9
     148 3.019439882e-10 0.059744558880 -2.716851346e-07 6.6
     149 3.292722540e-10 0.059508434806 -6.193156146e-07 6.2
     150 3.590739391e-10 0.059272310733 3.495619002e-07 6.5
     151 3.915729072e-10 0.059036186659 1.222864625e-07 6.9
     152 4.270132832e-10 0.058800062585 -4.221712535e-08 7.4
     153 4.656612873e-10 0.058563938511 -1.439555941e-07 6.8
     154 5.078072346e-10 0.058327814437 -1.829356875e-07 6.7
     155 5.537677160e-10 0.058091690363 -1.591641481e-07 6.8
     156 6.038879765e-10 0.057855566289 -7.264772339e-08 7.1
     157 6.585445080e-10 0.057619442216 7.660682388e-08 7.1
     158 7.181478783e-10 0.057383318142 2.885927336e-07 6.5
     159 7.831458144e-10 0.057147194068 5.633032377e-07 6.2
     160 8.540265665e-10 0.056911069994 -3.010137530e-07 6.5
     161 9.313225746e-10 0.056674945920 1.043142669e-07 7.0
     162 1.015614469e-09 0.056438821846 5.723448570e-07 6.2
     163 1.107535432e-09 0.056202697772 -8.306450194e-08 7.1
     164 1.207775953e-09 0.055966573698 5.155342444e-07 6.3
     165 1.317089016e-09 0.055730449625 1.091097324e-09 9.0
     166 1.436295757e-09 0.055494325551 -4.402707221e-07 6.4
     167 1.566291629e-09 0.055258201477 3.567051199e-07 6.4
     168 1.708053133e-09 0.055022077403 5.624481947e-08 7.2
     169 1.862645149e-09 0.054785953329 -1.711695874e-07 6.8
     170 2.031228938e-09 0.054549829255 -3.255506258e-07 6.5
     171 2.215070864e-09 0.054313705181 -4.069108348e-07 6.4
     172 2.415551906e-09 0.054077581108 -4.152627509e-07 6.4
     173 2.634178032e-09 0.053841457034 -3.506189168e-07 6.5
     174 2.872591513e-09 0.053605332960 -2.129918883e-07 6.7
     175 3.132583258e-09 0.053369208886 -2.394219267e-09 8.6
     176 3.416106266e-09 0.053133084812 2.811615281e-07 6.6
     177 3.725290298e-09 0.052896960738 -4.754652279e-07 6.3
     178 4.062457877e-09 0.052660836664 -4.082857386e-08 7.4
     179 4.430141728e-09 0.052424712591 4.667263255e-07 6.3
     180 4.831103812e-09 0.052188588517 -5.029512984e-08 7.3
     181 5.268356064e-09 0.051952464443 -4.839869736e-07 6.3
     182 5.745183026e-09 0.051716340369 2.526345019e-07 6.6
     183 6.265166516e-09 0.051480216295 -1.969240859e-08 7.7
     184 6.832212532e-09 0.051244092221 -2.087459234e-07 6.7
     185 7.450580597e-09 0.051007968147 -3.145456378e-07 6.5
     186 8.124915754e-09 0.050771844073 -3.371111525e-07 6.5
     187 8.860283457e-09 0.050535720000 -2.764620699e-07 6.6
     188 9.662207623e-09 0.050299595926 -1.326180039e-07 6.9
     189 1.053671213e-08 0.050063471852 9.440143545e-08 7.0
     190 1.149036605e-08 0.049827347778 4.045766331e-07 6.4
     191 1.253033303e-08 0.049591223704 -2.420870786e-07 6.6
     192 1.366442506e-08 0.049355099630 2.395534053e-07 6.6
     193 1.490116119e-08 0.049118975556 -2.252219731e-07 6.6
     194 1.624983151e-08 0.048882851483 4.277949056e-07 6.4
     195 1.772056691e-08 0.048646727409 1.448079123e-07 6.8
     196 1.932441525e-08 0.048410603335 -4.470065695e-08 7.3
     197 2.107342426e-08 0.048174479261 -1.407587265e-07 6.9
     198 2.298073210e-08 0.047938355187 -1.433942072e-07 6.8
     199 2.506066606e-08 0.047702231113 -5.263501346e-08 7.3
     200 2.732885013e-08 0.047466107039 1.314909412e-07 6.9
     201 2.980232239e-08 0.047229982966 4.089557365e-07 6.4
     202 3.249966302e-08 0.046993858892 -2.026429655e-07 6.7
     203 3.544113383e-08 0.046757734818 2.666382094e-07 6.6
     204 3.864883049e-08 0.046521610744 -1.427213314e-07 6.8
     205 4.214684851e-08 0.046285486670 -4.483846512e-07 6.3
     206 4.596146421e-08 0.046049362596 3.109955742e-07 6.5
     207 5.012133212e-08 0.045813238522 2.073633690e-07 6.7
     208 5.465770025e-08 0.045577114448 2.073183791e-07 6.7
     209 5.960464478e-08 0.045340990375 3.108231461e-07 6.5
     210 6.499932603e-08 0.045104866301 -4.225693533e-07 6.4
     211 7.088226765e-08 0.044868742227 -1.068421209e-07 7.0
     212 7.729766099e-08 0.044632618153 3.123191195e-07 6.5
     213 8.429369702e-08 0.044396494079 -8.979924182e-08 7.0
     214 9.192292842e-08 0.044160370005 -3.780712245e-07 6.4
     215 1.002426642e-07 0.043924245931 3.616102555e-07 6.4
     216 1.093154005e-07 0.043688121858 2.956315928e-07 6.5
     217 1.192092896e-07 0.043451997784 3.433583979e-07 6.5
     218 1.299986521e-07 0.043215873710 -3.936603858e-07 6.4
     219 1.417645353e-07 0.042979749636 -1.134241214e-07 6.9
     220 1.545953220e-07 0.042743625562 2.803691621e-07 6.6
     221 1.685873940e-07 0.042507501488 -9.497757625e-08 7.0
     222 1.838458568e-07 0.042271377414 -3.463649789e-07 6.5
     223 2.004853285e-07 0.042035253341 3.982655115e-07 6.4
     224 2.186308010e-07 0.041799129267 3.893573904e-07 6.4
     225 2.384185791e-07 0.041563005193 -3.573736389e-07 6.4
     226 2.599973041e-07 0.041326881119 -1.135260268e-07 6.9
     227 2.835290706e-07 0.041090757045 2.539798858e-07 6.6
     228 3.091906439e-07 0.040854632971 -1.007498562e-07 7.0
     229 3.371747881e-07 0.040618508897 -3.214330786e-07 6.5
     230 3.676917137e-07 0.040382384823 -4.081432381e-07 6.4
