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Type 'q()' to quit R. > #### 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): 0.05 0 0.04 > (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 1.83 0.00 1.83 > 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: 3.697822e-14" > all.equal(roundMpfr(st.nM, 64), as(st.n,"mpfr"), tolerance=1e-16)# (ditto) [1] "Mean relative difference: 3.697822e-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): 3.25 0.05 3.3 > > ## Is *equivalent* to 'new' stirlerr_simpl(n version = *) [not for 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(, *) : > 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 6.061346e-17 6.061718e-17 6.114046e-17 9.701665e-06 MM2 6.061346e-17 6.061718e-17 6.114046e-17 9.701665e-06 n0 9.701700e-06 9.701700e-06 9.701700e-06 6.046747e-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): 5.18 0.03 5.22 > > ## =========== "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): 0.47 0 0.47 > > > ### 2. relErr( stirlerr(.) ) ============================================================ > > ##' Very revealing plot showing the *relative* approximation error of stirlerr() > ##' > 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): 0.02 0 0.01 > > 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): 1.64 0 1.64 > > > 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 [232,] 2.334381e-03 2.12620358 -7.032e-17 [233,] 2.343204e-03 2.12436569 7.023e-17 [234,] 2.352061e-03 2.12252794 6.216e-18 [235,] 2.360951e-03 2.12069035 -7.983e-17 [236,] 2.369874e-03 2.11885291 4.616e-17 [237,] 2.378832e-03 2.11701562 2.356e-16 [238,] 2.387823e-03 2.11517849 1.543e-16 [239,] 2.396848e-03 2.11334150 2.587e-16 [240,] 2.405907e-03 2.11150467 -9.167e-17 [241,] 2.415001e-03 2.10966798 -4.971e-17 [242,] 2.424129e-03 2.10783146 -4.516e-17 [243,] 2.433292e-03 2.10599508 1.246e-16 [244,] 2.442489e-03 2.10415886 2.32e-17 [245,] 2.451720e-03 2.10232280 -2.199e-16 [246,] 2.460987e-03 2.10048688 7.847e-17 [247,] 2.470289e-03 2.09865113 2.089e-16 [248,] 2.479626e-03 2.09681553 2.274e-16 [249,] 2.488998e-03 2.09498009 4.04e-17 [250,] 2.498406e-03 2.09314480 -2.338e-16 [251,] 2.507849e-03 2.09130967 -5.553e-17 [252,] 2.517328e-03 2.08947470 -4.654e-18 [253,] 2.526843e-03 2.08763989 1.603e-16 [254,] 2.536393e-03 2.08580523 -1.976e-16 [255,] 2.545980e-03 2.08397074 2.586e-16 [256,] 2.555603e-03 2.08213640 9.756e-17 [257,] 2.565263e-03 2.08030223 1.591e-16 [258,] 2.574958e-03 2.07846821 2.651e-16 [259,] 2.584691e-03 2.07663436 9.114e-17 [260,] 2.594460e-03 2.07480066 1.145e-16 [261,] 2.604267e-03 2.07296713 -6.184e-17 [262,] 2.614110e-03 2.07113377 -1.977e-16 [263,] 2.623990e-03 2.06930056 2.149e-17 [264,] 2.633908e-03 2.06746752 1.831e-16 [265,] 2.643864e-03 2.06563464 -2.117e-17 [266,] 2.653857e-03 2.06380193 -2.172e-16 [267,] 2.663887e-03 2.06196938 -4.925e-17 [268,] 2.673956e-03 2.06013699 -1.836e-16 [269,] 2.684063e-03 2.05830478 -1.418e-16 [270,] 2.694208e-03 2.05647272 -1.394e-16 [271,] 2.704391e-03 2.05464084 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2.402737e-02 1.04555763 -2.705e-16 [851,] 2.411818e-02 1.04396090 8.742e-17 [852,] 2.420934e-02 1.04236493 3.099e-17 [853,] 2.430085e-02 1.04076973 -2.045e-17 [854,] 2.439270e-02 1.03917530 -4.708e-16 [855,] 2.448489e-02 1.03758164 -1.862e-16 [856,] 2.457744e-02 1.03598875 1.829e-17 [857,] 2.467033e-02 1.03439663 -2.73e-16 [858,] 2.476358e-02 1.03280528 1.807e-16 [859,] 2.485718e-02 1.03121471 -2.97e-16 [860,] 2.495113e-02 1.02962491 5.071e-17 [861,] 2.504544e-02 1.02803590 -4.632e-17 [862,] 2.514010e-02 1.02644766 6.223e-18 [863,] 2.523513e-02 1.02486021 -1.205e-16 [864,] 2.533051e-02 1.02327354 6.648e-17 [865,] 2.542625e-02 1.02168766 -1.26e-16 [866,] 2.552235e-02 1.02010257 -1.408e-16 [867,] 2.561882e-02 1.01851826 -2.085e-16 [868,] 2.571565e-02 1.01693475 -6.021e-17 [869,] 2.581285e-02 1.01535202 2.078e-16 [870,] 2.591041e-02 1.01377010 -5.141e-17 [871,] 2.600834e-02 1.01218897 -2.631e-16 [872,] 2.610665e-02 1.01060863 -4.697e-17 [873,] 2.620532e-02 1.00902910 -9.088e-17 [874,] 2.630437e-02 1.00745036 -1.693e-16 [875,] 2.640379e-02 1.00587243 -2.563e-16 [876,] 2.650359e-02 1.00429531 -2.685e-16 [877,] 2.660377e-02 1.00271899 -2.211e-16 [878,] 2.670432e-02 1.00114348 -2.12e-17 [879,] 2.680526e-02 0.99956878 5.782e-17 [880,] 2.690657e-02 0.99799489 1.259e-16 [881,] 2.700827e-02 0.99642182 5.866e-17 [882,] 2.711035e-02 0.99484955 5.209e-17 [883,] 2.721282e-02 0.99327811 -1.323e-16 [884,] 2.731568e-02 0.99170749 4.052e-17 [885,] 2.741892e-02 0.99013768 -3.958e-16 [886,] 2.752256e-02 0.98856870 -5.005e-17 [887,] 2.762658e-02 0.98700054 -2.743e-16 [888,] 2.773100e-02 0.98543321 3.239e-17 [889,] 2.783582e-02 0.98386670 -1.531e-16 [890,] 2.794103e-02 0.98230103 -2.255e-16 [891,] 2.804664e-02 0.98073618 6.722e-17 [892,] 2.815265e-02 0.97917217 -1.908e-16 [893,] 2.825905e-02 0.97760899 9.306e-17 [894,] 2.836587e-02 0.97604665 -1.783e-16 [895,] 2.847308e-02 0.97448515 -1.001e-16 [896,] 2.858070e-02 0.97292449 2.365e-16 [897,] 2.868873e-02 0.97136466 -6.861e-17 [898,] 2.879716e-02 0.96980569 1.277e-16 [899,] 2.890600e-02 0.96824755 -3.515e-16 [900,] 2.901526e-02 0.96669027 -2.804e-16 [901,] 2.912493e-02 0.96513383 -1.569e-16 [902,] 2.923501e-02 0.96357824 -1.36e-16 [903,] 2.934551e-02 0.96202351 6.619e-17 [904,] 2.945643e-02 0.96046963 -2.519e-18 [905,] 2.956776e-02 0.95891660 1.127e-16 [906,] 2.967952e-02 0.95736443 -3.469e-17 [907,] 2.979170e-02 0.95581313 -1.893e-17 [908,] 2.990430e-02 0.95426268 -2.731e-16 [909,] 3.001733e-02 0.95271310 -4.179e-16 [910,] 3.013079e-02 0.95116438 -4.116e-16 [911,] 3.024468e-02 0.94961653 -1.603e-16 [912,] 3.035899e-02 0.94806954 6.2e-17 [913,] 3.047374e-02 0.94652343 2.81e-16 [914,] 3.058892e-02 0.94497819 -3.512e-16 [915,] 3.070454e-02 0.94343382 -4.052e-17 [916,] 3.082059e-02 0.94189033 1.171e-16 [917,] 3.093708e-02 0.94034771 -1.07e-16 [918,] 3.105402e-02 0.93880598 2.727e-16 [919,] 3.117139e-02 0.93726513 1.25e-17 [920,] 3.128921e-02 0.93572515 -3.771e-16 [921,] 3.140747e-02 0.93418607 -1.724e-16 [922,] 3.152618e-02 0.93264787 1.627e-17 [923,] 3.164534e-02 0.93111056 5.401e-17 [924,] 3.176495e-02 0.92957414 -1.84e-18 [925,] 3.188501e-02 0.92803861 -2.467e-16 [926,] 3.200553e-02 0.92650397 5.867e-17 [927,] 3.212650e-02 0.92497023 -2.504e-16 [928,] 3.224793e-02 0.92343739 1.557e-16 [929,] 3.236982e-02 0.92190545 -2.768e-16 [930,] 3.249216e-02 0.92037441 1.197e-16 [931,] 3.261497e-02 0.91884427 6.503e-17 [932,] 3.273825e-02 0.91731504 -3.013e-17 [933,] 3.286199e-02 0.91578671 -1.18e-16 [934,] 3.298620e-02 0.91425929 -1.666e-16 [935,] 3.311087e-02 0.91273278 1.009e-16 [936,] 3.323602e-02 0.91120719 -1.158e-16 [937,] 3.336165e-02 0.90968251 6.961e-17 [938,] 3.348774e-02 0.90815874 -2.289e-16 [939,] 3.361432e-02 0.90663589 7.742e-17 [940,] 3.374137e-02 0.90511397 9.563e-17 [941,] 3.386890e-02 0.90359296 1.625e-17 [942,] 3.399691e-02 0.90207288 3.797e-17 [943,] 3.412541e-02 0.90055372 -2.957e-16 [944,] 3.425439e-02 0.89903549 -2.451e-16 [945,] 3.438387e-02 0.89751819 1.066e-16 [946,] 3.451383e-02 0.89600181 2.338e-16 [947,] 3.464428e-02 0.89448638 -8.598e-17 [948,] 3.477522e-02 0.89297187 -3.501e-16 [949,] 3.490666e-02 0.89145830 8.708e-17 [950,] 3.503860e-02 0.88994567 -1.509e-16 [951,] 3.517103e-02 0.88843398 1.641e-16 [952,] 3.530397e-02 0.88692324 -2.553e-16 [953,] 3.543741e-02 0.88541343 -3.422e-16 [954,] 3.557135e-02 0.88390457 -2.188e-16 [955,] 3.570580e-02 0.88239666 4.565e-16 [956,] 3.584076e-02 0.88088970 1.196e-17 [957,] 3.597622e-02 0.87938369 -6.376e-17 [958,] 3.611220e-02 0.87787864 -9.577e-17 [959,] 3.624870e-02 0.87637453 -1.308e-16 [960,] 3.638570e-02 0.87487139 -3.052e-16 [961,] 3.652323e-02 0.87336920 4.848e-17 [962,] 3.666128e-02 0.87186798 -1.073e-17 [963,] 3.679985e-02 0.87036772 -3.137e-16 [964,] 3.693894e-02 0.86886842 -2.551e-16 [965,] 3.707856e-02 0.86737008 3.083e-16 [966,] 3.721870e-02 0.86587272 -1.701e-16 [967,] 3.735938e-02 0.86437633 3.334e-16 [968,] 3.750058e-02 0.86288091 -2.782e-16 [969,] 3.764232e-02 0.86138646 4.781e-17 [970,] 3.778460e-02 0.85989299 -2.843e-16 [971,] 3.792742e-02 0.85840049 -2.075e-16 [972,] 3.807077e-02 0.85690898 -1.42e-16 [973,] 3.821466e-02 0.85541844 -4.378e-16 [974,] 3.835910e-02 0.85392889 -1.974e-16 [975,] 3.850409e-02 0.85244033 5.356e-17 [976,] 3.864962e-02 0.85095275 -3.243e-16 [977,] 