Last updated on 2024-11-05 17:50:12 CET.
| Flavor | Version | Tinstall | Tcheck | Ttotal | Status | Flags |
|---|---|---|---|---|---|---|
| r-devel-linux-x86_64-debian-clang | 0.5-9 | 20.12 | 369.06 | 389.18 | ERROR | |
| r-devel-linux-x86_64-debian-gcc | 0.5-9 | 12.66 | 267.48 | 280.14 | OK | |
| r-devel-linux-x86_64-fedora-clang | 0.5-9 | 687.16 | OK | |||
| r-devel-linux-x86_64-fedora-gcc | 0.5-9 | 661.03 | OK | |||
| r-devel-windows-x86_64 | 0.5-9 | 26.00 | 508.00 | 534.00 | OK | |
| r-patched-linux-x86_64 | 0.5-9 | 20.97 | 372.58 | 393.55 | OK | |
| r-release-linux-x86_64 | 0.5-9 | 18.36 | 372.15 | 390.51 | OK | |
| r-release-macos-arm64 | 0.5-9 | 242.00 | OK | |||
| r-release-macos-x86_64 | 0.5-9 | 805.00 | OK | |||
| r-release-windows-x86_64 | 0.5-9 | 27.00 | 518.00 | 545.00 | OK | |
| r-oldrel-macos-arm64 | 0.5-9 | 232.00 | OK | |||
| r-oldrel-macos-x86_64 | 0.5-9 | 624.00 | OK | |||
| r-oldrel-windows-x86_64 | 0.5-9 | 24.00 | 565.00 | 589.00 | OK |
Version: 0.5-9
Check: tests
Result: ERROR
Running ‘bd0-tst.R’ [5s/6s]
Running ‘chisq-nonc-ex.R’ [26s/33s]
Running ‘dgamma-tst.R’ [1s/1s]
Running ‘dnbinom-tst.R’ [25s/33s]
Running ‘dnchisq-tst.R’ [0s/1s]
Running ‘dt-ex.R’ [10s/15s]
Running ‘expm1x-tst.R’ [3s/4s]
Running ‘hyper-dist-ex.R’ [18s/22s]
Running ‘log4p1p-exp.R’ [3s/4s]
Running ‘pnbeta-tst.R’ [0s/1s]
Running ‘pnt-prec.R’ [7s/10s]
Running ‘pow-tst.R’ [4s/5s]
Running ‘ppois-ex.R’ [1s/2s]
Running ‘pqnorm_extreme.R’ [6s/7s]
Running ‘qPoisBinom-ex.R’ [1s/1s]
Running ‘qbeta-dist.R’ [6s/8s]
Running ‘qbeta-tst.R’ [0s/1s]
Running ‘qgamma-ex.R’ [13s/16s]
Running ‘stirlerr-tst.R’ [83s/94s]
Running ‘t-nonc-tst.R’ [14s/16s]
Running ‘wienergerm-pchisq-tst.R’ [0s/1s]
Running ‘wienergerm_nchisq.R’ [6s/6s]
Running the tests in ‘tests/stirlerr-tst.R’ failed.
Complete output:
> #### Testing stirlerr()
> #### =================== {previous 2nd part of this, now -->>> ./bd0-tst.R <<<---
> 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
>
> source(system.file(package="DPQ", "test-tools.R", mustWork=TRUE))
> ## => showProc.time(), ... list_() , loadList() , readRDS_() , save2RDS()
> ##_ options(conflicts.policy = list(depends.ok=TRUE, error=FALSE, warn=FALSE))
> require(sfsmisc) # masking 'list_' *and* gmp's factorize(), is.whole()
Loading required package: sfsmisc
Attaching package: 'sfsmisc'
The following object is masked _by_ '.GlobalEnv':
list_
The following objects are masked from 'package:gmp':
factorize, is.whole
> ##_ options(conflicts.policy = NULL)o
>
> ## plot1cuts() , etc: ---> ../inst/extraR/relErr-plots.R <<<<<<<
> ## ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
> source(system.file(package="DPQ", "extraR", "relErr-plots.R", mustWork=TRUE))
>
> do.pdf <- TRUE # (manually)
> do.pdf <- !dev.interactive(orNone = TRUE)
> do.pdf
[1] TRUE
> if(do.pdf) {
+ pdf.options(width = 9, height = 6.5) # for all pdf plots {9/6.5 = 1.38 ; 4/3 < 1.38 < sqrt(2) [A4]
+ pdf("stirlerr-tst.pdf")
+ }
>
> showProc.time()
Time (user system elapsed): 0.08 0 0.079
> (doExtras <- DPQ:::doExtras())
[1] FALSE
> (noLdbl <- (.Machine$sizeof.longdouble <= 8)) ## TRUE when --disable-long-double or 'M1mac' ..
[1] FALSE
> (M.mac <- grepl("aarch64-apple", R.version$platform)) # Mac with M1, M2, ... proc
[1] FALSE
>
> abs19 <- function(r) pmax(abs(r), 1e-19) # cut |err| to positive {for log-plots}
>
> options(width = 100, nwarnings = 1e5, warnPartialMatchArgs = FALSE)
>
>
>
> ##=== Really, dpois_raw() and dbinom_raw() *both* use stirlerr(x) for "all" 'x > 0'
> ## ~~~~~~~~~~~ ~~~~~~~~~~~~ =========== ===== !
>
> ## below, 6 "it's okay, but *far* from perfect:" ===> need more terms in stirlerr() [
> ## April 20: MM added more terms up to S10; 2024-01: up to S12 ..helps a little only
> x <- lseq(1/16, 6, length=2048)
> system.time(stM <- DPQmpfr::stirlerrM(Rmpfr::mpfr(x,2048))) # 1.7 sec elapsed
user system elapsed
3.052 0.028 3.418
> plot(x, stirlerr(x, use.halves=FALSE) - stM, type="l", log="x", main="absolute Error")
> plot(x, stirlerr(x, use.halves=FALSE) / stM - 1, type="l", log="x", main="relative Error")
> plot(x, abs(stirlerr(x, use.halves=FALSE) / stM - 1), type="l", log="xy",main="|relative Error|")
> drawEps.h(-(52:50))
> ## lgammacor() does *NOT* help, as it is *designed* for x >= 10 ... (but is interesting there)
> ##
> ## ==> Need another chebyshev() or rational-approx. for x in [.1, 7] or so !!
>
> ##=============> see also ../Misc/stirlerr-trms.R <===============
> ## ~~~~~~~~~~~~~~~~~~~~~~~
>
> cutoffs <- c(15,35,80,500) # cut points, n=*, in the stirlerr() "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, "R3")# rather use.halves=TRUE; but here , use.halves=FALSE
> plot(st.n ~ n, log="xy", type="b") ## looks good now (straight line descending {in log-log !}
> 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: 3.381400e-14 was 1.05884.........e-15
[1] "Mean relative difference: 3.317339e-14"
> all.equal(roundMpfr(st.nM, 64), as(st.n,"mpfr"), tolerance=1e-16)# (ditto)
[1] "Mean relative difference: 3.317339e-14"
>
>
> ## --- Look at the direct formula -- why is it not good for n ~= 5 ?
> ##
> ## Preliminary Conclusions :
> ## 1. there is *some* cancellation even for small n (how much?)
> ## 2. lgamma1p(n) does really not help much compared to lgamma(n+1) --- but a tiny bit in some cases
>
> ### 1. Investigating lgamma1p(n) vs lgamma(n+1) for n < 1 =============================================
>
> ##' @title Relative Error of lgamma(n+1) vs lgamma1p() vs MM's stirlerrD2():
> ##' @param n numeric, typically n << 1
> ##' @param precBits
> ##' @return relative error WRT mpfr(n, precBits)
> ##' @author Martin Maechler
> relE.lgam1 <- function(n, precBits = if(doExtras) 1024 else 320) {
+ M_LN2PI <- 1.837877066409345483 # ~ log(2*Const("pi",60)); very slightly more accurate than log(2*pi)
+ st <- lgamma(n +1) - (n +0.5)*log(n) + n - M_LN2PI/2
+ st. <- lgamma1p(n) - (n +0.5)*log(n) + n - M_LN2PI/2 # "lgamma1p"
+ st2 <- lgamma(n) + n*(1-(l.n <- log(n))) + (l.n - M_LN2PI)/2 # "MM2"
+ st0 <- -(l.n + M_LN2PI)/2 # "n0"
+ nM <- mpfr(n, precBits)
+ stM <- lgamma(nM+1) - (nM+0.5)*log(nM) + nM - log(2*Const("pi", precBits))/2
+ ## stM <- roundMpfr(stM, 128)
+ cbind("R3" = asNumeric(relErrV(stM, st))
+ , "lgamma1p"= asNumeric(relErrV(stM, st.))
+ , "MM2" = asNumeric(relErrV(stM, st2))
+ , "n0" = asNumeric(relErrV(stM, st0))
+ )
+ }
>
> n <- 2^-seq.int(1022, 1, by = -1/4)
> relEx <- relE.lgam1(n)
> showProc.time()
Time (user system elapsed): 6.067 0.051 6.956
>
> ## Is *equivalent* to 'new' stirlerr_simpl(n version = *) [not for <mpfr> though, see 'relEmat']:
> (simpVer <- eval(formals(stirlerr_simpl)$version))
[1] "R3" "lgamma1p" "MM2" "n0"
> if(is.null(simpVer)) { warning("got wrong old version of package 'DPQ':")
+ print(packageDescription("DPQ"))
+ stop("invalid outdated version package 'DPQ'")
+ }
> stir.allS <- function(n) sapply(simpVer, function(v) stirlerr_simpl(n, version=v))
> stirS <- stir.allS(n) # matrix
> nM <- mpfr(n, 256) # "high" precision = 256 should suffice!
> stirM <- stirlerr(nM)
> releS <- asNumeric(relErrV(stirM, stirS))
> all.equal(relEx, releS, tolerance = 0) # see TRUE on Linux
[1] TRUE
> stopifnot(all.equal(relEx, releS, tolerance = if(noLdbl) 2e-15 else 1e-15))
> simpVer3 <- simpVer[simpVer != "lgamma1p"] # have no mpfr-ified lgamma1p()!
> ## stirlerr_simpl(<mpfr>, *) :
> stirM2 <- sapplyMpfr(simpVer3, function(v) stirlerr_simpl(nM, version=v))
>
> ## TODO ?:
> ## apply(stirM2, 2, function(v) all.equal(v, stirS, check.class=FALSE))
> ## releS2 <- asNumeric(relErrV(stirM2, stirS))
> ## all.equal(relEx, releS, tolerance = 0) # see TRUE on Linux
> ## stopifnot(all.equal(relEx, releS, tolerance = 1e-15))
> relEmat <- matrix(NA, ncol(stirM2), ncol(stirS),
+ dimnames = list(simpVer3, colnames(stirS)))
> for(j in seq_len(ncol(stirM2)))
+ for(k in seq_len(ncol(stirS)))
+ relEmat[j,k] <- asNumeric(relErr(stirM2[,j], stirS[,k]))
> relEmat
R3 lgamma1p MM2 n0
R3 6.069585e-17 6.069957e-17 6.111712e-17 9.701665e-06
MM2 6.069585e-17 6.069957e-17 6.111712e-17 9.701665e-06
n0 9.701700e-06 9.701700e-06 9.701700e-06 6.054986e-17
> round(-log10(relEmat), 2) # well .. {why? / expected ?}
R3 lgamma1p MM2 n0
R3 16.22 16.22 16.21 5.01
MM2 16.22 16.22 16.21 5.01
n0 5.01 5.01 5.01 16.22
>
> cols <- c("gray30", adjustcolor(c(2,3,4), 1/2)); lwd <- c(1, 3,3,3)
> stopifnot((k <- length(cols)) == ncol(relEx), k == length(lwd))
> matplot(n, relEx, type = "l", log="x", col=cols, lwd=lwd, ylim = c(-1,1)*4.5e-16,
+ main = "relative errors of direct (approx.) formula for stirlerr(n), small n")
> mtext("really small errors are dominated by small (< 2^-53) errors of log(n)")
> ## very interesting: there are different intervals <---> log(n) Qpattern !!
> ## -- but very small difference, only for n >~= 1/1000 but not before
> drawEps.h(negative=TRUE) # abline(h= c(-4,-2:2, 4)*2^-53, lty=c(2,2,2, 1, 2,2,2), col="gray")
> legend("topleft", legend = colnames(relEx), col=cols, lwd=3)
>
> ## zoomed in a bit:
> n. <- 2^-seq.int(400,2, by = -1/4)
> relEx. <- relE.lgam1(n.)
> matplot(n., relEx., type = "l", log="x", col=cols, lwd=lwd, ylim = c(-1,1)*4.5e-16,
+ main = "relative errors of direct (approx.) formula for stirlerr(n), small n")
> drawEps.h(negative=TRUE)
> legend("topleft", legend = colnames(relEx.), col=cols, lwd=3)
>
> ##====> Absolute errors (and look at "n0") --------------------------------------
> matplot(n., abs19(relEx.), type = "l", log="xy", col=cols, lwd=lwd, ylim = c(4e-17, 5e-16),
+ main = quote(abs(relErr(stirlerr_simpl(n, '*')))))
> drawEps.h(); legend("top", legend = colnames(relEx.), col=cols, lwd=3)
> lines(n., abs19(relEx.[,"n0"]), type = "o", cex=1/4, col=cols[4], lwd=2)
>
> ## more zooom-in
> n.2 <- 2^-seq.int(85, 50, by= -1/100)
> stirS.2 <- sapply(c("R3", "lgamma1p", "n0"), function(v) stirlerr_simpl(n.2, version=v))
> releS.2 <- asNumeric(relErrV(stirlerr(mpfr(n.2, 320)), stirS.2))
>
> matplot(n.2, abs19(releS.2), type = "l", log="xy", col=cols, lwd=lwd, ylim = c(4e-17, 5e-16),
+ main = quote(abs(relErr(stirlerr_simpl(n, '*')))))
> drawEps.h(); legend("top", legend = colnames(releS.2), col=cols, lwd=3)
> abline(v = 5e-17, col=(cb <- adjustcolor("skyblue4", 1/2)), lwd=2, lty=3)
> axis(1, at=5e-17, col.axis=cb, line=-1/4, cex = 3/4)
>
> matplot(n.2, abs19(releS.2), type = "l", log="xy", col=cols, lwd=lwd, ylim = c(4e-17, 5e-16),
+ xaxt="n", xlim = c(8e-18, 1e-15), ## <<<<<<<<<<<<<<<<<<< Zoom-in
+ xlab = quote(n), main = quote(abs(relErr(stirlerr_simpl(n, '*')))))
> eaxis(1); drawEps.h(); legend("top", legend = colnames(releS.2), col=cols, lwd=3)
> abline(v = 5e-17, col=(cb <- adjustcolor("skyblue4", 1/2)), lwd=2, lty=3)
> mtext('stirlerr_simpl(*, "n0") is as good as others for n <= 5e-17', col=adjustcolor(cols[3], 2))
> ## ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
>
> ## ===> "all *but* "n0" approximations for "larger" small n:
> n2 <- 2^-seq.int(20,0, length.out=1000)
> relEx2 <- relE.lgam1(n2)[,c("R3", "lgamma1p", "MM2")] # "n0" is "bad": |relE| >= 2.2e-6 !