     231 4.009706570e-07 0.040146260750 -3.609537489e-07 6.4
     232 4.372616020e-07 0.039910136676 -1.799379790e-07 6.7
     233 4.768371582e-07 0.039674012602 1.348307426e-07 6.9
     234 5.199946082e-07 0.039437888528 -2.309643210e-07 6.6
     235 5.670581412e-07 0.039201764454 3.563334882e-07 6.4
     236 6.183812879e-07 0.038965640380 2.734302668e-07 6.6
     237 6.743495762e-07 0.038729516306 3.344816761e-07 6.5
     238 7.353834273e-07 0.038493392233 -2.537711148e-07 6.6
     239 8.019413140e-07 0.038257268159 1.001817764e-07 7.0
     240 8.745232040e-07 0.038021144085 -1.848126738e-07 6.7
     241 9.536743164e-07 0.037785020011 -3.156868191e-07 6.5
     242 1.039989216e-06 0.037548895937 -2.925440334e-07 6.5
     243 1.134116282e-06 0.037312771863 -1.154876044e-07 6.9
     244 1.236762576e-06 0.037076647789 2.153792568e-07 6.7
     245 1.348699152e-06 0.036840523716 -5.632337241e-08 7.2
     246 1.470766855e-06 0.036604399642 -1.638943095e-07 6.8
     247 1.603882628e-06 0.036368275568 -1.074536546e-07 7.0
     248 1.749046408e-06 0.036132151494 1.128786138e-07 6.9
     249 1.907348633e-06 0.035896027420 -2.381848667e-07 6.6
     250 2.079978433e-06 0.035659903346 3.148260195e-07 6.5
     251 2.268232565e-06 0.035423779272 3.067288682e-07 6.5
     252 2.473525152e-06 0.035187655198 -2.467805462e-07 6.6
     253 2.697398305e-06 0.034951531125 9.809280876e-08 7.0
     254 2.941533709e-06 0.034715407051 -9.217529851e-08 7.0
     255 3.207765256e-06 0.034479282977 -9.840819182e-08 7.0
     256 3.498092816e-06 0.034243158903 7.923719236e-08 7.1
     257 3.814697266e-06 0.034007034829 -2.523280687e-07 6.6
     258 4.159956866e-06 0.033770910755 2.978638337e-07 6.5
     259 4.536465130e-06 0.033534786681 -3.332980651e-07 6.5
     260 4.947050303e-06 0.033298662608 -8.305710231e-08 7.1
     261 5.394796609e-06 0.033062538534 -3.110326443e-07 6.5
     262 5.883067419e-06 0.032826414460 3.314533900e-07 6.5
     263 6.415530512e-06 0.032590290386 -1.553353934e-07 6.8
     264 6.996185632e-06 0.032354166312 2.279407816e-07 6.6
     265 7.629394531e-06 0.032118042238 1.638951952e-07 6.8
     266 8.319913732e-06 0.031881918164 3.135417902e-07 6.5
     267 9.072930260e-06 0.031645794091 3.656239145e-08 7.4
     268 9.894100606e-06 0.031409670017 -1.661172178e-08 7.8
     269 1.078959322e-05 0.031173545943 1.537749410e-07 6.8
     270 1.176613484e-05 0.030937421869 -7.675075331e-08 7.1
     271 1.283106102e-05 0.030701297795 -7.365042931e-08 7.1
     272 1.399237126e-05 0.030465173721 1.628068427e-07 6.8
     273 1.525878906e-05 0.030229049647 2.398762255e-08 7.6
     274 1.663982746e-05 0.029992925573 1.285369500e-07 6.9
     275 1.814586052e-05 0.029756801500 -1.216179055e-07 6.9
     276 1.978820121e-05 0.029520677426 -1.184344491e-07 6.9
     277 2.157918644e-05 0.029284553352 1.377656010e-07 6.9
     278 2.353226967e-05 0.029048429278 6.475007264e-08 7.2
     279 2.566212205e-05 0.028812305204 2.546562329e-07 6.6
     280 2.798474253e-05 0.028576181130 1.358057725e-07 6.9
     281 3.051757812e-05 0.028340057056 -2.762893481e-07 6.6
     282 3.327965493e-05 0.028103932983 1.553088330e-07 6.8
     283 3.629172104e-05 0.027867808909 -2.519742968e-07 6.6
     284 3.957640242e-05 0.027631684835 1.836182567e-07 6.7
     285 4.315837288e-05 0.027395560761 -1.887623422e-07 6.7
     286 4.706453935e-05 0.027159436687 -2.586992685e-07 6.6
     287 5.132424410e-05 0.026923312613 -2.666509724e-08 7.6
     288 5.596948506e-05 0.026687188539 -2.210356298e-08 7.7
     289 6.103515625e-05 0.026451064466 -2.296375028e-07 6.6
     290 6.655930986e-05 0.026214940392 -1.155410594e-07 6.9
     291 7.258344208e-05 0.025978816318 -1.934344165e-07 6.7
     292 7.915280485e-05 0.025742692244 5.977599671e-08 7.2
     293 8.631674575e-05 0.025506568170 1.410127990e-07 6.9
     294 9.412907870e-05 0.025270444096 6.554035969e-08 7.2
     295 1.026484882e-04 0.025034320022 -1.513968091e-07 6.8
     296 1.119389701e-04 0.024798195948 -7.984730210e-09 8.1
     297 1.220703125e-04 0.024562071875 1.377294045e-08 7.9
     298 1.331186197e-04 0.024325947801 -7.096806409e-08 7.1
     299 1.451668842e-04 0.024089823727 2.236347675e-07 6.7
     300 1.583056097e-04 0.023853699653 -3.406037585e-08 7.5
     301 1.726334915e-04 0.023617575579 1.071465752e-07 7.0
     302 1.882581574e-04 0.023381451505 1.915567744e-07 6.7
     303 2.052969764e-04 0.023145327431 -2.153880097e-07 6.7
     304 2.238779402e-04 0.022909203358 -1.943811740e-07 6.7
     305 2.441406250e-04 0.022673079284 -1.853585212e-07 6.7
     306 2.662372394e-04 0.022436955210 -1.734718722e-07 6.8
     307 2.903337683e-04 0.022200831136 -1.439218851e-07 6.8
     308 3.166112194e-04 0.021964707062 -8.196069157e-08 7.1
     309 3.452669830e-04 0.021728582988 2.710549296e-08 7.6
     310 3.765163148e-04 0.021492458914 1.979137868e-07 6.7
     311 4.105939528e-04 0.021256334841 3.784053959e-08 7.4
     312 4.477558805e-04 0.021020210767 -2.082511896e-08 7.7
     313 4.882812500e-04 0.020784086693 3.635452273e-08 7.4
     314 5.324744788e-04 0.020547962619 -1.675743397e-07 6.8