> cols <- c("gray30", adjustcolor(c(2,4), 1/2)); lwd <- c(1, 3,3)
> stopifnot((k <- length(cols)) == ncol(relEx2), k == length(lwd))
> matplot(n2, relEx2, type = "l", log="x", col=cols, lwd=lwd, ylim = c(-3,3)*1e-15, xaxt="n",
+ main = "relative errors of direct (approx.) formula for stirlerr(n), small n")
> eaxis(1, sub10=c(-3,0)); drawEps.h(negative=TRUE)
> legend("topleft", legend = colnames(relEx2), col=cols, lwd=3)
> ##==> "MM" is *NOT* good for n < 1 *but*
>
> ## "the same" -- even larger small n:
> n3 <- seq(.01, 5, length=1000)
> relEx3 <- relE.lgam1(n3)[,c("R3", "lgamma1p", "MM2")] # "no" is "bad" ..
> stopifnot((k <- length(cols)) == ncol(relEx3), k == length(lwd))
>
> matplot(n3, relEx3, type = "l", col=cols, lwd=lwd,
+ main = "relative errors of direct (approx.) formula for stirlerr(n), small n")
> legend("topleft", legend = colnames(relEx3), col=cols, lwd=3)
> drawEps.h(negative=TRUE)
>
> matplot(n3, abs19(relEx3), type = "l", col=cols, lwd=lwd,
+ log="y", ylim = 2^-c(54, 44), yaxt = "n", ylab = quote(abs(relE)), xlab=quote(n),
+ main = "|relative errors| of direct (approx.) formula for stirlerr(n), small n")
> eaxis(2, cex.axis=0.9); legend("topleft", legend = colnames(relEx3), col=cols, lwd=3)
> drawEps.h()
> lines(n3, smooth.spline(abs(relEx3)[,1], df=12)$y, lwd=3, col=cols[1])
> lines(n3, smooth.spline(abs(relEx3)[,2], df=12)$y, lwd=3, col=adjustcolor(cols[2], 1/2))
> lines(n3, smooth.spline(abs(relEx3)[,3], df=12)$y, lwd=4, col=adjustcolor(cols[3], offset = rep(.2,4)))
> ## ===> from n >~= 1, "MM2" is definitely better up to n = 5 !!
>
> ## Check log() only :
> plot(n, asNumeric(relErrV(log(mpfr(n, 256)), log(n))), ylim = c(-1,1)*2^-53,
+ log="x", type="l", xaxt="n") ## ===> indeed --- log(n) approximation pattern !!
> eaxis(1) ; drawEps.h(negative=TRUE)
> showProc.time()
Time (user system elapsed): 10.756 0.059 11.923
>
> ## =========== "R3" vs "lgamma1p" -------------------------- which is better?
>
> ## really for the very small n, all is dominated by -(n+0.5)*log(n); and lgamma1p() is unnecessary!
> i <- 1:20; ni <- n[i]
> lgamma1p(ni)
[1] -1.284347e-308 -1.527355e-308 -1.816342e-308 -2.160006e-308 -2.568695e-308 -3.054710e-308
[7] -3.632683e-308 -4.320013e-308 -5.137390e-308 -6.109421e-308 -7.265367e-308 -8.640026e-308
[13] -1.027478e-307 -1.221884e-307 -1.453073e-307 -1.728005e-307 -2.054956e-307 -2.443768e-307
[19] -2.906147e-307 -3.456010e-307
> - (ni +0.5)*log(ni) + ni
[1] 354.1982 354.1116 354.0249 353.9383 353.8516 353.7650 353.6783 353.5917 353.5051 353.4184
[11] 353.3318 353.2451 353.1585 353.0718 352.9852 352.8986 352.8119 352.7253 352.6386 352.5520
>
> ## much less extreme:
> n2 <- lseq(2^-12, 1/2, length=1000)
> relE2 <- relE.lgam1(n2)[,-4]
>
> cols <- c("gray30", adjustcolor(2:3, 1/2)); lwd <- c(1,3,3)
> matplot(n2, relE2, type = "l", log="x", col=cols, lwd=lwd)
> legend("topleft", legend=colnames(relE2), col=cols, lwd=2, lty=1:3)
> drawEps.h(negative=TRUE)
>
> matplot(n2, abs19(relE2), type = "l", log="xy", col=cols, lwd=lwd, ylim = c(6e-17, 1e-15),
+ xaxt = "n"); eaxis(1, sub10=c(-2,0))
> legend("topleft", legend=colnames(relE2), col=cols, lwd=2, lty=1:3)
> drawEps.h()
> ## "MM2" is *worse* here, n < 1/2
> for(j in 1:3) lines(n2, smooth.spline(abs(relE2[,j]), df=10)$y, lwd=3,
+ col=adjustcolor(cols[j], 1.5, offset = rep(-1/4, 4)))
> ## "lgammap very slightly better in [0.002, 0.05] ...
> ## "TODO": draw 0.90-quantile curves {--> cobs::cobs() ?} instead of mean-curves?
>
> ## which is better? ... "random difference"
> d.absrelE <- abs(relE2[,"R3"]) - abs(relE2[,"lgamma1p"])
> plot (n2, d.absrelE, type = "l", log="x", # no clear picture ...
+ main = "|relE_R3| - |relE_lgamma1p|", axes=FALSE, frame.plot=TRUE)
> eaxis(1, sub10=c(-2,1)); eaxis(2); axis(3, at=max(n2)); abline(v = max(n2), lty=3, col="gray")
> ## 'lgamma1p' very slightly better:
> lines(n2, smooth.spline(d.absrelE, df=12)$y, lwd=3, col=2)
>
> ## not really small n at all == here see, how "bad" the direct formula gets for 1 < n < 10 or so
> n3 <- lseq(2^-14, 2^2, length=800)
> relE3 <- relE.lgam1(n3)[, -4]
>
> matplot(n3, relE3, type = "l", log="x", col=cols, lty=1, lwd = c(1,3),
+ main = quote(rel.lgam1(n)), xlab=quote(n))
>
> matplot(n3, abs19(relE3), type = "l", log="xy", col=cols, lwd = c(1,3), xaxt="n",
+ main = quote(abs(rel.lgam1(n))), xlab=quote(n), ylim = c(2e-17, 4e-14))
> drawEps.h(); eaxis(1, sub10=c(-2,3))
> legend("topleft", legend=colnames(relE3), col=cols, lwd=2)
> ## very small difference --- draw the 3 smoothers :
> for(j in 1:3) {
+ ll <- lowess(log(n3), abs19(relE3[, j]), f= 1/12)
+ with(ll, lines(exp(x), y, col=adjustcolor(cols[j], 1.5), lwd=3))
+ }
> ## ==> lgamma1p(.) very slightly in n ~ 10^-4 -- 10^-2 --- but not where it matters: n ~ 0.1 -- 1 !!
> ## "MM2" gets best from n >~ 1 !
> abline(v=1, lty=3, col = adjustcolor(1, 3/4))
> showProc.time()
Time (user system elapsed): 1.142 0.004 1.472
>
>
> ### 2. relErr( stirlerr(.) ) ============================================================
>
> ##' Very revealing plot showing the *relative* approximation error of stirlerr(<dblprec>)
> ##'
> p.stirlerrDev <- function(n, precBits = if(doExtras) 2048L else 512L,
+ stnM = stirlerr(mpfr(n, precBits), use.halves=use.halves, verbose=verbose),
+ abs = FALSE,
+ ## cut points, n=*, in the stirlerr() algorithm; "FIXME": sync with ../R/dgamma.R <<<<
+ scheme = c("R3", "R4.4_0"),
+ cutoffs = switch(match.arg(scheme)
+ , R3 = c(15, 35, 80, 500)
+ , R4.4_0 = c(4.9, 5.0, 5.1, 5.2, 5.3, 5.4, 5.7,
+ 6.1, 6.5, 7, 7.9, 8.75, 10.5, 13,
+ 20, 26, 60, 200, 3300, 17.4e6)
+ ## {FIXME: need to sync} <==> ../man/stirlerr.Rd <==> ../R/dgamma.R
+ ),
+ use.halves = missing(cutoffs),
+ direct.ver = c("R3", "lgamma1p", "MM2", "n0"),
+ verbose = getOption("verbose"),
+ type = "b", cex = 1,
+ col = adjustcolor(1, 3/4), colnB = adjustcolor("orange4", 1/3),
+ log = if(abs) "xy" else "x",
+ xlim=NULL, ylim = if(abs) c(8e-18, max(abs(N(relE)))))
+ {
+ op <- par(las = 1, mgp=c(2, 0.6, 0))
+ on.exit(par(op))
+ require("Rmpfr"); require("sfsmisc")
+ st <- stirlerr(n, scheme=scheme, cutoffs=cutoffs, use.halves=use.halves, direct.ver=direct.ver,
+ verbose=verbose)
+ relE <- relErrV(stnM, st) # eps0 = .Machine$double.xmin
+ N <- asNumeric
+ form <- if(abs) abs(N(relE)) ~ n else N(relE) ~ n
+ plot(form, log=log, type=type, cex=cex, col=col, xlim=xlim, ylim=ylim,
+ ylab = quote(relErrV(stM, st)), axes=FALSE, frame.plot=TRUE,
+ main = sprintf("stirlerr(n, cutoffs) rel.error [wrt stirlerr(Rmpfr::mpfr(n, %d))]",
+ precBits))
+ eaxis(1, sub10=3)
+ eaxis(2)
+ mtext(paste("cutoffs =", deparse1(cutoffs)))
+ ylog <- par("ylog")
+ ## FIXME: improve this ---> drawEps.h() above
+ if(ylog) {
+ epsC <- c(1,2,4,8)*2^-52
+ epsCxp <- expression(epsilon[C],2*epsilon[C], 4*epsilon[C], 8*epsilon[C])
+ } else {
+ epsC <- (-2:2)*2^-52
+ epsCxp <- expression(-2*epsilon[C],-epsilon[C], 0, +epsilon[C], +2*epsilon[C])
+ }
+ dy <- diff(par("usr")[3:4])
+ if(diff(range(if(ylog) log10(epsC) else epsC)) > dy/50) {
+ lw <- rep(1/2, 5); lw[if(ylog) 1 else 3] <- 2
+ abline( h=epsC, lty=3, lwd=lw)
+ axis(4, at=epsC, epsCxp, las=2, cex.axis = 3/4, mgp=c(3/4, 1/4, 0), tck=0)
+ } else ## only x-axis
+ abline(h=if(ylog) epsC else 0, lty=3, lwd=2)
+ abline(v = cutoffs, col=colnB)
+ axis(3, at=cutoffs, col=colnB, col.axis=colnB,
+ labels = formatC(cutoffs, digits=3, width=1))
+ invisible(relE)
+ } ## p.stirlerrDev()
>
> showProc.time()
Time (user system elapsed): 0.006 0 0.007
>
> n <- lseq(2^-10, 1e10, length=4096)
> n <- lseq(2^-10, 5000, length=4096)
> ## store "expensive" stirlerr() result, and re-use many times below:
> nM <- mpfr(n, if(doExtras) 2048 else 512)
> st.nM <- stirlerr(nM, use.halves=FALSE) ## << on purpose
>
> p.stirlerrDev(n=n, stnM=st.nM, use.halves = FALSE) # default cutoffs= c(15, 40, 85, 600)
> p.stirlerrDev(n=n, stnM=st.nM, use.halves = FALSE, ylim = c(-1,1)*1e-12) # default cutoffs= c(15, 40, 85, 600)
>
> ## show the zoom-in region in next plot
> yl2 <- 3e-14*c(-1,1)
> abline(h = yl2, col=adjustcolor("tomato", 1/4), lwd=3, lty=2)
>
> if(do.pdf) { dev.off() ; pdf("stirlerr-relErr_1.pdf") }
>
> ## drop n < 7:
> p.stirlerrDev(n=n, stnM=st.nM, xlim = c(7, max(n)), use.halves=FALSE) # default cutoffs= c(15, 40, 85, 600)
> abline(h = yl2, col=adjustcolor("tomato", 1/4), lwd=3, lty=2)
>
> ## The first plot clearly shows we should do better:
> ## Current code is switching to less terms too early, loosing up to 2 decimals precision
> if(FALSE) # no visible difference {use.halves = T / F }:
+ p.stirlerrDev(n=n, stnM=st.nM, ylim = yl2, use.halves = FALSE)
> p.stirlerrDev(n=n, stnM=st.nM, ylim = yl2, use.halves = TRUE)# exact at n/2 (n <= ..)
> abline(h = yl2, col=adjustcolor("tomato", 1/4), lwd=3, lty=2)
>
> showProc.time()
Time (user system elapsed): 3.059 0.008 3.184
>
>
> if(do.pdf) { dev.off(); pdf("stirlerr-relErr_6-fin-1.pdf") }
>
> ### ~19.April 2021: "This is close to *the* solution" (but see 'cuts' below)
> cuts <- c(7, 12, 20, 26, 60, 200, 3300)
> ## ~~~~~~~~~~~~~~~~~~~~~~~~~~~~
> st. <- stirlerr(n=n , cutoffs = cuts, verbose=TRUE)
stirlerr(n, cutoffs = 7,12,20,26,60,200,3300) : case I (n <= 7), using direct formula for n= num [1:2354] 0.000977 0.00098 0.000984 0.000988 0.000991 ...
case II (n > 7 ), 7 cutoffs: ( 7, 12, 20, 26, 60, 200, 3300 ): n in cutoff intervals:
(7,12] (12,20] (20,26] (26,60] (60,200] (200,3.3e+03] (3.3e+03,Inf]
143 135 69 222 319 743 111
> st.nM <- stirlerr(n=nM, cutoffs = cuts, use.halves=FALSE) ## << on purpose
> relE <- asNumeric(relErrV(st.nM, st.))
> head(cbind(n, relE), 20)
n relE
[1,] 0.0009765625 -5.082434e-18
[2,] 0.0009802536 2.562412e-16
[3,] 0.0009839587 -1.229734e-16
[4,] 0.0009876777 1.527219e-16
[5,] 0.0009914108 8.168569e-17
[6,] 0.0009951581 1.722690e-16
[7,] 0.0009989195 -8.277734e-17
[8,] 0.0010026951 -1.706971e-16
[9,] 0.0010064850 9.143135e-17
[10,] 0.0010102892 1.008544e-16
[11,] 0.0010141077 2.920866e-17
[12,] 0.0010179408 -3.724266e-17
[13,] 0.0010217883 -2.367176e-16
[14,] 0.0010256503 -2.103384e-16
[15,] 0.0010295269 4.023359e-17
[16,] 0.0010334182 2.699887e-16
[17,] 0.0010373242 2.584752e-16
[18,] 0.0010412450 7.004536e-17
[19,] 0.0010451806 2.086952e-17
[20,] 0.0010491311 5.520460e-17
> ## nice printout :
> print(cbind(n = format(n, drop0trailing = TRUE),
+ stirlerr= format(st.,scientific=FALSE, digits=4),
+ relErr = signif(relE, 4))
+ , quote=FALSE)
n stirlerr relErr
[1,] 9.765625e-04 2.55398004 -5.082e-18
[2,] 9.802536e-04 2.55211721 2.562e-16
[3,] 9.839587e-04 2.55025446 -1.23e-16
[4,] 9.876777e-04 2.54839178 1.527e-16
[5,] 9.914108e-04 2.54652917 8.169e-17
[6,] 9.951581e-04 2.54466664 1.723e-16
[7,] 9.989195e-04 2.54280419 -8.278e-17
[8,] 1.002695e-03 2.54094181 -1.707e-16
[9,] 1.006485e-03 2.53907950 9.143e-17
[10,] 1.010289e-03 2.53721728 1.009e-16
[11,] 1.014108e-03 2.53535513 2.921e-17
[12,] 1.017941e-03 2.53349305 -3.724e-17
[13,] 1.021788e-03 2.53163106 -2.367e-16
[14,] 1.025650e-03 2.52976914 -2.103e-16
[15,] 1.029527e-03 2.52790730 4.023e-17
[16,] 1.033418e-03 2.52604553 2.7e-16
[17,] 1.037324e-03 2.52418385 2.585e-16
[18,] 1.041245e-03 2.52232224 7.005e-17
[19,] 1.045181e-03 2.52046071 2.087e-17
[20,] 1.049131e-03 2.51859926 5.52e-17
[21,] 1.053096e-03 2.51673789 2.175e-16
[22,] 1.057077e-03 2.51487659 1.455e-16
[23,] 1.061072e-03 2.51301538 1.371e-16
[24,] 1.065083e-03 2.51115425 -7.119e-17
[25,] 1.069108e-03 2.50929319 -3.203e-18
[26,] 1.073149e-03 2.50743222 -1.061e-16
[27,] 1.077206e-03 2.50557133 5.392e-17
[28,] 1.081277e-03 2.50371052 1.841e-16
[29,] 1.085364e-03 2.50184978 9.259e-17
[30,] 1.089466e-03 2.49998913 8.189e-17
[31,] 1.093584e-03 2.49812857 5.493e-17
[32,] 1.097718e-03 2.49626808 -3.044e-17
[33,] 1.101867e-03 2.49440767 -2.861e-16
[34,] 1.106031e-03 2.49254735 -7.993e-17
[35,] 1.110212e-03 2.49068711 6.438e-17
[36,] 1.114408e-03 2.48882695 6.849e-17
[37,] 1.118620e-03 2.48696687 -7.436e-17
[38,] 1.122848e-03 2.48510688 -6.001e-17
[39,] 1.127092e-03 2.48324697 2.464e-16
[40,] 1.131352e-03 2.48138715 2.368e-17
[41,] 1.135628e-03 2.47952741 6.221e-17
[42,] 1.139921e-03 2.47766775 6.605e-17
[43,] 1.144229e-03 2.47580818 6.717e-17
[44,] 1.148554e-03 2.47394869 -1.866e-16
[45,] 1.152895e-03 2.47208929 1.801e-16
[46,] 1.157253e-03 2.47022997 3.256e-16
[47,] 1.161627e-03 2.46837074 -1.514e-16
[48,] 1.166018e-03 2.46651159 -1.983e-18
[49,] 1.170425e-03 2.46465253 -1.39e-16
[50,] 1.174849e-03 2.46279355 1.634e-16
[51,] 1.179289e-03 2.46093467 8.901e-17
[52,] 1.183747e-03 2.45907586 1.956e-16
[53,] 1.188221e-03 2.45721715 3.932e-17
[54,] 1.192712e-03 2.45535852 2.802e-16
[55,] 1.197220e-03 2.45349998 -9.067e-18
[56,] 1.201745e-03 2.45164153 -1.149e-16
[57,] 1.206287e-03 2.44978317 4.78e-17
[58,] 1.210847e-03 2.44792489 -1.19e-16
[59,] 1.215423e-03 2.44606671 4.1e-17
[60,] 1.220017e-03 2.44420861 -2.046e-16
[61,] 1.224629e-03 2.44235060 -5.823e-17
[62,] 1.229257e-03 2.44049268 1.566e-16
[63,] 1.233903e-03 2.43863485 -1.991e-16
[64,] 1.238567e-03 2.43677711 7.107e-17
[65,] 1.243249e-03 2.43491947 -5.887e-17
[66,] 1.247948e-03 2.43306191 -1.285e-16
[67,] 1.252665e-03 2.43120444 -1.121e-16
[68,] 1.257399e-03 2.42934706 1.501e-16
[69,] 1.262152e-03 2.42748978 4.505e-17
[70,] 1.266922e-03 2.42563258 -4.205e-17
[71,] 1.271711e-03 2.42377548 1.292e-16
[72,] 1.276518e-03 2.42191847 -2.172e-17
[73,] 1.281342e-03 2.42006156 3.472e-16
[74,] 1.286186e-03 2.41820473 -1.962e-16
[75,] 1.291047e-03 2.41634800 -1.519e-16
[76,] 1.295927e-03 2.41449136 -1.752e-16
[77,] 1.300825e-03 2.41263482 -1.535e-17
[78,] 1.305742e-03 2.41077837 2.245e-17
[79,] 1.310677e-03 2.40892201 2.469e-16
[80,] 1.315631e-03 2.40706575 7.826e-17
[81,] 1.320604e-03 2.40520958 1.237e-16
[82,] 1.325595e-03 2.40335351 -1.321e-16
[83,] 1.330605e-03 2.40149754 1.9e-16
[84,] 1.335635e-03 2.39964166 -4.563e-17
[85,] 1.340683e-03 2.39778587 3.198e-17
[86,] 1.345750e-03 2.39593018 2.481e-16
[87,] 1.350837e-03 2.39407459 2.077e-16
[88,] 1.355943e-03 2.39221909 -7.773e-17
[89,] 1.361068e-03 2.39036370 4.031e-17
[90,] 1.366212e-03 2.38850839 1.41e-16
[91,] 1.371376e-03 2.38665319 1.129e-16
[92,] 1.376559e-03 2.38479809 1.782e-17
[93,] 1.381762e-03 2.38294308 -7.837e-17
[94,] 1.386985e-03 2.38108817 -6.103e-17
[95,] 1.392227e-03 2.37923336 -2.204e-18
[96,] 1.397489e-03 2.37737865 -3.985e-17
[97,] 1.402772e-03 2.37552404 9.959e-18
[98,] 1.408074e-03 2.37366953 4.462e-17
[99,] 1.413396e-03 2.37181512 9.567e-17
[100,] 1.418738e-03 2.36996081 2.97e-17
[101,] 1.424100e-03 2.36810660 5.213e-17
[102,] 1.429483e-03 2.36625249 -1.177e-16
[103,] 1.434886e-03 2.36439848 8.449e-17
[104,] 1.440309e-03 2.36254457 1.037e-16
[105,] 1.445753e-03 2.36069077 1.778e-16
[106,] 1.451218e-03 2.35883706 2.247e-17
[107,] 1.456703e-03 2.35698346 8.409e-17
[108,] 1.462209e-03 2.35512997 7.423e-17
[109,] 1.467736e-03 2.35327657 2.406e-16
[110,] 1.473283e-03 2.35142328 -1.224e-16
[111,] 1.478852e-03 2.34957010 -8.128e-17
[112,] 1.484441e-03 2.34771701 1.423e-16
[113,] 1.490052e-03 2.34586404 2.358e-16
[114,] 1.495684e-03 2.34401116 1.73e-16
[115,] 1.501337e-03 2.34215839 2.433e-17
[116,] 1.507012e-03 2.34030573 -1.479e-16
[117,] 1.512708e-03 2.33845317 1.394e-18
[118,] 1.518425e-03 2.33660072 2.797e-16
[119,] 1.524165e-03 2.33474838 9.285e-17
[120,] 1.529925e-03 2.33289614 -2.374e-17
[121,] 1.535708e-03 2.33104401 1.553e-16
[122,] 1.541513e-03 2.32919198 -5.306e-17
[123,] 1.547339e-03 2.32734006 -6.804e-17
[124,] 1.553188e-03 2.32548825 -5.164e-17
[125,] 1.559058e-03 2.32363655 -5.4e-17
[126,] 1.564951e-03 2.32178496 -2.546e-16
[127,] 1.570866e-03 2.31993348 9.653e-17
[128,] 1.576803e-03 2.31808210 1.293e-16
[129,] 1.582763e-03 2.31623084 -1.441e-16
[130,] 1.588746e-03 2.31437968 -8.2e-17
[131,] 1.594750e-03 2.31252864 -1.644e-16
[132,] 1.600778e-03 2.31067770 7.274e-17
[133,] 1.606829e-03 2.30882688 -1.791e-16
[134,] 1.612902e-03 2.30697617 -2.591e-16
[135,] 1.618998e-03 2.30512557 1.926e-16
[136,] 1.625118e-03 2.30327508 5.897e-17
[137,] 1.631260e-03 2.30142470 1.636e-16
[138,] 1.637426e-03 2.29957444 2.642e-16
[139,] 1.643615e-03 2.29772429 8.958e-17
[140,] 1.649827e-03 2.29587425 -5.138e-17
[141,] 1.656063e-03 2.29402432 2.726e-18
[142,] 1.662322e-03 2.29217451 1.624e-16
[143,] 1.668605e-03 2.29032482 2.081e-17
[144,] 1.674912e-03 2.28847524 -8.873e-17
[145,] 1.681243e-03 2.28662577 -1.437e-17
[146,] 1.687597e-03 2.28477642 2.161e-16
[147,] 1.693976e-03 2.28292718 2.451e-16
[148,] 1.700379e-03 2.28107806 1.237e-16
[149,] 1.706806e-03 2.27922906 8.671e-17
[150,] 1.713257e-03 2.27738017 1.924e-17
[151,] 1.719732e-03 2.27553140 -2.288e-16
[152,] 1.726232e-03 2.27368275 1.424e-16
[153,] 1.732757e-03 2.27183421 -6.027e-17
[154,] 1.739306e-03 2.26998579 2.736e-16
[155,] 1.745880e-03 2.26813749 -8.571e-18
[156,] 1.752479e-03 2.26628931 2.038e-17
[157,] 1.759103e-03 2.26444125 1.725e-16
[158,] 1.765752e-03 2.26259331 4.62e-17
[159,] 1.772426e-03 2.26074549 8.967e-17
[160,] 1.779125e-03 2.25889779 -1.87e-16
[161,] 1.785850e-03 2.25705021 2.779e-16
[162,] 1.792600e-03 2.25520275 4.373e-17
[163,] 1.799375e-03 2.25335541 7.847e-17
[164,] 1.806176e-03 2.25150819 1.403e-16
[165,] 1.813003e-03 2.24966110 4.483e-17
[166,] 1.819856e-03 2.24781412 -5.16e-17
[167,] 1.826734e-03 2.24596727 2.658e-16
[168,] 1.833639e-03 2.24412055 1.531e-16
[169,] 1.840569e-03 2.24227394 1.182e-16
[170,] 1.847526e-03 2.24042746 -1.273e-16
[171,] 1.854509e-03 2.23858111 1.501e-16
[172,] 1.861519e-03 2.23673487 8.684e-17
[173,] 1.868554e-03 2.23488877 6.754e-17
[174,] 1.875617e-03 2.23304279 -1.128e-16
[175,] 1.882706e-03 2.23119693 4.346e-17
[176,] 1.889822e-03 2.22935120 1.813e-16
[177,] 1.896965e-03 2.22750560 1.635e-16
[178,] 1.904135e-03 2.22566012 1.071e-16
[179,] 1.911332e-03 2.22381478 3.274e-16
[180,] 1.918557e-03 2.22196955 -8.211e-17
[181,] 1.925808e-03 2.22012446 5.732e-17
[182,] 1.933087e-03 2.21827950 8.192e-17
[183,] 1.940394e-03 2.21643466 -1.008e-16
[184,] 1.947728e-03 2.21458995 -1.671e-16
[185,] 1.955089e-03 2.21274537 -2.104e-17
[186,] 1.962479e-03 2.21090093 1.535e-16
[187,] 1.969897e-03 2.20905661 1.078e-16
[188,] 1.977342e-03 2.20721242 1.371e-16
[189,] 1.984816e-03 2.20536837 -2.04e-16
[190,] 1.992318e-03 2.20352444 2.151e-16
[191,] 1.999848e-03 2.20168065 -1.861e-16
[192,] 2.007407e-03 2.19983699 1.494e-16
[193,] 2.014995e-03 2.19799346 1.132e-16
[194,] 2.022611e-03 2.19615006 -1.717e-16
[195,] 2.030255e-03 2.19430680 2.281e-16
[196,] 2.037929e-03 2.19246367 2.363e-16
[197,] 2.045632e-03 2.19062068 2.262e-16
[198,] 2.053364e-03 2.18877782 -1.272e-17
[199,] 2.061125e-03 2.18693510 2.496e-16
[200,] 2.068915e-03 2.18509251 1.261e-16
[201,] 2.076735e-03 2.18325005 -6.126e-17
[202,] 2.084585e-03 2.18140774 5.53e-17
[203,] 2.092464e-03 2.17956556 2.26e-16
[204,] 2.100373e-03 2.17772351 -1.686e-16
[205,] 2.108311e-03 2.17588161 4.373e-17
[206,] 2.116280e-03 2.17403984 -9.225e-17
[207,] 2.124279e-03 2.17219821 1.872e-16
[208,] 2.132308e-03 2.17035672 1.467e-16
[209,] 2.140368e-03 2.16851536 -1.073e-16
[210,] 2.148457e-03 2.16667415 1.736e-16
[211,] 2.156578e-03 2.16483308 6.527e-17
[212,] 2.164729e-03 2.16299214 2.314e-16
[213,] 2.172911e-03 2.16115135 1.237e-16
[214,] 2.181124e-03 2.15931070 -1.756e-16
[215,] 2.189368e-03 2.15747019 2.711e-16
[216,] 2.197643e-03 2.15562982 1.188e-16
[217,] 2.205950e-03 2.15378960 -9.713e-17
[218,] 2.214287e-03 2.15194951 1.068e-16
[219,] 2.222657e-03 2.15010957 -5.873e-18
[220,] 2.231058e-03 2.14826978 2.034e-16
[221,] 2.239490e-03 2.14643012 1.031e-16
[222,] 2.247955e-03 2.14459062 2.195e-16