     315 5.806675366e-04 0.020311838545 1.696075673e-07 6.8
     316 6.332224388e-04 0.020075714471 -9.599203099e-08 7.0
     317 6.905339660e-04 0.019839590397 -1.676104142e-07 6.8
     318 7.530326296e-04 0.019603466323 -3.122729808e-08 7.5
     319 8.211879055e-04 0.019367342250 -3.779823365e-08 7.4
     320 8.955117609e-04 0.019131218176 -1.576477511e-07 6.8
     321 9.765625000e-04 0.018895094102 -6.902105554e-09 8.2
     322 1.064948958e-03 0.018658970028 7.899459276e-08 7.1
     323 1.161335073e-03 0.018422845954 1.293463403e-07 6.9
     324 1.266444878e-03 0.018186721880 -1.651599366e-07 6.8
     325 1.381067932e-03 0.017950597806 -9.328012252e-08 7.0
     326 1.506065259e-03 0.017714473733 3.008480975e-08 7.5
     327 1.642375811e-03 0.017478349659 -8.903183435e-08 7.1
     328 1.791023522e-03 0.017242225585 -8.893825099e-08 7.1
     329 1.953125000e-03 0.017006101511 5.866337671e-08 7.2
     330 2.129897915e-03 0.016769977437 7.502565791e-08 7.1
     331 2.322670146e-03 0.016533853363 3.812254845e-09 8.4
     332 2.532889755e-03 0.016297729289 -1.115639670e-07 7.0
     333 2.762135864e-03 0.016061605216 6.307704026e-08 7.2
     334 3.012130518e-03 0.015825481142 -1.675984262e-08 7.8
     335 3.284751622e-03 0.015589357068 -1.223429025e-08 7.9
     336 3.582047044e-03 0.015353232994 1.188536267e-07 6.9
     337 3.906250000e-03 0.015117108920 -1.217006762e-07 6.9
     338 4.259795831e-03 0.014880984846 -1.314145734e-07 6.9
     339 4.645340293e-03 0.014644860772 -1.288061289e-07 6.9
     340 5.065779510e-03 0.014408736698 -5.759581900e-08 7.2
     341 5.524271728e-03 0.014172612625 -1.110605872e-07 7.0
     342 6.024261037e-03 0.013936488551 2.552848177e-08 7.6
     343 6.569503244e-03 0.013700364477 -7.038977512e-08 7.2
     344 7.164094088e-03 0.013464240403 -8.014783437e-08 7.1
     345 7.812500000e-03 0.013228116329 6.514375728e-08 7.2
     346 8.519591661e-03 0.012991992255 -1.226700763e-08 7.9
     347 9.290680586e-03 0.012755868181 3.829301320e-09 8.4
     348 1.013155902e-02 0.012519744108 -1.720012310e-08 7.8
     349 1.104854346e-02 0.012283620034 2.105845986e-08 7.7
     350 1.204852207e-02 0.012047495960 1.170781816e-08 7.9
     351 1.313900649e-02 0.011811371886 6.422402798e-08 7.2
     352 1.432818818e-02 0.011575247812 9.486124830e-08 7.0
     353 1.562500000e-02 0.011339123738 3.894983802e-08 7.4
     354 1.703918332e-02 0.011102999664 3.208461419e-08 7.5
     355 1.858136117e-02 0.010866875591 3.155170258e-08 7.5
     356 2.026311804e-02 0.010630751517 1.297325658e-08 7.9
     357 2.209708691e-02 0.010394627443 -3.001927595e-08 7.5
     358 2.409704415e-02 0.010158503369 7.470776553e-08 7.1
     359 2.627801298e-02 0.009922379295 2.881775985e-08 7.5
     360 2.865637635e-02 0.009686255221 4.355807892e-08 7.4
     361 3.125000000e-02 0.009450131147 3.486255773e-08 7.5
     362 3.407836665e-02 0.009214007073 -4.540194665e-08 7.3
     363 3.716272234e-02 0.008977883000 6.057274149e-08 7.2
     364 4.052623608e-02 0.008741758926 -4.748040361e-08 7.3
     365 4.419417382e-02 0.008505634852 -4.947996146e-08 7.3
     366 4.819408829e-02 0.008269510778 5.303485540e-09 8.3
     367 5.255602595e-02 0.008033386704 2.750288552e-09 8.6
     368 5.731275270e-02 0.007797262630 7.649781142e-09 8.1
     369 6.250000000e-02 0.007561138556 2.575243663e-08 7.6
     370 6.815673329e-02 0.007325014483 2.588699921e-08 7.6
     371 7.432544469e-02 0.007088890409 -3.873516885e-08 7.4
     372 8.105247217e-02 0.006852766335 -2.868915550e-08 7.5
     373 8.838834765e-02 0.006616642261 8.820975506e-09 8.1
     374 9.638817659e-02 0.006380518187 -1.249981429e-08 7.9
     375 1.051120519e-01 0.006144394113 -2.542299282e-08 7.6
     376 1.146255054e-01 0.005908270039 -3.411815497e-08 7.5
     377 1.250000000e-01 0.005672145966 1.097270552e-08 8.0
     378 1.363134666e-01 0.005436021892 -5.883867171e-09 8.2
     379 1.486508894e-01 0.005199897818 -1.778870562e-08 7.7
     380 1.621049443e-01 0.004963773744 -2.477515548e-08 7.6
     381 1.767766953e-01 0.004727649670 -2.316978920e-08 7.6
     382 1.927763532e-01 0.004491525596 -1.391214588e-08 7.9
     383 2.102241038e-01 0.004255401522 4.032062573e-09 8.4
     384 2.292510108e-01 0.004019277448 1.322726373e-10 9.9
     385 2.500000000e-01 0.003783153375 1.667484262e-09 8.8
     386 2.726269332e-01 0.003547029301 -6.843555944e-09 8.2
     387 2.973017788e-01 0.003310905227 -5.830291139e-09 8.2
     388 3.242098887e-01 0.003074781153 -4.568762257e-09 8.3
     389 3.535533906e-01 0.002838657079 -6.372726125e-09 8.2
     390 3.855527064e-01 0.002602533005 4.293519651e-09 8.4
     391 4.204482076e-01 0.002366408931 4.466780612e-09 8.4
     392 4.585020216e-01 0.002130284858 -3.886752475e-09 8.4
     393 5.000000000e-01 0.001894160784 1.495882973e-09 8.8
     >
     > ## } ## only when we find inaccurate regions
     > showProc.time()
     Time (user system elapsed): 0.1 0 0.09
     >
     >
     > ## Oops: another qgamma() / qchisq() problem: mostly NaN's == all solved now
     > curve(qgamma(x, 20), 1e-16, 1e-10, log='x')