[223,] 2.256452e-03 2.14275125 -8.599e-18
[224,] 2.264980e-03 2.14091204 -1.438e-16
[225,] 2.273541e-03 2.13907296 1.433e-16
[226,] 2.282135e-03 2.13723404 1.483e-16
[227,] 2.290760e-03 2.13539526 1.834e-17
[228,] 2.299419e-03 2.13355663 -1.028e-16
[229,] 2.308110e-03 2.13171814 1.411e-16
[230,] 2.316834e-03 2.12987980 -1.209e-16
[231,] 2.325591e-03 2.12804162 -1.114e-16
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[1777,] 7.934288e-01 0.10083799 -3.283e-16
[1778,] 7.964277e-01 0.10048340 -7.719e-17
[1779,] 7.994380e-01 0.10012993 -1.285e-15
[1780,] 8.024596e-01 0.09977758 -1.14e-15
[1781,] 8.054927e-01 0.09942633 -1.6e-15
[1782,] 8.085372e-01 0.09907619 -4.611e-16
[1783,] 8.115932e-01 0.09872716 -1.016e-15
[1784,] 8.146608e-01 0.09837923 2.247e-16
[1785,] 8.177399e-01 0.09803240 -2.668e-16
[1786,] 8.208307e-01 0.09768667 -3.951e-16
[1787,] 8.239332e-01 0.09734204 -2.652e-15
[1788,] 8.270474e-01 0.09699850 -3.016e-15
[1789,] 8.301734e-01 0.09665604 1.873e-17
[1790,] 8.333112e-01 0.09631468 5.087e-16
[1791,] 8.364609e-01 0.09597440 -3.853e-16
[1792,] 8.396225e-01 0.09563520 -5.77e-17
[1793,] 8.427960e-01 0.09529709 2.825e-16
[1794,] 8.459815e-01 0.09496005 -6.892e-17
[1795,] 8.491790e-01 0.09462409 1.059e-15
[1796,] 8.523887e-01 0.09428920 -2.288e-15
[1797,] 8.556104e-01 0.09395538 4.936e-16
[1798,] 8.588444e-01 0.09362263 -7.284e-16
[1799,] 8.620906e-01 0.09329094 4.545e-16
[1800,] 8.653490e-01 0.09296032 -1.448e-15
[1801,] 8.686197e-01 0.09263076 3.391e-16
[1802,] 8.719029e-01 0.09230225 -1.105e-15
[1803,] 8.751984e-01 0.09197480 2.261e-16
[1804,] 8.785064e-01 0.09164840 4.227e-16
[1805,] 8.818268e-01 0.09132305 -7.896e-17
[1806,] 8.851599e-01 0.09099875 -6.838e-16
[1807,] 8.885055e-01 0.09067550 3.989e-16
[1808,] 8.918638e-01 0.09035328 -7.927e-16
[1809,] 8.952348e-01 0.09003211 -8.374e-16
[1810,] 8.986185e-01 0.08971198 -1.243e-15
[1811,] 9.020150e-01 0.08939288 3.522e-16
[1812,] 9.054243e-01 0.08907481 -1.826e-15
[1813,] 9.088465e-01 0.08875777 4.213e-17
[1814,] 9.122817e-01 0.08844176 -6.081e-16
[1815,] 9.157299e-01 0.08812677 -7.828e-16
[1816,] 9.191910e-01 0.08781281 -5.659e-16
[1817,] 9.226653e-01 0.08749987 6.681e-16
[1818,] 9.261527e-01 0.08718794 -8.353e-16
[1819,] 9.296533e-01 0.08687703 -5.122e-16
[1820,] 9.331671e-01 0.08656713 3.776e-17
[1821,] 9.366941e-01 0.08625824 -4.529e-16
[1822,] 9.402346e-01 0.08595036 2.028e-16
[1823,] 9.437884e-01 0.08564348 -5.387e-16
[1824,] 9.473556e-01 0.08533761 -1.227e-15
[1825,] 9.509363e-01 0.08503273 8.258e-16
[1826,] 9.545305e-01 0.08472885 -1.687e-15
[1827,] 9.581384e-01 0.08442597 -1.437e-15
[1828,] 9.617598e-01 0.08412408 -9.479e-16
[1829,] 9.653950e-01 0.08382318 2.578e-16
[1830,] 9.690439e-01 0.08352326 1.359e-15
[1831,] 9.727066e-01 0.08322433 -2.198e-15
[1832,] 9.763831e-01 0.08292639 6.445e-16
[1833,] 9.800735e-01 0.08262942 -1.232e-15
[1834,] 9.837779e-01 0.08233343 -1.827e-15
[1835,] 9.874963e-01 0.08203841 6.758e-17
[1836,] 9.912287e-01 0.08174437 -8.212e-16
[1837,] 9.949753e-01 0.08145129 8.057e-16
[1838,] 9.987360e-01 0.08115919 1.383e-15
[1839,] 1.002511 0.08086804 -4.681e-16
[1840,] 1.006300 0.08057786 3.158e-16
[1841,] 1.010104 0.08028864 -1.174e-15
[1842,] 1.013921 0.08000038 -2.007e-15
[1843,] 1.017754 0.07971307 -3.548e-15
[1844,] 1.021601 0.07942671 7.136e-16
[1845,] 1.025462 0.07914130 -6.84e-16
[1846,] 1.029338 0.07885684 -1.343e-15
[1847,] 1.033228 0.07857333 3.561e-16
[1848,] 1.037134 0.07829075 1.673e-15
[1849,] 1.041054 0.07800912 -2.801e-16
[1850,] 1.044989 0.07772842 -1.075e-15
[1851,] 1.048938 0.07744866 2.329e-15
[1852,] 1.052903 0.07716983 -1.756e-15
[1853,] 1.056883 0.07689193 1.56e-15
[1854,] 1.060877 0.07661496 1.04e-15
[1855,] 1.064887 0.07633891 -7.338e-16
[1856,] 1.068912 0.07606379 4.737e-16
[1857,] 1.072952 0.07578958 -1.814e-15
[1858,] 1.077008 0.07551630 -2.419e-15
[1859,] 1.081078 0.07524392 3.003e-16
[1860,] 1.085165 0.07497246 -6.242e-16
[1861,] 1.089266 0.07470191 -2.671e-15
[1862,] 1.093383 0.07443227 -7.352e-16
[1863,] 1.097516 0.07416353 -1.457e-15
[1864,] 1.101664 0.07389570 2.275e-15
[1865,] 1.105828 0.07362876 2.535e-15
[1866,] 1.110008 0.07336273 -1.42e-15
[1867,] 1.114203 0.07309758 -1.182e-15
[1868,] 1.118415 0.07283334 -3.873e-16
[1869,] 1.122642 0.07256998 2.524e-15
[1870,] 1.126885 0.07230751 -2.493e-16
[1871,] 1.131144 0.07204592 -3.711e-15
[1872,] 1.135420 0.07178522 -4.222e-16
[1873,] 1.139711 0.07152540 8.291e-16
[1874,] 1.144019 0.07126646 -7.988e-16
[1875,] 1.148343 0.07100839 -1.265e-16
[1876,] 1.152684 0.07075120 -1.558e-15
[1877,] 1.157040 0.07049487 -9.087e-16
[1878,] 1.161414 0.07023942 1.726e-15
[1879,] 1.165803 0.06998483 -5.183e-17
[1880,] 1.170210 0.06973111 -1.422e-15
[1881,] 1.174633 0.06947824 2.518e-15
[1882,] 1.179073 0.06922624 -7.119e-16
[1883,] 1.183529 0.06897509 9.497e-16
[1884,] 1.188002 0.06872480 -7.462e-16
[1885,] 1.192493 0.06847536 -2.578e-15
[1886,] 1.197000 0.06822676 2.651e-15
[1887,] 1.201524 0.06797902 -8.521e-16
[1888,] 1.206066 0.06773211 -2.211e-15
[1889,] 1.210624 0.06748606 1.676e-16
[1890,] 1.215200 0.06724084 1.965e-16
[1891,] 1.219793 0.06699645 1.441e-15
[1892,] 1.224404 0.06675291 -2.952e-15
[1893,] 1.229031 0.06651019 -2.416e-15
[1894,] 1.233677 0.06626830 -1.054e-15
[1895,] 1.238340 0.06602725 3.144e-15
[1896,] 1.243020 0.06578702 -1.917e-15
[1897,] 1.247718 0.06554761 -2.721e-15
[1898,] 1.252434 0.06530902 -1.939e-15
[1899,] 1.257168 0.06507125 -2.131e-15
[1900,] 1.261920 0.06483429 1.484e-15
[1901,] 1.266690 0.06459815 9.309e-16
[1902,] 1.271477 0.06436282 -3.867e-16
[1903,] 1.276283 0.06412830 -2.843e-15
[1904,] 1.281107 0.06389459 1.186e-15
[1905,] 1.285949 0.06366168 -4.652e-16
[1906,] 1.290810 0.06342957 -1.214e-15
[1907,] 1.295689 0.06319826 1.793e-16
[1908,] 1.300586 0.06296775 -7.672e-16
[1909,] 1.305502 0.06273803 1.388e-15
[1910,] 1.310436 0.06250911 4.136e-16
[1911,] 1.315389 0.06228098 3.217e-16
[1912,] 1.320361 0.06205363 -3.569e-15
[1913,] 1.325352 0.06182707 -1.09e-15
[1914,] 1.330361 0.06160129 1.838e-15
[1915,] 1.335389 0.06137630 5.138e-16
[1916,] 1.340437 0.06115208 -3.195e-15
[1917,] 1.345503 0.06092864 2.198e-15
[1918,] 1.350589 0.06070597 -5.86e-16
[1919,] 1.355694 0.06048408 9.79e-16
[1920,] 1.360818 0.06026295 3.056e-15
[1921,] 1.365961 0.06004259 2.699e-15
[1922,] 1.371124 0.05982300 4.03e-17
[1923,] 1.376306 0.05960416 -1.909e-15
[1924,] 1.381508 0.05938609 -1.619e-16
[1925,] 1.386730 0.05916878 7.599e-17
[1926,] 1.391972 0.05895222 2.312e-15
[1927,] 1.397233 0.05873642 -2.011e-15
[1928,] 1.402514 0.05852137 1.88e-16
[1929,] 1.407815 0.05830706 -1.501e-15
[1930,] 1.413136 0.05809350 8.322e-16
[1931,] 1.418477 0.05788069 -2.59e-15
[1932,] 1.423839 0.05766862 -1.602e-15
[1933,] 1.429220 0.05745729 -4.045e-15
[1934,] 1.434622 0.05724670 -5.737e-15
[1935,] 1.440045 0.05703684 -2.58e-15
[1936,] 1.445488 0.05682772 2.042e-15
[1937,] 1.450951 0.05661932 1.187e-15
[1938,] 1.456435 0.05641166 -3.626e-15
[1939,] 1.461940 0.05620472 2.141e-16
[1940,] 1.467466 0.05599851 9.299e-16
[1941,] 1.473013 0.05579302 3.592e-15
[1942,] 1.478580 0.05558824 4.964e-15
[1943,] 1.484169 0.05538419 -5.402e-15
[1944,] 1.489778 0.05518085 3.912e-16
[1945,] 1.495409 0.05497823 -6.289e-15
[1946,] 1.501061 0.05477631 2.456e-15
[1947,] 1.506735 0.05457511 4.12e-15
[1948,] 1.512430 0.05437461 2.574e-15
[1949,] 1.518147 0.05417482 -2.724e-15
[1950,] 1.523885 0.05397573 1.141e-15
[1951,] 1.529644 0.05377734 -1.981e-15
[1952,] 1.535426 0.05357965 -2.582e-15
[1953,] 1.541229 0.05338265 -9.349e-16
[1954,] 1.547055 0.05318635 1.021e-15
[1955,] 1.552902 0.05299073 -1.744e-15
[1956,] 1.558772 0.05279581 -1.408e-15
[1957,] 1.564663 0.05260158 -2.171e-15
[1958,] 1.570577 0.05240803 -8.398e-16
[1959,] 1.576514 0.05221516 -2.951e-15
[1960,] 1.582472 0.05202298 1.318e-15
[1961,] 1.588454 0.05183147 -2.924e-15
[1962,] 1.594458 0.05164064 -2.468e-16
[1963,] 1.600484 0.05145049 -5.71e-16
[1964,] 1.606533 0.05126100 8.692e-16
[1965,] 1.612606 0.05107219 2.484e-17
[1966,] 1.618701 0.05088404 1.327e-15
[1967,] 1.624819 0.05069657 -1.618e-15
[1968,] 1.630960 0.05050975 -2.173e-15
[1969,] 1.637125 0.05032360 -1.718e-15
[1970,] 1.643313 0.05013811 -2.544e-15
[1971,] 1.649524 0.04995327 1.617e-16
[1972,] 1.655759 0.04976909 -3.492e-15
[1973,] 1.662017 0.04958556 -7.871e-16
[1974,] 1.668299 0.04940269 -3.591e-16
[1975,] 1.674604 0.04922046 2.503e-16
[1976,] 1.680934 0.04903889 -3.363e-15
[1977,] 1.687287 0.04885795 -3.969e-16
[1978,] 1.693665 0.04867766 -2.458e-15
[1979,] 1.700066 0.04849802 -4.059e-15
[1980,] 1.706492 0.04831901 -2.201e-15
[1981,] 1.712942 0.04814063 3.177e-16
[1982,] 1.719416 0.04796290 -4.841e-15
[1983,] 1.725915 0.04778579 -6.591e-15
[1984,] 1.732439 0.04760932 1.521e-15
[1985,] 1.738987 0.04743347 -3.311e-15
[1986,] 1.745560 0.04725826 1.542e-15
[1987,] 1.752157 0.04708366 6.898e-16
[1988,] 1.758780 0.04690969 9.416e-17
[1989,] 1.765428 0.04673634 5.156e-15
[1990,] 1.772100 0.04656361 -1.637e-15
[1991,] 1.778798 0.04639150 -1.54e-15
[1992,] 1.785522 0.04622000 -5.768e-16
[1993,] 1.792270 0.04604912 -5.491e-15
[1994,] 1.799045 0.04587884 1.75e-16
[1995,] 1.805844 0.04570918 2.485e-15
[1996,] 1.812670 0.04554012 -7.234e-15
[1997,] 1.819521 0.04537167 -3.603e-15
[1998,] 1.826399 0.04520381 -1.532e-15
[1999,] 1.833302 0.04503657 -2.359e-15
[2000,] 1.840231 0.04486991 -4.735e-15
[2001,] 1.847187 0.04470386 -1.351e-16
[2002,] 1.854168 0.04453840 -8.021e-16
[2003,] 1.861177 0.04437354 -3.029e-15
[2004,] 1.868211 0.04420926 2.626e-15
[2005,] 1.875273 0.04404558 9.737e-16
[2006,] 1.882360 0.04388248 -1.996e-15
[2007,] 1.889475 0.04371997 -4.155e-15
[2008,] 1.896617 0.04355804 4.594e-15
[2009,] 1.903785 0.04339670 1.529e-15
[2010,] 1.910981 0.04323593 2.103e-15
[2011,] 1.918204 0.04307574 6.296e-16
[2012,] 1.925454 0.04291613 8.562e-16
[2013,] 1.932732 0.04275709 -1.49e-15
[2014,] 1.940037 0.04259863 -1.85e-15
[2015,] 1.947370 0.04244073 -8.186e-15
[2016,] 1.954730 0.04228340 1.494e-15
[2017,] 1.962119 0.04212664 6.517e-15
[2018,] 1.969535 0.04197044 2.445e-16
[2019,] 1.976979 0.04181481 -6.152e-15
[2020,] 1.984451 0.04165974 -3.198e-15
[2021,] 1.991952 0.04150522 9.752e-18
[2022,] 1.999481 0.04135127 -1.122e-15
[2023,] 2.007038 0.04119787 1.619e-15
[2024,] 2.014624 0.04104502 -3.315e-15
[2025,] 2.022239 0.04089272 -2.242e-15
[2026,] 2.029883 0.04074098 3.008e-15
[2027,] 2.037555 0.04058978 -5.591e-15
[2028,] 2.045256 0.04043912 -3.44e-15
[2029,] 2.052987 0.04028902 3.512e-15
[2030,] 2.060746 0.04013945 -1.344e-15
[2031,] 2.068535 0.03999043 -1.161e-14
[2032,] 2.076354 0.03984194 2.467e-16
[2033,] 2.084202 0.03969399 -4.227e-15
[2034,] 2.092079 0.03954658 -1.864e-16
[2035,] 2.099987 0.03939970 4.424e-15
[2036,] 2.107924 0.03925335 5.364e-15
[2037,] 2.115891 0.03910753 -4.937e-15
[2038,] 2.123889 0.03896224 2.674e-15
[2039,] 2.131916 0.03881748 -4.884e-15
[2040,] 2.139974 0.03867324 3.795e-15
[2041,] 2.148063 0.03852952 -1.041e-15
[2042,] 2.156182 0.03838633 3.173e-15
[2043,] 2.164332 0.03824365 -1.094e-14
[2044,] 2.172512 0.03810149 -1.747e-15
[2045,] 2.180724 0.03795985 -4.156e-15
[2046,] 2.188966 0.03781872 -8.631e-16
[2047,] 2.197240 0.03767811 8.76e-15
[2048,] 2.205544 0.03753800 -2.333e-17
[2049,] 2.213881 0.03739840 -4.512e-15
[2050,] 2.222249 0.03725931 1.637e-15
[2051,] 2.230648 0.03712073 1.094e-15
[2052,] 2.239079 0.03698265 -1.016e-15
[2053,] 2.247542 0.03684507 1.954e-15
[2054,] 2.256037 0.03670799 -7.343e-15
[2055,] 2.264564 0.03657141 8.815e-15
[2056,] 2.273124 0.03643533 -3.75e-15
[2057,] 2.281715 0.03629974 -1.163e-16
[2058,] 2.290340 0.03616464 2.287e-15
[2059,] 2.298996 0.03603004 2.322e-15
[2060,] 2.307686 0.03589593 2.393e-15
[2061,] 2.316408 0.03576231 -4.001e-15
[2062,] 2.325163 0.03562917 7.919e-15
[2063,] 2.333952 0.03549652 -6.088e-15
[2064,] 2.342774 0.03536435 7.692e-15
[2065,] 2.351628 0.03523266 -1.908e-15
[2066,] 2.360517 0.03510145 4.311e-15
[2067,] 2.369439 0.03497072 -1.847e-15
[2068,] 2.378395 0.03484047 -6.013e-16
[2069,] 2.387384 0.03471070 2.413e-15
[2070,] 2.396408 0.03458139 -5.676e-15
[2071,] 2.405466 0.03445256 -6.746e-15
[2072,] 2.414557 0.03432420 -6.183e-15
[2073,] 2.423684 0.03419631 -7.864e-15
[2074,] 2.432845 0.03406889 -1.31e-14
[2075,] 2.442040 0.03394193 5.082e-15
[2076,] 2.451270 0.03381543 6.294e-16
[2077,] 2.460535 0.03368940 -4.532e-15
[2078,] 2.469835 0.03356383 5.593e-17
[2079,] 2.479170 0.03343872 1.786e-15
[2080,] 2.488541 0.03331406 -3.706e-15
[2081,] 2.497947 0.03318986 3.486e-17
[2082,] 2.507388 0.03306612 9.84e-16
[2083,] 2.516866 0.03294282 6.226e-15
[2084,] 2.526378 0.03281998 -2.4e-15
[2085,] 2.535927 0.03269759 1.471e-15
[2086,] 2.545512 0.03257565 5.955e-16
[2087,] 2.555134 0.03245415 -3.394e-15
[2088,] 2.564791 0.03233310 -6.065e-16
[2089,] 2.574485 0.03221249 7.551e-15
[2090,] 2.584216 0.03209232 2.621e-15
[2091,] 2.593984 0.03197260 6.198e-16
[2092,] 2.603788 0.03185331 5.632e-15
[2093,] 2.613630 0.03173446 6.96e-15
[2094,] 2.623508 0.03161605 -1.768e-15
[2095,] 2.633425 0.03149807 1.225e-14
[2096,] 2.643378 0.03138052 -4.228e-15
[2097,] 2.653369 0.03126340 4.802e-15
[2098,] 2.663398 0.03114672 5.122e-15
[2099,] 2.673465 0.03103046 9.716e-15
[2100,] 2.683570 0.03091463 -2.628e-16
[2101,] 2.693713 0.03079922 -3.655e-15
[2102,] 2.703894 0.03068424 -1.083e-14
[2103,] 2.714114 0.03056968 -2.751e-15
[2104,] 2.724373 0.03045554 -5.199e-15
[2105,] 2.734670 0.03034182 -9.545e-15
[2106,] 2.745006 0.03022852 -2.001e-14
[2107,] 2.755381 0.03011564 -9.947e-15
[2108,] 2.765796 0.03000317 -2.573e-15
[2109,] 2.776250 0.02989111 1.086e-14
[2110,] 2.786743 0.02977946 1.437e-14
[2111,] 2.797276 0.02966823 2.901e-15
[2112,] 2.807849 0.02955740 1.141e-14
[2113,] 2.818462 0.02944699 -2.503e-15
[2114,] 2.829115 0.02933697 -9.145e-15
[2115,] 2.839808 0.02922737 -5.264e-15
[2116,] 2.850542 0.02911817 1.52e-14
[2117,] 2.861316 0.02900936 -1.32e-14
[2118,] 2.872131 0.02890096 -1.929e-15
[2119,] 2.882986 0.02879296 6.094e-15
[2120,] 2.893883 0.02868536 -1.819e-14
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[2122,] 2.915801 0.02847134 -4.923e-15
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[2125,] 2.948988 0.02815326 1.284e-15
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[3992,] 3.377348e+03 0.00002467 -2.527e-16
[3993,] 3.390114e+03 0.00002458 -5.583e-17
[3994,] 3.402927e+03 0.00002449 -1.72e-16
[3995,] 3.415789e+03 0.00002440 -6.724e-17
[3996,] 3.428700e+03 0.00002430 -1.401e-16
[3997,] 3.441659e+03 0.00002421 -9.898e-17
[3998,] 3.454668e+03 0.00002412 -1.595e-16
[3999,] 3.467725e+03 0.00002403 -1.374e-16
[4000,] 3.480832e+03 0.00002394 -3.276e-17
[4001,] 3.493989e+03 0.00002385 -6.231e-17
[4002,] 3.507195e+03 0.00002376 -1.285e-16
[4003,] 3.520451e+03 0.00002367 -6.731e-17
[4004,] 3.533757e+03 0.00002358 -2.32e-17
[4005,] 3.547114e+03 0.00002349 -1.568e-16
[4006,] 3.560521e+03 0.00002340 -2.222e-16
[4007,] 3.573978e+03 0.00002332 -1.65e-16
[4008,] 3.587487e+03 0.00002323 -2.566e-17
[4009,] 3.601047e+03 0.00002314 -6.435e-17
[4010,] 3.614657e+03 0.00002305 -1.187e-16
[4011,] 3.628320e+03 0.00002297 -1.807e-16
[4012,] 3.642034e+03 0.00002288 -2.159e-16
[4013,] 3.655799e+03 0.00002279 -6.109e-17
[4014,] 3.669617e+03 0.00002271 -1.484e-16
[4015,] 3.683487e+03 0.00002262 -3.794e-17
[4016,] 3.697410e+03 0.00002254 -3.247e-17
[4017,] 3.711385e+03 0.00002245 -2.219e-16
[4018,] 3.725413e+03 0.00002237 -8.905e-17
[4019,] 3.739494e+03 0.00002228 -1.673e-17
[4020,] 3.753628e+03 0.00002220 -1.991e-16
[4021,] 3.767815e+03 0.00002212 -2.642e-17
[4022,] 3.782056e+03 0.00002203 -5.082e-17
[4023,] 3.796351e+03 0.00002195 -7.433e-17
[4024,] 3.810701e+03 0.00002187 -6.705e-17
[4025,] 3.825104e+03 0.00002179 -3.841e-18
[4026,] 3.839562e+03 0.00002170 -7.373e-17
[4027,] 3.854074e+03 0.00002162 -1.13e-16
[4028,] 3.868641e+03 0.00002154 -1.671e-16
[4029,] 3.883263e+03 0.00002146 -4.394e-17
[4030,] 3.897941e+03 0.00002138 -1.024e-16
[4031,] 3.912674e+03 0.00002130 -1.103e-16
[4032,] 3.927463e+03 0.00002122 -1.02e-16
[4033,] 3.942307e+03 0.00002114 3.6e-17
[4034,] 3.957208e+03 0.00002106 -1.234e-16
[4035,] 3.972165e+03 0.00002098 -1.88e-17
[4036,] 3.987179e+03 0.00002090 5.146e-17
[4037,] 4.002249e+03 0.00002082 -3.797e-18
[4038,] 4.017376e+03 0.00002074 -2.494e-16
[4039,] 4.032561e+03 0.00002067 -1.753e-16
[4040,] 4.047802e+03 0.00002059 -2.058e-16
[4041,] 4.063102e+03 0.00002051 -8.551e-17
[4042,] 4.078459e+03 0.00002043 -1.01e-16
[4043,] 4.093874e+03 0.00002036 -8.708e-17
[4044,] 4.109348e+03 0.00002028 -4.11e-17
[4045,] 4.124880e+03 0.00002020 -2.088e-16
[4046,] 4.140471e+03 0.00002013 -1.721e-16
[4047,] 4.156121e+03 0.00002005 -2.429e-16
[4048,] 4.171829e+03 0.00001998 -1.634e-16
[4049,] 4.187598e+03 0.00001990 -9.396e-17
[4050,] 4.203426e+03 0.00001983 -1.056e-16
[4051,] 4.219313e+03 0.00001975 1.626e-18
[4052,] 4.235261e+03 0.00001968 -6.848e-17
[4053,] 4.251269e+03 0.00001960 -1.644e-16
[4054,] 4.267337e+03 0.00001953 -3.362e-17
[4055,] 4.283467e+03 0.00001945 -1.294e-16
[4056,] 4.299657e+03 0.00001938 1.393e-17
[4057,] 4.315908e+03 0.00001931 -1.734e-17
[4058,] 4.332221e+03 0.00001924 -1.47e-16
[4059,] 4.348595e+03 0.00001916 -1.942e-16
[4060,] 4.365032e+03 0.00001909 -8.93e-17
[4061,] 4.381530e+03 0.00001902 -9.382e-17
[4062,] 4.398091e+03 0.00001895 -1.668e-16
[4063,] 4.414715e+03 0.00001888 -4.669e-17
[4064,] 4.431401e+03 0.00001881 -2.186e-16
[4065,] 4.448150e+03 0.00001873 -5.586e-17
[4066,] 4.464963e+03 0.00001866 -2.021e-16
[4067,] 4.481839e+03 0.00001859 -3.78e-17
[4068,] 4.498779e+03 0.00001852 -4.886e-17
[4069,] 4.515783e+03 0.00001845 -1.409e-16
[4070,] 4.532851e+03 0.00001838 -3.731e-17
[4071,] 4.549984e+03 0.00001832 -2.195e-16
[4072,] 4.567182e+03 0.00001825 -1.645e-17
[4073,] 4.584444e+03 0.00001818 -1.875e-17
[4074,] 4.601772e+03 0.00001811 7.383e-17
[4075,] 4.619165e+03 0.00001804 -1.858e-16
[4076,] 4.636624e+03 0.00001797 -6.538e-17
[4077,] 4.654149e+03 0.00001791 -4.84e-17
[4078,] 4.671740e+03 0.00001784 -9.022e-17
[4079,] 4.689398e+03 0.00001777 -4.602e-17
[4080,] 4.707123e+03 0.00001770 -9.452e-17
[4081,] 4.724914e+03 0.00001764 -1.064e-17
[4082,] 4.742773e+03 0.00001757 -7.592e-17
[4083,] 4.760699e+03 0.00001750 -5.355e-17
[4084,] 4.778693e+03 0.00001744 -1.003e-16
[4085,] 4.796755e+03 0.00001737 -7.795e-17
[4086,] 4.814885e+03 0.00001731 -2.51e-17
[4087,] 4.833084e+03 0.00001724 7.088e-17
[4088,] 4.851352e+03 0.00001718 -1.708e-16
[4089,] 4.869688e+03 0.00001711 -8.182e-17
[4090,] 4.888094e+03 0.00001705 4.589e-18
[4091,] 4.906570e+03 0.00001698 -3.178e-17
[4092,] 4.925115e+03 0.00001692 -1.18e-16
[4093,] 4.943731e+03 0.00001686 -1.146e-16
[4094,] 4.962416e+03 0.00001679 4.272e-17
[4095,] 4.981173e+03 0.00001673 -8.038e-17
[4096,] 5.000000e+03 0.00001667 1.57e-17
>
> p.stirlerrDev(n=n, stnM=st.nM, cex=1/4, type="o", cutoffs = cuts)
> ## and zoom in:
> p.stirlerrDev(n=n, stnM=st.nM, cex=1/4, type="o", cutoffs = cuts, ylim = yl2)
> p.stirlerrDev(n=n, stnM=st.nM, cex=1/4, type="o", cutoffs = cuts, ylim = yl2/20)
>
> if(do.pdf) { dev.off(); pdf("stirlerr-relErr_6-fin-2.pdf") }
>
> ## zoom in ==> {good for n >= 10}
> p.stirlerrDev(n=n, stnM=st.nM, cex=1/4, type="o", ylim = 2e-15*c(-1,1),
+ cutoffs = cuts)## old default cutoffs = c(15,35, 80, 500)
>
> if(do.pdf) { dev.off(); pdf("stirlerr-relErr_6-fin-3.pdf") }
> showProc.time()
Time (user system elapsed): 3.33 0.02 3.532
>
>
> ##-- April 20: have more terms up to S10 in stirlerr() --> can use more cutoffs
> n <- n5m <- lseq(1/64, 5000, length=4096)
> nM <- mpfr(n, if(doExtras) 2048L # a *lot* accuracy for stirlerr(nM,*)
+ else 512L)
> ct10.1 <- c( 5.4, 7.5, 8.5, 10.625, 12.125, 20, 26, 60, 200, 3300)# till 2024-01-19
> ct10.2 <- c( 5.4, 7.9, 8.75,10.5 , 13, 20, 26, 60, 200, 3300)
> cuts <-
+ ct12.1 <- c(5.22, 6.5, 7.0, 7.9, 8.75,10.5 , 13, 20, 26, 60, 200, 3300)
> ## ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
> ## 5.25 is "too small" but the direct formula is already really bad there, ...