     > curve(qgamma(x, 20), 1e-300, .99 , log='xy') # and add the critical region from above:
     > abline(v=c(1e-16, 1e-10), col="light blue")
     > curve(qgamma(x, 20), 1e-26, 1e-07, log='x')
     > ##-> now using log=TRUE in same region:
     > curve(qgamma(x, 20, log=TRUE), -38, -16)## no problem!!
     > curve(qgamma(exp(x), 20), add=TRUE, col="green3", n=2001)
     > ## had problem here, but no longer !
     >
     > ##--> Further fix for qgamma: when 'x' is very small: use "log=TRUE of log(x)"!
     >
     > ## had bug (gave NaN), but no longer:
     > (q_12 <- qgamma(1e-12, 20))
     [1] 2.330042
     > all.equal(1e-12, pgamma(q_12, 20), tol=0)# show rel.err (Lnx 64-bit: 4.04e-16)
     [1] "Mean relative difference: 4.038968e-16"
     > stopifnot(
     + all.equal(1e-12, pgamma(q_12, 20), tolerance = 1e-14)
     + )
     >
     >
     > ## --- Nice graphic : --- but amazingly *S..L..O..W*
     >
     > p.qgammaSml <- function(from= 1e-110, to = 1e-5, ylim = c(0.4, 1000),
     + n = 201, k.lab = 3,
     + a1 = c(10, seq(10.1,20, by=.2), 21:105),
     + a2 = seq(110,330, by=10),
     + a3 = seq(350,1600, by=50))
     + {
     + ## Purpose: nice qgamma() lines ``for small x'' aka p
     + ## ----------------------------------------------------------------------
     + ## Author: Martin Maechler, Date: 22 Mar 2004, 14:23
     + x <- exp(seq(log(from), log(to), length = n))
     +
     + op <- par(las=1, lab = c(10,10, 7), xaxs = "i", mex = 0.8)
     + on.exit(par(op))
     + plot(x, qgamma(x, a1[1]), log="xy", ylim=ylim, type='l', xaxt = "n",
     + main = paste("qgamma(x, a) for very small x, a in [",
     + formatC(a1[1]),", ",formatC(max(a1,a2,a3)),"] - log-log", sep=''),
     + sub = R.version.string)
     + lab.x <- pretty(log10(c(from,to)), 20)
     + axis(1, at=10^lab.x, lab = paste("10^",formatC(lab.x),sep=''))
     + if(is.nan(qgamma(1e-12, 20)))
     + text(1e-60, 20, "all NaN", cex = 2)
     + if(!is.finite(qgamma(1e-140, 155)))
     + text(1e-240, 5, "all +Inf", cex = 2)
     +
     + lines.txt <- function(a.s, col = par("col")) {
     + col <- rep(col, length=length(a.s))
     + for(i in seq(along=a.s)) {
     + qx <- qgamma(x, (a <- a.s[i]))
     + if(i %% k.lab == 0 &&
     + any(ifi <- is.finite(qx) & qx >= ylim[1])) {
     + ik <- (i%%(2*k.lab))/k.lab # = 0 or 1
     + j <- quantile(which(ifi), c(.02,(1:3)/4+ ik/10, .98))
     + ## "segments" around the labels :
     + i0 <- 1
     + for(jj in j) {
     + ii <- i0:(jj-1)
     + i2 <- jj + -1:1
     + lines(x[ii], qx[ii], col=col[i])
     + lines(x[i2], qx[i2], col=col[i], type = 'c')
     + i0 <- jj+1
     + }
     + text(x[j], qx[j], formatC(a), col= "gray40", cex = 0.8)
     + }
     + else
     + lines(x, qx, col=col[i])
     +
     + }
     + }
     + oo <- options(warn = -1)
     + lines.txt(a1[-1])
     + lines.txt(a2, col= 2)
     + lines.txt(a3, col= rainbow(length(a3), .8, .8,
     + start = (max(a3)-min(a3))/(1+max(a3))))
     + invisible(options(oo))
     + }
     >
     > showProc.time()
     Time (user system elapsed): 0.01 0.01 0.04
     >
     > p.qgammaSml()
     > p.qgammaSml(1e-300)
     > p.qgammaSml(1e-300,1e-50, a2= seq(100,360, by=4), a3=seq(350,1500, by=10))
     >
     > showProc.time()
     Time (user system elapsed): 1.42 0 1.42
     >
     > ## The "upper" problematic corner:
     > p.qgammaSml(1e-19, 1e-3, a2=NULL,a3=NULL, ylim=c(.1,20))
     > p.qgammaSml(1e-19, 1e-3, a2=seq(1,12, by=.04), ylim=c(.1,20),a3=NULL,k.lab=10)
     > ## now shows the problem (quite well):
     > ## could it be in pgamma()'s inaccuracy, leading to qgamma() bias ?
     > aa <- c(seq(9, 22, by=0.25),seq(22.3,40,by=0.4))
     > caa <- formatC(range(aa))
     > sfsmisc::mult.fig(2)
     > curve(pgamma(x, sh=aa[1]), 0.5, 20, log = 'xy', ylim = c(1e-60, .2),
     + main = sprintf("pgamma(x, a) for a in [%s,%s]", caa[1],caa[2]))
     > for(sh in aa) curve(pgamma(x, sh), add = TRUE, col=2)
     > abline(h=c(1e-15), col="light blue", lty=2)
     >
     > curve(pgamma(x, sh=aa[1]), 0.5, 20, log = 'xy', ylim = c(1e-15, .8),
     + main = sprintf("pgamma(x, a) for a in [%s,%s]", caa[1],caa[2]))
     > for(sh in aa) curve(pgamma(x, sh), add = TRUE, col=2)
     > ## the "border curve" between "Pearson" and "Continued fraction (upper tail)"
     > ## in pgamma.c :
     > curve(pgamma(max(1,x), x), add = TRUE, col=4)
     > ## ==> pgamma() is perfect here {series expansion up to eps_C accuracy}!
     >
     > aa <- c(seq(9, 22, by=0.25),seq(22.3,40.4,by=0.4))
     > p.qgammaSml(1e-24, 1e-5, a1=aa, a2=NULL,a3=NULL, ylim=c(.8,8))
     > ## -------- save the above?
     > aa1 <- c(aa,seq(40.5,90, by=0.5))
     > p.qgammaSml(1e-60, 1e-5, a1=aa1, a2=NULL,a3=NULL, ylim=c(.9, 16))
     > aa2 <- c(aa1, seq(91,150, by= 1))
     > p.qgammaSml(1e-90, 1e-5, a1=aa2, a2=NULL,a3=NULL, ylim=c(.9, 35))
     > aa3 <- c(aa2, seq(150,250, by= 2), seq(253, 400, by=5))
     > p.qgammaSml(1e-200, 1e-5, a1=aa3, a2=NULL,a3=NULL, ylim=c(.9, 100))
     > p.qgammaSml(1e-200, 1e-5, a1=aa3, a2=NULL,a3=NULL, ylim=c(.9, 200),k.lab=9e9)