> st.nM <- roundMpfr(stirlerr(nM, use.halves=FALSE, ## << on purpose;
+ verbose=TRUE), precBits = 128)
stirlerr(n): As 'n' is "mpfr", using "mpfr" & stirlerrM():
> ## NB: for x=xM <mpfr>; `cutoffs` are *not* used.
> p.stirlerrDev(n=n, stnM=st.nM, cex=1/4, type="o", scheme = "R4.4_0")
> p.stirlerrDev(n=n, stnM=st.nM, cex=1/4, type="o", scheme = "R4.4_0", ylim = c(-1,1)*3e-15)
> p.stirlerrDev(n=n, stnM=st.nM, cex=1/4, type="o", scheme = "R4.4_0", ylim = c(-1,1)*1e-15)
>
> p.stirlerrDev(n=n, stnM=st.nM, cex=1/4, type="o", scheme = "R4.4_0", abs=TRUE)
> axis(1,at= 2:6, col=NA, col.axis=(cola <- "lightblue"), line=-3/4)
> abline(v = 2:6, lty=3, col=cola)
> if(FALSE)## using exact values sferr_halves[] *instead* of MPFR ones: ==> confirmation they lay on top
+ lines((0:30)/2, abs(stirlerr((0:30)/2, cutoffs=cuts, verbose=TRUE)/DPQ:::sferr_halves - 1), type="o", col=2,lwd=2)
>
> if(FALSE) ## nice (but unneeded) printout :
+ print(cbind(n = format(n, drop0trailing = TRUE),
+ stirlerr= format(st.,scientific=FALSE, digits=4),
+ relErr = signif(relE, 4))
+ , quote=FALSE)
>
> showProc.time()
Time (user system elapsed): 2.787 0.016 2.991
>
> <0c>
> ## ========== where should the cutoffs be ? ===================================================
>
> .stirl.cutoffs <- function(scheme)
+ eval(do.call(substitute, list(formals(stirlerr)$cutoffs, list(scheme = scheme))))
> drawCuts <- function(scheme, axis=NA, lty = 3, col = "skyblue", ...) {
+ abline(v = (ct <- .stirl.cutoffs(scheme)), lty=lty, col=col, ...)
+ if(is.finite(axis)) axisCuts(side = axis, at = ct, col=col, ...)
+ }
> axisCuts <- function(scheme, side = 3, at = .stirl.cutoffs(scheme), col = "skyblue", line = -3/4, ...)
+ axis(side, at=at, labels=formatC(at), col.axis = col, col=NA, col.ticks=NA, line=line, ...)
> mtextCuts <- function(cutoffs, scheme, ...) {
+ if(!missing(scheme)) cutoffs <- .stirl.cutoffs(scheme)
+ mtext(paste("cutoffs =", deparse1(cutoffs)), ...)
+ }
>
>
> if(do.pdf) { dev.off(); pdf("stirlerr-tst_order_k.pdf") }
>
> mK <- 20L # := max(k)
> ## order = k = 1:mK terms in series approx:
> k <- 1:mK
> n <- 2^seq(1, 28, by=1/16)
> nM <- mpfr(n, 1024)
> stnM <- stirlerr(nM) # the "true" values
>
> stirlOrd <- sapply(k, function(k.) stirlerr(n, order = k.))
> stirlO_lgcor <- cbind(stirlOrd, sapply(5:6, function(nal) lgammacor(n, nalgm = nal)))
> relE <- asNumeric(stirlO_lgcor/stnM -1) # "true" relative error
>
> ## use a "smooth" but well visible polette :
> palROBG <- colorRampPalette(c("red", "darkorange2", "blue", "seagreen"), space = "Lab")
> palette(adjustcolor(palROBG(mK+2), 3/4))
> ## -- 2 lgamcor()'s
>
> (tit.k <- substitute(list( stirlerr(n, order=k) ~~"error", k == 1:mK), list(mK = mK)))
list(stirlerr(n, order = k) ~ ~"error", k == 1:20L)
> (tit.kA <- substitute(list(abs(stirlerr(n, order=k) ~~"error"), k == 1:mK), list(mK = mK)))
list(abs(stirlerr(n, order = k) ~ ~"error"), k == 1:20L)
> lgammacorTit <- function(...) mtext("+ lgammacor(x, 5) [bad] + lgammacor(x, 6) [good]", col=1, ...)
>
> matplotB(n, relE, cex=2/3, ylim = c(-1,1)*1e-13, col=k,
+ log = "x", xaxt="n", main = tit.k)
> lgammacorTit()
> eaxis(1, nintLog = 20)
> drawCuts("R4.4_0")
>
> ## zoom in (ylim)
> matplotB(n, relE, cex=2/3, ylim = c(-1,1)*5e-15, col=k,
+ log = "x", xaxt="n", main = tit.k)
> lgammacorTit()
> eaxis(1, nintLog = 20); abline(h = (-2:2)*2^-53, lty=3, lwd=1/2)
> drawCuts("R4.4_0", axis = 3)
>
> ## log-log |rel.Err| -- "linear"
> matplotB(n, abs19(relE), cex=2/3, col=k, ylim = c(8e-17, 1e-7), log = "xy", main=tit.kA)
> mtext(paste("k =", deparse(k))) ; abline(h = 2^-(53:51), lty=3, lwd=1/2)
> lgammacorTit(line=-1)
> drawCuts("R4.4_0", axis = 3)
>
> ## zoom in -- still "large" {no longer carry the lgammacor() .. }:
> n2c <- 2^seq(2, 8, by=1/256)
> nMc <- mpfr(n2c, 1024)
> stnMc <- stirlerr(nMc) # the "true" values
> stirlOrc <- sapply(k, function(k.) stirlerr(n2c, order = k.))
> relEc <- asNumeric(stirlOrc/stnMc -1) # "true" relative error
>
> matplotB(n2c, relEc, cex=2/3, ylim = c(-1,1)*1e-13, col=k,
+ log = "x", xaxt="n", main = tit.k)
> eaxis(1, sub10 = 2)
> drawCuts("R4.4_0", axis=3)
>
> ## log-log |rel.Err| -- "linear"
> matplotB(n2c, abs19(relEc), cex=2/3, col=k, ylim = c(8e-17, 1e-3), log = "xy", main=tit.kA)
> mtext(paste("k =", deparse(k))) ; abline(h = 2^-(53:51), lty=3, lwd=1/2)
> drawCuts("R4.4_0", axis = 3)
>
>
> ## zoom into the critical n region
> nc <- seq(3.5, 11, by=1/128)
> ncM <- mpfr(nc, 256)
> stncM <- stirlerr(ncM) # the "true" values
> stirlO.c <- sapply(k, function(k) stirlerr(nc, order = k))
> relEc <- asNumeric(stirlO.c/stncM -1) # "true" relative error
>
>
> ## log-log |rel.Err| -- "linear"
> matplotB(nc, abs19(relEc), cex=2/3, col=k, ylim = c(2e-17, 1e-8),
+ log = "xy", xlab = quote(n), main = quote(abs(relErr(stirlerr(n, order==k)))))
> mtext(paste("k =", deparse(k))) ; drawEps.h(lwd = 1/2)
> lines(nc, abs19(asNumeric(stirlerr_simpl(nc, "R3" )/stncM - 1)), lwd=1.5, col=adjustcolor("thistle", .6))
> lines(nc, abs19(asNumeric(stirlerr_simpl(nc, "MM2")/stncM - 1)), lwd=4, col=adjustcolor(20, .4))
> ## lines(nc, abs19(asNumeric(stirlerr_simpl(nc,"MM2")/stncM - 1)), lwd=3, col=adjustcolor("purple", 2/3))
> legend(10^par("usr")[1], 1e-9, legend=paste0("k=", k), bty="n", lwd=2,
+ col=k, lty=1:5, pch= c(1L:9L, 0L, letters)[seq_along(k)])
> drawCuts("R4.4_0", axis=3)
>
> ## Zoom-in [only]
> matplotB(nc, abs19(relEc), cex=2/3, col=k, ylim = c(4e-17, 1e-11), xlim = c(4.8, 6.5),
+ log = "xy", xlab = quote(n), main = quote(abs(relErr(stirlerr(n, order==k)))))
> mtext(paste("k =", deparse(k))) ; drawEps.h(lwd = 1/2)
> lines(nc, abs19(asNumeric(stirlerr_simpl(nc, "R3" )/stncM - 1)), lwd=1.5, col=adjustcolor("thistle", .6))
> lines(nc, abs19(asNumeric(stirlerr_simpl(nc, "MM2")/stncM - 1)), lwd=4, col=adjustcolor(20, .4))
>
> k. <- k[-(1:6)]
> legend("bottomleft", legend=paste0("k=", k.), bty="n", lwd=2,
+ col=k., lty=1:5, pch= c(1L:9L, 0L, letters)[k.])
> drawCuts("R4.4_0", axis=3)
>
> showProc.time()
Time (user system elapsed): 3.711 0.016 4.303
>
> ##--- Accuracy of "R4.4_0" -------------------------------------------------------
>
> for(nc in list(seq(4.75, 28, by=1/512), # for a bigger pix
+ seq(4.75, 9, by=1/1024)))
+ {
+ ncM <- mpfr(nc, 1024)
+ stncM <- stirlerr(ncM) # the "true" values
+ stirl.440 <- stirlerr(nc, scheme = "R4.4_0")
+ stirl.3 <- stirlerr(nc, scheme = "R3")
+ relE440 <- asNumeric(relErrV(stncM, stirl.440))
+ relE3 <- asNumeric(relErrV(stncM, stirl.3 ))
+ ##
+ plot(nc, abs19(relE440), xlab=quote(n), main = quote(abs(relErr(stirlerr(n, '"R4.4_0"')))),
+ type = "l", log = "xy", ylim = c(4e-17, 1e-13))
+ mtextCuts(scheme="R4.4_0", cex=4/5)
+ drawCuts("R4.4_0", lty=2, lwd=2, axis=4)
+ drawEps.h()
+ if(max(nc) <= 10) abline(v = 5+(0:20)/10, lty=3, col=adjustcolor(4, 1/2))
+ if(TRUE) { # but just so ...
+ c3 <- adjustcolor("royalblue", 1/2)
+ lines(nc, pmax(abs(relE3), 1e-18), col=c3)
+ title(quote(abs(relErr(stirlerr(n, '"R3"')))), adj=1, col.main = c3)
+ drawCuts("R3", lty=4, col=c3); mtextCuts(scheme="R3", adj=1, col=c3)
+ }
+ addOrd <- TRUE
+ addOrd <- dev.interactive(orNone=TRUE)
+ if(addOrd) {
+ if(max(nc) >= 100) {
+ i <- (15 <= nc & nc <= 85) # (no-op in [4.75, 7] !)
+ ni <- nc[i]
+ } else { i <- TRUE; ni <- nc }
+ for(k in 8:17) lines(ni, abs19(asNumeric(relErrV(stncM[i], stirlerr(ni, order=k)))), col=adjustcolor(k, 1/3))
+ title(sub = "stirlerr(*, order k = 8:17)")
+ }
+ } ## for(nc ..)
>
> if(FALSE)
+ lines(nc, abs19(relE440))# *re* draw!