     > p.qgammaSml(1e-60, 1e-5, a1=aa3, a2=NULL,a3=NULL, ylim=c(.9, 200),k.lab=9e9)
     >
     > showProc.time()
     Time (user system elapsed): 4.79 0.05 4.85
     >
     > ## lower a \> 10
     >
     > curve(qgamma(x, 19), 1e-14, 1e-9, log='x')
     > curve(qgamma(x, 18), 1e-14, 1e-9, log='x')
     > curve(qgamma(x, 15), 1e-11, 5e-9, log='x')
     > curve(qgamma(x, 13), 5e-10, 1e-8, log='x')
     > curve(qgamma(x, 11), 1e-8, 5e-8, log='x')
     > curve(qgamma(x, 10.5), 4.2e-8, 6e-8, log='x')
     > curve(qgamma(x, 10.3), 6e-8, 7e-8, log='x')
     > curve(qgamma(x, 10.2), 7.1e-8, 7.6e-8, log='x')
     > curve(qgamma(x, 10.15),7.7e-8, 7.9e-8, log='x')
     > curve(qgamma(x, 10.14),7.88e-8,7.92e-8, log='x',n=10001)
     >
     > ## no more problems for smaller a!! here:
     > curve(qgamma(x, 10.13), 1e-10, 5e-4, log='x',n=20001)
     > curve(qgamma(x, 10.12), 1e-10, 5e-4, log='x',n=20001)
     > curve(qgamma(x, 10.1), 1e-10, 5e-4, log='x',n=20001)
     >
     > showProc.time()
     Time (user system elapsed): 0.9 0 0.9
     >
     > ##--- the "+Inf" / premature "0" case:
     > curve(qgamma(x, 155, log=TRUE), -1500, 0, log='y', n=2001,col=2)
     > curve(qgamma(x, 1e3, log=TRUE), -1500, 0, log='y', n=2001,col=2)
     > ## now works, but slowly and with kink
     > curve(qgamma (x, 1e5, log=TRUE), -3e5, 0, log='y', n=2001,col=2,lwd=3)
     > curve(qgammaAppr(x, 1e5, log=TRUE), add = TRUE, n=2001, col="blue",lwd=.4)
     > ## --- curves are almost "identical"
     > ## ===> the kink *does* come from the initial approx... hmm
     >
     > ## still "identical"
     > curve(qgamma (x, 1e4, log=TRUE), -3e4, 0, log='y', n=2001,col=2)
     > curve(qgammaAppr(x, 1e4, log=TRUE), add = TRUE, n=2001, col="tomato3")
     >
     > ## now see some difference (approx. has kink at ~ -165)
     > curve(qgamma (x, 100, log=TRUE), -200, 0, log='y', n=2001,col=2)
     > curve(qgammaAppr(x, 100, log=TRUE), add = TRUE, n=2001, col="tomato3")
     > ##
     > (kk <- 100 * 2/1.24)# 161.29
     [1] 161.2903
     > curve(qgamma (x, 100, log=TRUE), -1.1*kk, -.95*kk, log='y', n=2001,col=2)
     > curve(qgammaAppr(x, 100, log=TRUE), add = TRUE, n=2001, col="tomato3")
     > abline(v = -kk, col='blue', lty=2)# exactly: kink is at a * 2 / 1.24 = a / .62
     > curve(qgammaAppr(x - 100/.62, 100,log=TRUE), -1e-3, +1e-3)
     >
     > showProc.time()
     Time (user system elapsed): 0.24 0 0.22
     >
     > p.qgammaLog <- function(alpha, xl.f = 1.5, xr.f = 0.4, n = 2001)
     + {
     + ## Purpose:
     + ## ----------------------------------------------------------------------
     + ## Arguments:
     + ## ----------------------------------------------------------------------
     + ## Author: Martin Maechler, Date: 30 Mar 2004, 18:44
     + kk <- -alpha / .62 # = (alpha * 2) / (-1.24)
     + curve(qgamma(x, alpha, log=TRUE), xl.f*kk, xr.f*kk, log='y',
     + n=n, col=2, lwd=3.6, lty = 4,
     + main= paste("qgamma(x, alpha=",formatC(alpha,digits=10),", log = TRUE)"))
     + lines(kk, qgamma(kk, alpha, log=TRUE), type = 'h', lty = 3)
     + curve(qgamma (exp(x), alpha), add = TRUE, col="orange", n=n, lwd= 2)
     + curve(qgammaAppr(x, alpha, log=TRUE), add = TRUE, col=3, n=n,lwd = .4)
     + }
     >
     > showProc.time()
     Time (user system elapsed): 0.01 0 0
     >
     > p.qgammaLog(25)
     > p.qgammaLog(16)# ~ [-25, -20]
     > p.qgammaLog(12, 1.2, 0.8)# small problem remaining
     > p.qgammaLog(11, 1.2, 0.8)# even smaller
     > p.qgammaLog(10.5, 1.1, 0.9)# even smaller
     > p.qgammaLog(10.25, 1.1, 0.9)# even smaller
     > ## 2019-08: __nothing__ visible from here on:
     > p.qgammaLog(10.18, 1.02, 0.98)# even smaller
     > p.qgammaLog(10.15, 1.02, 0.98)# even smaller
     > p.qgammaLog(10.14, 1.001, 0.999)# even smaller
     > p.qgammaLog(10.139, 1.0002, 0.9998)#
     > p.qgammaLog(10.138, 1.0002, 0.9998)#
     > p.qgammaLog(10.137, 1.00001, 0.99999)#
     > p.qgammaLog(10.13699, 1.0000001, 0.9999999)#
     > p.qgammaLog(10.1369899, 1.0000001, 0.9999999)#
     > p.qgammaLog(10.1369894, 1.0000001, 0.9999999)#
     > p.qgammaLog(10.1369893, 1.0000001, 0.9999999)# even smaller at -16.34998
     >
     > showProc.time()
     Time (user system elapsed): 0.75 0.02 0.8
     >
     > ##-- here is the boundary --- for 64-bit AMD Opteron ---
     > ## and for 32-bit AMD Athlon
     >
     > p.qgammaLog(10.1369892, 1.0000001, 0.9999999)# no more
     > p.qgammaLog(10.136989, 1.0000001, 0.9999999)#
     > p.qgammaLog(10.136988, 1.0000001, 0.9999999)#
     > p.qgammaLog(10.136985, 1.0000001, 0.9999999)#
     > p.qgammaLog(10.13698, 1.0000001, 0.9999999)#
     > p.qgammaLog(10.13697, 1.0000001, 0.9999999)#
     > p.qgammaLog(10.13695, 1.0000001, 0.9999999)#
     > p.qgammaLog(10.1368, 1.000001, 0.999999)#
     > p.qgammaLog(10.1365, 1.000001, 0.999999)#
     > p.qgammaLog(10.136, 1.000001, 0.999999)#
     > p.qgammaLog(10.125, 1.1, 0.9)# --- see it now
     > p.qgammaLog(10, 1.2, 0.8)
     > p.qgammaLog(9)
     >
     > showProc.time()
     Time (user system elapsed): 0.61 0.01 0.62
     >
     > ## For large alpha: show difference to see problem better
     > ## ---> for alpha >= 10, the x problem starts *roughly* at x = -0.8*alpha
     > ##
     >
     > sfsmisc::mult.fig(2)
     > curve(qgammaAppr(x, 5, log=TRUE), - 8.1, -8, n=2001)
     > curve(qgammaAppr(x- 5/.62, 5, log=TRUE), -1e-15, 0)
     >
     > ## is the kink from pgamma() ? : no: this looks fine,
     > curve(pgamma(x, 1e5, log=TRUE), 1, 2e5, log='x', n=2001,col=2)
     > ## and this does too:
     > curve( dgamma(x, 1e5), .5e5, 2e5); par(new=TRUE)
     > curve( dgamma(x, 1e5, log=TRUE), .5e5, 2e5, col=2, yaxt="n")
     > axis(4,col.axis=2); par(new=TRUE)
     > curve( pgamma(x, 1e5), .5e5, 2e5, n=2001, col=3); par(new=TRUE)
     > curve( pgamma(x, 1e5, log=TRUE), .5e5, 2e5, n=2001, col=4); par(new=TRUE)
     > curve(-pgamma(x, 1e5, log=TRUE,lower=FALSE), .5e5, 2e5, n=2001, col=4)
     > ## all looking nice
     >
     >
     > x <- 10^seq(2,6, length=4001)
     > qx <- qgamma(pgamma(x, 1e5, log=TRUE), 1e5, log=TRUE)
     > plot(x, qx, type ='l', col=2, asp = 1); abline(0,1, lty=3)
     >
     > showProc.time()
     Time (user system elapsed): 0.1 0.02 0.11
     > <0c>
     > ###------------- Approximations of qgamma() ------
     > ##
     >
     > ## source("/u/maechler/R/MM/NUMERICS/dpq-functions/qchisqAppr.R")
     > ##--> qchisqAppr()
     > ##--> qchisqWH [ = Wilson Hilferty ]
     > ##--> qchisqKG [ = Kennedy & Gentle's improvements "a la AS 91" ]
     > ## dyn.load("/u/maechler/R/MM/NUMERICS/dpq-functions/qchisq_appr.so")
     >
     > ## Consider the two different implementations of
     > ## lgamma1p(a) := lgamma(1+a) == log(gamma(1+a) == log(a*gamma(a)) "stable":
     >
     > if(!exists("lseq", mode="function"))
     + lseq <- if(requireNamespace("sfsmisc")) sfsmisc::lseq else
     + function(from, to, length) exp(seq(log(from), log(to), length.out = length))
     >
     > if(require("Rmpfr")) { ##---------------- MPFR numbers -------------------------
     +
     + .mpfr.all.eq <- Rmpfr::all.equal
     + AllEq <- function(target, current, ...)
     + .mpfr.all.eq(target, current, ...,
     + formatFUN = function(x, ...) Rmpfr::format(x, digits = 9))
     +
     + print(gammaE <- Const("gamma",200)); pi. <- Const("pi",200)
     + print(a0 <- (gammaE^2 + pi.^2/6)/2)
     + print(psi2.1 <- -2*zeta(mpfr(3,200)))# == psigamma(1,2) =~ -2.4041138
     + print(a1 <- (psi2.1 - gammaE*(pi.^2/2 + gammaE^2))/6)
     +
     + x <- lseq(1e-30, 0.8, length = if(doExtras) 1000 else 125)
     + x. <- mpfr(x, 200)
     + xct. <- log(x. * gamma(x.)) ## using MPFR arithmetic .. no overflow ...
     + xc2. <- log(x.) + lgamma(x.)## (ditto)
     + print(AllEq(xct., xc2., tol = 0)) # 3.15779......e-57
     + xct <- as.numeric(xct.)
     + stopifnot(exprs = {
     + AllEq(xct., xc2., tol = 1e-45)
     + AllEq(xct , xc2., tol = 1e-15)
     + ##
     + all.equal(lgamma1p(x), lgamma1p(x, tol= 1e-16), tol=0)
     + ## -> no difference; i.e., default tol = 1e-14 seems fine enough!
     + })
     + showProc.time()
     +
     + m.appr <- cbind(log(x*gamma(x)), lgamma(1+x), log(x) + lgamma(x),
     + lgamma1p.(x, k=1, cut=3e-6),
     + lgamma1p.(x, k=2, cut=1e-4),
     + lgamma1p.(x, k=3, cut=8e-4),
     + lgamma1p(x))#, tol= 1e-14), # = default
     +
     + eMat <- m.appr - xct # absolute error
     + ## Relative errors:
     + str(reMat. <- m.appr/xct. - 1)
     + str(reMat <- as(reMat., "array")) # as(., "matrix") fails in older versions
     +
     + matplot(x, eMat , log="x", type="l", lty=1) #-> problematic log(x) + lgamma(x) for "large"
     + matplot(x, abs( eMat), log="xy", type="l", lty=1) #-> but good for small; lgamma1p is much better
     + matplot(x, abs(reMat), log="xy", type="l", lty=1)
     + abline(v= 3.47548562941137e-08, col = "gray80", lwd=3)#<- the cutoff value of lgamma1p()
     + ##---> should use earlier cutoff!
     + ## zoom in:
     +
     + matplot(x, abs(reMat), log="xy", type="l", col=1:7, lty=1,
     + lwd=2, xlim=c(8e-9, 1e-3), ylim = c(1e-18, 1e-7), axes=FALSE, frame=TRUE,
     + main = expression(lgamma1p(x) == log(Gamma(x+1)) ~~~ "approximations"
     + ~~~ abs(rel.Err(.))))
     + eaxis(1); eaxis(2)
     + abline(v= 3.47548562941137e-08, col = "gray80", lwd=3)#<- the cutoff value of lgamma1p()
     + abline(h= c(1,2,4)*.Machine$double.eps, lty=3, col="skyblue")
     + legend("topright", col=1:7, lty=1,lwd=2,
     + c("log(x*gamma(x))", "lgamma(1+x)", "log(x) + lgamma(x)",
     + "lgamma1p.(x, k=1, c=3e-6)",
     + "lgamma1p.(x, k=2, c=1e-4)",
     + "lgamma1p.(x, k=3, c=8e-4)",
     + "lgamma1p(x)"), bty="n", ncol=2)
     + abline(v = c(3e-6, 1e-4, 8e-4), col=4:6, lty=2, lwd=1/2)
     +
     + ## FIXME: do the same for the lgaamma1p_series()
     +
     + ## rm(x., xct., xc2., reMat., eMat, AllEq)
     + detach("package:Rmpfr")
     + showProc.time()
     +
     + } ## if( MPFR ) ----------------------------------------------------------------
     Loading required package: Rmpfr
     Loading required package: gmp
    