>
> showProc.time()
Time (user system elapsed): 11.923 0.008 13.367
>
>
> palette("Tableau")
>
> ## Focus more:
> stirlerrPlot <- function(nc, k, res=NULL, legend.xy = "left", full=TRUE, precB = 1024,
+ ylim = c(3e-17, 2e-13), cex = 5/4) {
+ stopifnot(require("Rmpfr"), require("graphics"))
+ if(is.list(res) && all(c("nc", "k", "relEc","splarelE") %in% names(res))) { ## do *not* recompute
+ list2env(res, envir = environment())
+ } else { ## compute
+ stopifnot(is.finite(nc), nc > 0, length(nc) >= 100, k == as.integer(k), 0 <= k, k <= 20)
+ ncM <- mpfr(nc, precB)
+ stncM <- stirlerr(ncM) # the "true" values
+ stirlO.c <- sapply(k, function(k) stirlerr(nc, order = k))
+ relEc <- asNumeric(stirlO.c/stncM -1) # "true" relative error
+ ## log |rel.Err| -- "linear"
+ ## smooth() on log-scale {and transform back}:
+ splarelE <- apply(log(abs19(relEc)), 2, function(y) exp(smooth.spline(y, df=4)$y))
+ ## the direct formulas (default "R3", "MM2"):
+ arelEs0 <- abs(asNumeric(stirlerr_simpl(nc )/stncM - 1))
+ arelEs2 <- abs(asNumeric(stirlerr_simpl(nc, "MM2")/stncM - 1))
+ }
+ pch. <- c(1L:9L, 0L, letters)[k]
+ if(full)
+ matplotB(nc, abs19(relEc), col=k, pch = pch., cex=cex, ylim=ylim, log = "y",
+ xlab = quote(n), main = quote(abs(relErr(stirlerr(n, order==k)))))
+ else ## smooth only
+ matplotB(nc, splarelE, col=adjustcolor(k,2/3), pch=pch., lwd=2, cex=cex, ylim=ylim, log = "y",
+ xlab = quote(n), main = quote(abs(relErr(stirlerr(n, order==k)))))
+ mtext(paste("k =", deparse(k))) ; abline(h = 2^-(53:51), lty=3, lwd=1/2)
+ legend(legend.xy, legend=paste0("k=", k), bty="n", lwd=2, col=k, lty=1:5, pch = pch.)
+ abline(v = 5+(0:20)/10, lty=3, col=adjustcolor(10, 1/2))
+ drawCuts("R4.4_0", axis=3)
+ if(full) {
+ matlines(nc, splarelE, col=adjustcolor(k,2/3), lwd=4)
+ lines(nc, pmax(arelEs0, 1e-19), lwd=1.5, col=adjustcolor( 2, 0.2))
+ lines(nc, pmax(arelEs2, 1e-19), lwd=1, col=adjustcolor(10, 0.2))
+ }
+ lines(nc, smooth.spline(arelEs0, df=12)$y, lwd=3, col= adjustcolor( 2, 1/2))
+ lines(nc, smooth.spline(arelEs2, df=12)$y, lwd=3, col= adjustcolor(10, 1/2))
+ invisible(list(nc=nc, k=k, relEc = relEc, splarelE = splarelE, arelEs0=arelEs0, arelEs2=arelEs2))
+ }
>
> rr1 <- stirlerrPlot(nc = seq(4.75, 9.0, by=1/1024),
+ k = 7:20)
> stirlerrPlot(res = rr1, full=FALSE, ylim = c(8e-17, 1e-13))
> if(interactive())
+ stirlerrPlot(res = rr1)
>
> rr <- stirlerrPlot(nc = seq(5, 6.25, by=1/2048), k = 9:18)
> stirlerrPlot(res = rr, full=FALSE, ylim = c(8e-17, 1e-13))
>
> showProc.time()
Time (user system elapsed): 8.074 0.103 8.708
> <0c>
>
> palette("default")
>
> if(do.pdf) { dev.off(); pdf("stirlerr-tst_order_k-vs-k1.pdf") }
>
> ##' Find 'cuts', i.e., a region c(k) +/- s(k) i.e. intervals [c(k) - s(k), c(k) + s(k)]
> ##' where c(k) is such that relE(n=c(k), k) ~= eps)
>
> ##' 1. Find the c1(k) such that |relE(n, k)| ~= c1(k) * n^{-2k}
> findC1 <- function(n, ks, e1 = 1e-15, e2 = 1e-5, res=NULL, precBits = 1024, do.plot = TRUE, ...)
+ {
+ if(is.list(res) && all(c("n", "ks", "arelE") %in% names(res))) { ## do *not* recompute, take from 'res':
+ list2env(res, envir = environment())
+ } else { ## compute
+ stopifnot(require("Rmpfr"),
+ is.numeric(ks), ks == (k. <- as.integer(ks)), length(ks <- k.) >= 1,
+ length(e1) == 1L, length(e2) == 1L, is.finite(c(e1,e2)), e1 >= 0, e2 >= e1,
+ 0 <= ks, ks <= 20, is.numeric(n), n > 0, is.finite(n), length(n) >= 100)
+ nM <- mpfr(n, precBits)
+ stirM <- stirlerr(nM) # the "true" values
+ stirlOrd <- sapply(ks, function(k) stirlerr(n, order = k))
+ arelE <- abs(asNumeric(stirlOrd/stirM -1)) # "true" relative error
+ }
+ arelE19 <- pmax(arelE, 1e-19)
+ ## log |rel.Err| -- "linear"
+ ## on log-scale {and transform back; for linear fit, only use values inside [e1, e2]
+ if(do.plot) # experi
+ matplotB(n, arelE19, log="xy", ...)
+ ## matplot(n, arelE19, type="l", log="xy", xlim = c(min(n), 20))
+
+ ## re-compute these, as they *also* depend on (e1, e2)
+ c1 <- vapply(seq_along(ks), function(i) {
+ k <- ks[i]
+ y <- arelE19[,i]
+ iUse <- e1 <= y & y <= e2
+ if(sum(iUse) < 10) stop("only", sum(iUse), "values in [e1,e2]")
+ ## .lm.fit(cbind(1, log(n[iUse])), log(y[iUse]))$coefficients
+ ## rather, we *know* the error is c* n^{-2k} , i.e.,
+ ## log |relE| = log(c) - 2k * log(n)
+ ## <==> c = exp( log|relE| + 2k * log(n))
+ exp(mean(log(y[iUse]) + 2*k * log(n[iUse])))
+ }, numeric(1))
+ if(do.plot) {
+ drawEps.h()
+ for(i in seq_along(ks))
+ lines(n, c1[i] * n^(-2*ks[i]), col=adjustcolor(i, 1/3), lwd = 4, lty = 2)
+ }
+ invisible(list(n=n, ks=ks, arelE = arelE, c1 = c1))
+ } ## findC1()
>
> c1.Res <- findC1(n = 2^seq(2, 26, by=1/128), ks = 1:18)
> (s.c1.fil <- paste0("stirlerr-c1Res-", myPlatform(), ".rds"))
[1] "stirlerr-c1Res-R-d_87286_unix_DbnGNU_Ltrx_.rds"
> saveRDS(c1.Res, file = s.c1.fil)
>
>
> if(!exists("c1.Res")) {
+ c1.Res <- readRDS(s.c1.fil)
+ ## re-do the "default" plot of findC1():
+ findC1(res = c1.Res, xaxt="n"); eaxis(1, sub10=2)
+ }
>
> ## the same, zoomed in:
> findC1(res = c1.Res, xlim = c(4, 40), ylim = c(2e-17, 1e-12))
> ks <- c1.Res$ks; pch. <- c(1L:9L, 0L, letters)[ks]
> legend("left", legend=paste0("k=", ks), bty="n", lwd=2, col=ks, lty=1:5, pch = pch.)
>
> ## smaller set : larger e1 :
> c1.r2 <- findC1(res = c1.Res, xlim = c(4, 30), ylim = c(4e-17, 1e-13), e1 = 4e-15)
> legend("left", legend=paste0("k=", ks), bty="n", lwd=2, col=ks, lty=1:5, pch = pch.)
>
> print(digits = 4,
+ cbind(ks, c1. = c1.Res$c1, c1.2 = c1.r2$c1,
+ relD = round(relErrV(c1.r2$c1, c1.Res$c1), 4)))
ks c1. c1.2 relD
[1,] 1 3.326e-02 3.331e-02 -0.0017
[2,] 2 9.532e-03 9.511e-03 0.0022
[3,] 3 7.042e-03 7.049e-03 -0.0009
[4,] 4 9.845e-03 9.811e-03 0.0035
[5,] 5 2.172e-02 2.169e-02 0.0015
[6,] 6 7.028e-02 6.967e-02 0.0088
[7,] 7 3.061e-01 3.041e-01 0.0066
[8,] 8 1.771e+00 1.744e+00 0.0156
[9,] 9 1.275e+01 1.259e+01 0.0128
[10,] 10 1.156e+02 1.129e+02 0.0235
[11,] 11 1.242e+03 1.224e+03 0.0152
[12,] 12 1.638e+04 1.592e+04 0.0294
[13,] 13 2.466e+05 2.427e+05 0.0162
[14,] 14 4.434e+06 4.331e+06 0.0238
[15,] 15 8.994e+07 8.846e+07 0.0167
[16,] 16 2.134e+09 2.082e+09 0.0253
[17,] 17 5.608e+10 5.515e+10 0.0169
[18,] 18 1.704e+12 1.655e+12 0.0292
>
> c1.. <- c1.r2[c("ks", "c1")] # just the smallest part is needed here:
> (s.c1.fil <- paste0("stirlerr-c1r2-", myPlatform(), ".rds"))
[1] "stirlerr-c1r2-R-d_87286_unix_DbnGNU_Ltrx_.rds"
> saveRDS(c1.., file = s.c1.fil) # was "stirlerr-c1.rds"
>
> ## 2. Now, find the n(k) +/- se.n(k) intervals
> ## Use these c1 from above
>
> ##' Given c1-results relErr(n,k); |relE(n,k) ~= c1 * n^{-2k} , find n such that
> ##' |relE(n,k)| line {in log-log scale} cuts y = eps, i.e., n* such that |relE(n,k)| <= eps for all n >= n*
> n.ep <- function(eps, c1Res, ks = c1Res$ks, c1 = c1Res$c1, ...) {
+ stopifnot(is.finite(eps), length(eps) == 1L, eps > 0,
+ length(ks) == length(c1), is.numeric(c1), is.integer(ks), ks >= 1)
+ ## n: given k, the location where the |relE(n,k)| line {in log-log} cuts y = eps
+ ## |relE(n,k) ~= c1 * n^{-2k} <==>
+ ## log|relE(n,k)| ~= log(c1) - 2k* log(n) <==>
+ ## c := mean{ exp( log|relE(n,k)| + 2k* log(n) ) } ------- see findC1()
+ ## now, solve for n :
+ ## c1 * n^{-2k} == eps
+ ## log(c1) - 2k* log(n) == log(eps)
+ ## log(n) == (log(eps) - log(c1)) / (-2k) <==>
+ ## n == exp((log(c1) - log(eps)) / 2k)
+ exp((log(c1) - log(eps))/(2*ks))
+ }
>
> ## get c1..
> if(!exists("c1..")) c1.. <- readRDS("stirlerr-c1.rds")
>
> ne2 <- n.ep(2^-51, c1Res = c1..) ## ok
> ne1 <- n.ep(2^-52, c1Res = c1..)
> ne. <- n.ep(2^-53, c1Res = c1..)
>
> form <- function(n) format(signif(n, 3), scientific=FALSE)
> data.frame(k = ks, ne2 = form(ne2), ne1 = form(ne1), ne. = form(ne.),
+ cutoffs = form(rev(.stirl.cutoffs("R4.4_0")[-1])))
k ne2 ne1 ne. cutoffs
1 1 8660000.00 12200000.00 17300000.00 17400000.00
2 2 2150.00 2560.00 3040.00 3700.00
3 3 159.00 178.00 200.00 200.00
4 4 46.60 50.80 55.40 81.00
5 5 23.40 25.10 26.90 36.00
6 6 15.20 16.10 17.10 25.00
7 7 11.50 12.10 12.70 19.00
8 8 9.43 9.85 10.30 14.00
9 9 8.20 8.53 8.86 11.00
10 10 7.42 7.68 7.95 9.50
11 11 6.89 7.11 7.34 8.80
12 12 6.53 6.72 6.92 8.25
13 13 6.27 6.44 6.62 7.60
14 14 6.10 6.25 6.41 7.10
15 15 5.98 6.12 6.26 6.50
16 16 5.90 6.03 6.16 6.50
17 17 5.85 5.97 6.10 6.50
18 18 5.83 5.95 6.06 6.50
>
> ## ------- Linux F 36/38 x86_64 (nb-mm5|v-lynne)
> ## k ne2 ne1 ne. cutoffs
> ## 1 8660000.00 12200000.00 17300000.00 17400000.00
> ## 2 2150.00 2560.00 3040.00 3700.00
> ## 3 159.00 178.00 200.00 200.00
> ## 4 46.60 50.80 55.40 81.00
> ## 5 23.40 25.10 26.90 36.00
> ## 6 15.20 16.10 17.10 25.00
> ## 7 11.50 12.10 12.70 19.00
> ## 8 9.43 9.85 10.30 14.00
> ## 9 8.20 8.53 8.86 11.00
> ## 10 7.42 7.68 7.95 9.50
> ## 11 6.89 7.11 7.34 8.80
> ## 12 6.53 6.72 6.92 8.25
> ## 13 6.27 6.44 6.62 7.60
> ## 14 6.10 6.25 6.41 7.10
> ## 15 5.98 6.12 6.26 6.50 * (not used)
> ## 16 5.90 6.03 6.16 6.50 * " "
> ## 17 5.85 5.97 6.10 6.50 * " "
> ## 18 5.83 5.95 6.06 6.50 << used all the way down to 5.25
>
> ## ok --- correct order of magnitude ! --- good!
>
>
> ## 2b. find *interval* around the 'n(eps)' values
>
> ## -- Try simply
> d.k <- ne. - ne1
> ## interval
> int.k <- cbind(ne1 - d.k,
+ ne1 + d.k)
> ## look at e.g.