     Attaching package: 'gmp'
    
     The following objects are masked from 'package:base':
    
     %*%, apply, crossprod, matrix, tcrossprod
    
     C code of R package 'Rmpfr': GMP using 64 bits per limb
    
    
     Attaching package: 'Rmpfr'
    
     The following object is masked from 'package:gmp':
    
     outer
    
     The following object is masked from 'package:DPQ':
    
     log1mexp
    
     The following objects are masked from 'package:stats':
    
     dbinom, dgamma, dnbinom, dnorm, dpois, dt, pnorm
    
     The following objects are masked from 'package:base':
    
     cbind, pmax, pmin, rbind
    
     1 'mpfr' number of precision 200 bits
     [1] 0.57721566490153286060651209008240243104215933593992359880576723
     1 'mpfr' number of precision 200 bits
     [1] 0.98905599532797255539539565150063470793918352072821409044319567
     1 'mpfr' number of precision 200 bits
     [1] -2.404113806319188570799476323022899981529972584680997763584544
     1 'mpfr' number of precision 200 bits
     [1] -0.90747907608088628901656016735627511492861144907256376094133062
     Error in target == current : comparison of these types is not implemented
     Calls: print ... .mpfr.all.eq -> .mpfr.all.eq -> .local -> all.equal.numeric
     Execution halted
    Running the tests in 'tests/stirlerr-tst.R' failed.
    Complete output:
     > #### Testing stirlerr(), bd0(), ebd0(), dpois_raw(), ...
     > #### ===============================================
     >
     > require(DPQ)
     Loading required package: DPQ
     > for(pkg in c("Rmpfr", "DPQmpfr"))
     + if(!requireNamespace(pkg)) {
     + cat("no CRAN package", sQuote(pkg), " ---> no tests here.\n")
     + q("no")
     + }
     Loading required namespace: Rmpfr
     Loading required namespace: DPQmpfr
     > require("Rmpfr")
     Loading required package: Rmpfr
     Loading required package: gmp
    
     Attaching package: 'gmp'
    
     The following objects are masked from 'package:base':
    
     %*%, apply, crossprod, matrix, tcrossprod
    
     C code of R package 'Rmpfr': GMP using 64 bits per limb
    
    
     Attaching package: 'Rmpfr'
    
     The following object is masked from 'package:gmp':
    
     outer
    
     The following object is masked from 'package:DPQ':
    
     log1mexp
    
     The following objects are masked from 'package:stats':
    
     dbinom, dgamma, dnbinom, dnorm, dpois, dt, pnorm
    
     The following objects are masked from 'package:base':
    
     cbind, pmax, pmin, rbind
    
     >
     > cutoffs <- c(15,35,80,500) # cut points, n=*, in the above "algorithm"
     > ##
     > n <- c(seq(1,15, by=1/4),seq(16, 25, by=1/2), 26:30, seq(32,50, by=2), seq(55,1000, by=5),
     + 20*c(51:99), 50*(40:80), 150*(27:48), 500*(15:20))
     > st.n <- stirlerr(n)# rather use.halves=TRUE, just here , use.halves=FALSE)
     > plot(st.n ~ n, log="xy", type="b") ## looks good now
     > nM <- mpfr(n, 2048)
     > st.nM <- stirlerr(nM, use.halves=FALSE) ## << on purpose
     > all.equal(asNumeric(st.nM), st.n)# TRUE
     [1] TRUE
     > all.equal(st.nM, as(st.n,"mpfr"))# .. difference: 1.05884..............................e-15
     Error in target == current : comparison of these types is not implemented
     Calls: all.equal -> all.equal -> .local -> all.equal.numeric
     Execution halted
Flavor: r-devel-windows-x86_64-old

Version: 0.5-0
Check: examples
Result: ERROR
    Running examples in ‘DPQ-Ex.R’ failed
    The error most likely occurred in:
    
    > ### Name: numer-utils
    > ### Title: Numerical Utilities - Functions, Constants
    > ### Aliases: M_LN2 M_SQRT2 M_minExp M_cutoff G_half all_mpfr any_mpfr logr
    > ### modf okLongDouble
    > ### Keywords: math
    >
    > ### ** Examples
    >
    > (Ms <- ls("package:DPQ", pattern = "^M"))
    [1] "M_LN2" "M_SQRT2" "M_cutoff" "M_minExp"
    > lapply(Ms, function(nm) { cat(nm,": "); print(get(nm)) }) -> .tmp
    M_LN2 : [1] 0.6931472
    M_SQRT2 : [1] 1.414214
    M_cutoff : [1] 3.196577e+18
    M_minExp : [1] -708.3964
    >
    > logr(1:3, a=1e-10)
    [1] -1.000000e-10 -5.000000e-11 -3.333333e-11
    >
    > okLongDouble() # typically TRUE, but not e.g. in a valgrinded R-devel of Oct.2019
    [1] FALSE
    > ## Here is typically the "boundary":
    > rr <- uniroot(function(x) okLongDouble(x) - 1/2, c(11350, 11400), tol=1e-7)
    Error in uniroot(function(x) okLongDouble(x) - 1/2, c(11350, 11400), tol = 1e-07) :
     f() values at end points not of opposite sign
    Execution halted
Flavor: r-release-macos-arm64

Version: 0.5-0
Check: running R code from vignettes
Result: ERROR
    Errors in running code in vignettes:
    when running code in ‘log1pmx-etc.Rnw’
     ...
    
     When sourcing ‘log1pmx-etc.R’:
    Error: log1pmx-etc.R:74:8: unexpected input
    73: lcurve(log1pmx, -.01, .01); rect(-.002,log1pmx(-.002), .002, 0); zoomTo0(2e-3,1e-5)
    74: lcurve(\
     ^
    Execution halted
    
     ‘Noncentral-Chisq.Rnw’ using ‘UTF-8’... [0s/0s] OK
     ‘comp-beta.Rnw’ using ‘UTF-8’... [0s/0s] OK
     ‘log1pmx-etc.Rnw’ using ‘UTF-8’... failed
Flavor: r-oldrel-macos-x86_64

Version: 0.5-0
Check: re-building of vignette outputs
Result: NOTE
    Error(s) in re-building vignettes:
     ...
    --- re-building ‘Noncentral-Chisq.Rnw’ using Sweave
    --- finished re-building ‘Noncentral-Chisq.Rnw’
    
    --- re-building ‘comp-beta.Rnw’ using Sweave
    --- finished re-building ‘comp-beta.Rnw’
    
    --- re-building ‘log1pmx-etc.Rnw’ using Sweave
    Loading required package: DPQ
    
    Error: processing vignette 'log1pmx-etc.Rnw' failed with diagnostics:
     chunk 5 (label = log1pmx-curves)
    Error in log1pmx-etc.Rnw:433:8: unexpected input
    432: lcurve(log1pmx, -.01, .01); rect(-.002,log1pmx(-.002), .002, 0); zoomTo0(2e-3,1e-5)
    433: lcurve(\
     ^
    
    --- failed re-building ‘log1pmx-etc.Rnw’
    
    SUMMARY: processing the following file failed:
     ‘log1pmx-etc.Rnw’
    
    Error: Vignette re-building failed.
    Execution halted
Flavor: r-oldrel-macos-x86_64

Version: 0.5-0
Check: re-building of vignette outputs
Result: WARN
    Error(s) in re-building vignettes:
    --- re-building 'Noncentral-Chisq.Rnw' using Sweave
    --- finished re-building 'Noncentral-Chisq.Rnw'
    
    --- re-building 'comp-beta.Rnw' using Sweave
    --- finished re-building 'comp-beta.Rnw'
    
    --- re-building 'log1pmx-etc.Rnw' using Sweave
    Loading required package: DPQ
    
    Error: processing vignette 'log1pmx-etc.Rnw' failed with diagnostics:
     chunk 5 (label = log1pmx-curves)
    Error in log1pmx-etc.Rnw:433:8: unexpected input
    432: lcurve(log1pmx, -.01, .01); rect(-.002,log1pmx(-.002), .002, 0); zoomTo0(2e-3,1e-5)
    433: lcurve(\
     ^
    
    --- failed re-building 'log1pmx-etc.Rnw'
    
    SUMMARY: processing the following file failed:
     'log1pmx-etc.Rnw'
    
    Error: Vignette re-building failed.
    Execution halted
Flavor: r-oldrel-windows-ix86+x86_64