> data.frame(k=ks, `n(k)` = form(ne1), int = form(int.k))
k n.k. int.1 int.2
1 1 12200000.00 7180000.00 17300000.00
2 2 2560.00 2070.00 3040.00
3 3 178.00 156.00 200.00
4 4 50.80 46.20 55.40
5 5 25.10 23.30 26.90
6 6 16.10 15.20 17.10
7 7 12.10 11.40 12.70
8 8 9.85 9.41 10.30
9 9 8.53 8.19 8.86
10 10 7.68 7.41 7.95
11 11 7.11 6.88 7.34
12 12 6.72 6.52 6.92
13 13 6.44 6.27 6.62
14 14 6.25 6.10 6.41
15 15 6.12 5.98 6.26
16 16 6.03 5.90 6.16
17 17 5.97 5.85 6.10
18 18 5.95 5.83 6.06
> ## k n.k. int.1 int.2
> ## 1 12200000.00 7180000.00 17300000.00
> ## 2 2560.00 2070.00 3040.00
> ## 3 178.00 156.00 200.00
> ## 4 50.80 46.20 55.40
> ## 5 25.10 23.30 26.90
> ## 6 16.10 15.20 17.10
> ## 7 12.10 11.40 12.70
> ## 8 9.85 9.41 10.30
> ## 9 8.53 8.19 8.86
> ## 10 7.68 7.41 7.95
> ## 11 7.11 6.88 7.34
> ## 12 6.72 6.52 6.92
> ## 13 6.44 6.27 6.62
> ## 14 6.25 6.10 6.41
> ## 15 6.12 5.98 6.26
> ## 16 6.03 5.90 6.16
> ## 17 5.97 5.85 6.10
> ## 18 5.95 5.83 6.06
>
> ##' as function {well, *not* computing c1.k from scratch
> nInt <- function(k, c1.k, ep12 = 2^-(52:53)) {
+ if(length(k) == 1L) { # special convention to call for *one* k, with c1.k vector
+ stopifnot(k == (k <- as.integer(k)), k >= 1, length(c1.k) >= k)
+ c1.k <- c1.k[k]
+ }
+
+ ## see n.ep() above
+ n_ <- function(eps, k, c1) {
+ stopifnot(is.finite(eps), length(eps) == 1L, eps > 0,
+ length(k) == length(c1), is.numeric(c1), is.integer(k), k >= 1)
+ exp((log(c1) - log(eps))/(2*k))
+ }
+
+ ne1 <- n_(ep12[1], k, c1.k)
+ ne. <- n_(ep12[2], k, c1.k)
+ d.k <- ne. - ne1
+ stopifnot(d.k > 0)
+ ## interval: {"fudge" 0.5 / 2.5} from results -- also 'noLdbl' gives *quite* different pic!:
+ odd <- k %% 2 == 1
+ cbind(ne1 - ifelse( odd & noLdbl, 1.5, 0.5) * d.k,
+ ne1 + ifelse(!odd , 8, 2.5) * d.k)
+ }
>
> nInt(k= 1, c1..$c1)
[,1] [,2]
[1,] 9711838 24932455
> nInt(k= 2, c1..$c1)
[,1] [,2]
[1,] 2316.237 6430.581
> nInt(k=18, c1..$c1)
[,1] [,2]
[1,] 5.888326 6.870896
>
> nints.k <- nInt(ks, c1..$c1)
> ## for printing
> form(as.data.frame( nints.k ))
V1 V2
1 9710000.00 24900000.00
2 2320.00 6430.00
3 167.00 232.00
4 48.50 87.50
5 24.20 29.60
6 15.70 23.80
7 11.70 13.60
8 9.63 13.30
9 8.36 9.36
10 7.54 9.85
11 7.00 7.68
12 6.62 8.29
13 6.36 6.88
14 6.17 7.51
15 6.05 6.48
16 5.96 7.09
17 5.91 6.28
18 5.89 6.87
>
> ## (-.5 , +2.5) ## originally ( -1, +1)
> ## 1 9710000.00 24900000.00 # 7180000.00 17300000.00
> ## 2 2320.00 3770.00 # 2070.00 3040.00
> ## 3 167.00 232.00 # 156.00 200.00
> ## 4 48.50 62.30 # 46.20 55.40
> ## 5 24.20 29.60 # 23.30 26.90
> ## 6 15.70 18.50 # 15.20 17.10
> ## 7 11.70 13.60 # 11.40 12.70
> ## 8 9.63 10.90 # 9.41 10.30
> ## 9 8.36 9.36 # 8.19 8.86
> ## 10 7.54 8.36 # 7.41 7.95
> ## 11 7.00 7.68 # 6.88 7.34
> ## 12 6.62 7.21 # 6.52 6.92
> ## 13 6.36 6.88 # 6.27 6.62
> ## 14 6.17 6.64 # 6.10 6.41
> ## 15 6.05 6.48 # 5.98 6.26
> ## 16 5.96 6.36 # 5.90 6.16
> ## 17 5.91 6.28 # 5.85 6.10
> ## 18 5.89 6.24 # 5.83 6.06
>
> ## 3. Then for each of the intervals, compare order k vs k+1
> ## -- ==> optimal cutoff { how much platform dependency ?? }
>
> ### Here, compute only
> find1cuts <- function(k, c1,
+ n = nInt(k, c1), # the *set* of n's or the 'range'
+ len.n = 1000,
+ precBits = 1024, nM = mpfr(n, precBits),
+ stnM = stirlerr(nM),
+ stirlOrd = sapply(k+(0:1), function(.k.) stirlerr(n, order = .k.)),
+ relE = asNumeric(stirlOrd/stnM -1), # "true" relative error for the {k, k+1}
+ do.spl=TRUE, df.spline = 9, # df = 5 gives 2 cutpoints for k==1
+ do.low=TRUE, f.lowess = 0.2,
+ do.cobs = require("cobs", quietly=TRUE), tau = 0.90)
+ {
+ ## check relErrV( stirlerr(n, order=k ) vs
+ ## stirlerr(n, order=k+1)
+ if(length(n) == 2L)
+ if(n[1] < n[2]) n <- seq(n[1], n[2], length.out = len.n)
+ else stop("'n' must be *increasing")
+ force(relE)
+ y <- abs19(relE)
+ ## NB: all smoothing --- as in stirlerrPlot() above -- should happen in log-space
+ ## (log(abs19(relEc)), 2, function(y) exp(smooth.spline(y, df=4)$y))
+ ly <- log(y) # == log(abs19(relE)) == log(max(|r|, 1e-19))
+ if(do.spl) {##
+ s1 <- exp(smooth.spline(ly[,1], df=df.spline)$y)
+ s2 <- exp(smooth.spline(ly[,2], df=df.spline)$y)
+ }
+ if(do.low) { ## lowess
+ s1l <- exp(lowess(ly[,1], f=f.lowess)$y)
+ s2l <- exp(lowess(ly[,2], f=f.lowess)$y)
+ }
+ ## also use cobs() splines for the 90% quantile !!
+ EE <- environment() # so can set do.cobs to FALSE in case of error
+ if(do.cobs) { ## <==> require("cobs") # yes, this is in tests/
+ cobsF <- function(Y) cobs(n, Y, tau=tau, nknots = 6, lambda = -1,
+ print.warn=FALSE, print.mesg=FALSE)
+ ## sparseM::chol(<matrix.csr>) now gives error when it gave warning {about singularity}
+ cobsF <- function(Y) {
+ r <- tryCatch(cobs(n, Y, tau=tau, nknots = 6, lambda = -1,
+ print.warn=FALSE, print.mesg=FALSE),
+ error = identity)
+ if(inherits(r, "error")) {
+ assign("do.cobs", FALSE, envir = EE) # and return
+ list(fitted = FALSE)
+ }
+ else
+ r
+ }
+ cs1 <- exp(cobsF(ly[,1])$fitted)
+ cs2 <- exp(cobsF(ly[,2])$fitted)
+ }
+ smooths <- list(spl = if(do.spl ) cbind(s1, s2 ),
+ low = if(do.low ) cbind(s1l,s2l),
+ cobs= if(do.cobs) cbind(cs1,cs2))
+ ## diffL <- list(spl = if(do.spl ) s2 -s1,
+ ## low = if(do.low ) s2l-s1l,
+ ## cobs= if(do.cobs) cs2-cs1)
+ ### FIXME: simplification does not always *work* -- (i, n.) are not always ok
+ ### ------ notably within R-devel-no-ldouble {hence probably macOS M1 ... ..}
+ sapply(smooths, function(s12) { d <- s12[,2] - s12[,1]
+ ## typically a (almost or completely) montone increasing function, crossing zero *once*
+ ## compute cutpoint:
+ ## i := the first n[i] with d(n[i]) >= 0
+ i <- which(d >= 0)[1]
+ if(length(i) == 1L && !is.na(i) && i > 1L) {
+ i_ <- i - 1L # ==> d(n[i_]) < 0
+ ## cutpoint must be in [n[i_], n[i]] --- do linear interpolation
+ n. <- n[i_] - (n[i] - n[i_])* d[i_] / (d[i] - d[i_])
+ } else {
+ if(length(i) != 1L) i <- -length(i)
+ n. <- NA_integer_
+ }
+ c(i=i, n.=n.)
+ }) -> n.L
+
+ list(k=k, n=n, relE = unname(relE), smooths=smooths, i.n = n.L)
+ } ## find1cuts()
>
> k. <- 1:15
> system.time(
+ ## Failed on lynne [2024-06-04, R 4.4.1 beta] with
+ ## Error in .local(x, ...) : insufficient space ---> SparseM :: chol(<..>)
+ ## ==> now we catch this inside find1cuts():
+ resL <- lapply(setNames(,k.), function(k) find1cuts(k=k, c1=c1..$c1))
+ ) ## -- warnings, notably from cobs() not converging
user system elapsed
24.641 0.015 29.172
There were 34 warnings (use warnings() to see them)
> ## needs 12 sec (!!) user system elapsed =
> (s.find15.fil <- paste0("stirlerr-find1_1-15_", myPlatform(), ".rds"))
[1] "stirlerr-find1_1-15_R-d_87286_unix_DbnGNU_Ltrx_.rds"
> ## now we catch cobs() errors {from SparseM::chol},
> ## okCuts <- !inherits(resL, "error")
> ## if(okCuts) {
> saveRDS(resL, file = s.find15.fil) # was "stirlerr-find1_1-15.rds"
> ## } else traceback()
> ## 11: stop(mess)
> ## 10: .local(x, ...)
> ## 9: chol(e, tmpmax = tmpmax, nsubmax = nsubmax, nnzlmax = nnzlmax)
> ## 8: chol(e, tmpmax = tmpmax, nsubmax = nsubmax, nnzlmax = nnzlmax)
> ## 7: rq.fit.sfnc(Xeq, Yeq, Xieq, Yieq, tau = tau, rhs = rhs, control = rqCtrl)
> ## 6: drqssbc2(x, y, w, pw = pw, knots = knots, degree = degree, Tlambda = if (select.lambda) lambdaSet else lambda,
> ## constraint = constraint, ptConstr = ptConstr, maxiter = maxiter,
> ## trace = trace - 1, nrq, nl1, neqc, niqc, nvar, tau = tau,
> ## select.lambda = select.lambda, give.pseudo.x = keep.x.ps,
> ## rq.tol = rq.tol, tol.0res = tol.0res, print.warn = print.warn)
> ## 5: cobs(n, Y, tau = tau, nknots = 6, lambda = -1, print.warn = FALSE,
> ## print.mesg = FALSE) at stirlerr-tst.R!udBzpT#32
> ## 4: cobsF(ly[, 1]) at stirlerr-tst.R!udBzpT#34
> ## 3: find1cuts(k = k, c1 = c1..$c1) at #1
> ## 2: FUN(X[[i]], ...)
> ## 1: lapply(setNames(, k.), function(k) find1cuts(k = k, c1 = c1..$c1))
>
> ok1cutsLst <- function(res) {
+ stopifnot(is.list(res), sapply(res, is.list)) # must be list of lists
+ cobsL <- lapply(lapply(res, `[[`, "smooths"), `[[`, "cobs")
+ vapply(cobsL, is.array, NA)
+ }
> (resLok <- ok1cutsLst(resL))
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
> ## 1 2 3 4 5 6 ...... 15
> ## FALSE TRUE TRUE TRUE TRUE TRUE ...... TRUE
>
>
> if(FALSE) {
+ ## e.g. in R-devel-no-ldouble:
+ (r1 <- find1cuts(k=1, c1=c1..$c1))$i.n # list --- no longer ok {SparseM::chol -> insufficient space}
+ (r2 <- find1cuts(k=2, c1=c1..$c1))$i.n # "good"
+ (r3 <- find1cuts(k=3, c1=c1..$c1))$i.n # 3-vector: only 3x 'i' == 1
+ }
>
> ## if(okCuts) {
> mult.fig(15, main = "stirlerr(n, order=k) vs order = k+1")$old.par -> opar
> invisible(lapply(resL[resLok], plot1cuts))
> ## plus the "last" ones {also showing that k=15 is worse here anyway than k=17}
> str(r17 <- find1cuts(k=17, n = seq(5.1, 6.5, length.out = 1500), c1=c1..$c1))
List of 5
$ k : num 17
$ n : num [1:1500] 5.1 5.1 5.1 5.1 5.1 ...
$ relE : num [1:1500, 1:2] 5.31e-14 5.29e-14 5.25e-14 5.21e-14 5.19e-14 ...
$ smooths:List of 3
..$ spl : num [1:1500, 1:2] 5.29e-14 5.26e-14 5.23e-14 5.19e-14 5.16e-14 ...
.. ..- attr(*, "dimnames")=List of 2
.. .. ..$ : NULL
.. .. ..$ : chr [1:2] "s1" "s2"
..$ low : num [1:1500, 1:2] 5.29e-14 5.26e-14 5.23e-14 5.20e-14 5.17e-14 ...
.. ..- attr(*, "dimnames")=List of 2
.. .. ..$ : NULL
.. .. ..$ : chr [1:2] "s1l" "s2l"
..$ cobs: NULL
$ i.n : int [1:2, 1:3] 1 NA 1 NA NA NA
..- attr(*, "dimnames")=List of 2
.. ..$ : chr [1:2] "i" "n."
.. ..$ : chr [1:3] "spl" "low" "cobs"
Warning messages:
1: In min(sol1["k", i.keep]) :
no non-missing arguments to min; returning Inf
2: In min(sol1["k", i.keep]) :
no non-missing arguments to min; returning Inf
> plot1cuts(r17) # no-ldouble is *very* different than normal: k *much better* than k+1
Error in stopifnot(is.list(smooths), (nS <- length(smooths)) >= 1L, sapply(smooths, :
'list' object cannot be coerced to type 'integer'
Calls: plot1cuts -> stopifnot -> eval -> eval -> stopifnot
Execution halted
Flavor: r-devel-linux-x86_64-debian-clang