3.879571e-02 0.84946616 -3.412e-16 [978,] 3.894234e-02 0.84798056 1.557e-16 [979,] 3.908953e-02 0.84649596 1.216e-16 [980,] 3.923728e-02 0.84501235 -8.871e-17 [981,] 3.938558e-02 0.84352973 5.891e-17 [982,] 3.953445e-02 0.84204812 3.148e-16 [983,] 3.968388e-02 0.84056751 2.21e-17 [984,] 3.983387e-02 0.83908789 -4.926e-17 [985,] 3.998443e-02 0.83760928 -1.519e-16 [986,] 4.013556e-02 0.83613168 -1.871e-16 [987,] 4.028726e-02 0.83465509 -2.027e-17 [988,] 4.043953e-02 0.83317951 1.291e-16 [989,] 4.059238e-02 0.83170493 1.072e-16 [990,] 4.074581e-02 0.83023138 -1.685e-16 [991,] 4.089982e-02 0.82875883 -9.306e-17 [992,] 4.105440e-02 0.82728731 6.137e-17 [993,] 4.120958e-02 0.82581680 -2.687e-17 [994,] 4.136534e-02 0.82434732 -1.502e-16 [995,] 4.152169e-02 0.82287886 7.17e-17 [996,] 4.167862e-02 0.82141142 1.373e-16 [997,] 4.183616e-02 0.81994502 -2.858e-16 [998,] 4.199428e-02 0.81847964 -1.54e-16 [999,] 4.215301e-02 0.81701529 -3.8e-17 [1000,] 4.231234e-02 0.81555198 -6.678e-17 [1001,] 4.247226e-02 0.81408970 2.762e-16 [1002,] 4.263280e-02 0.81262845 4.856e-17 [1003,] 4.279393e-02 0.81116825 4.078e-17 [1004,] 4.295568e-02 0.80970909 -1.661e-16 [1005,] 4.311804e-02 0.80825096 2.196e-17 [1006,] 4.328101e-02 0.80679389 1.227e-16 [1007,] 4.344460e-02 0.80533786 8.701e-17 [1008,] 4.360881e-02 0.80388288 -1.174e-16 [1009,] 4.377364e-02 0.80242895 9.071e-17 [1010,] 4.393909e-02 0.80097607 1.286e-16 [1011,] 4.410517e-02 0.79952424 2.322e-16 [1012,] 4.427187e-02 0.79807347 9.065e-18 [1013,] 4.443920e-02 0.79662376 -5.117e-17 [1014,] 4.460717e-02 0.79517511 -4.1e-16 [1015,] 4.477577e-02 0.79372752 -5.002e-16 [1016,] 4.494501e-02 0.79228100 -4.408e-16 [1017,] 4.511489e-02 0.79083554 3.923e-16 [1018,] 4.528541e-02 0.78939114 1.281e-17 [1019,] 4.545657e-02 0.78794782 4.667e-17 [1020,] 4.562839e-02 0.78650557 -2.425e-16 [1021,] 4.580085e-02 0.78506439 4.724e-16 [1022,] 4.597396e-02 0.78362429 -1.708e-17 [1023,] 4.614773e-02 0.78218527 -8.247e-17 [1024,] 4.632215e-02 0.78074732 -7.897e-17 [1025,] 4.649724e-02 0.77931046 -9.085e-18 [1026,] 4.667298e-02 0.77787468 1.327e-16 [1027,] 4.684939e-02 0.77643998 -7.516e-17 [1028,] 4.702647e-02 0.77500637 -1.455e-16 [1029,] 4.720421e-02 0.77357385 -2.664e-16 [1030,] 4.738263e-02 0.77214242 3.007e-16 [1031,] 4.756172e-02 0.77071208 -3.419e-17 [1032,] 4.774149e-02 0.76928284 -6.811e-17 [1033,] 4.792194e-02 0.76785470 1.776e-16 [1034,] 4.810307e-02 0.76642765 -6.59e-17 [1035,] 4.828488e-02 0.76500171 1.191e-16 [1036,] 4.846739e-02 0.76357686 -3.237e-16 [1037,] 4.865058e-02 0.76215312 -2.161e-16 [1038,] 4.883446e-02 0.76073049 2.318e-16 [1039,] 4.901904e-02 0.75930897 -1.716e-16 [1040,] 4.920432e-02 0.75788855 -4.697e-16 [1041,] 4.939030e-02 0.75646925 -2.228e-16 [1042,] 4.957698e-02 0.75505106 1.406e-16 [1043,] 4.976436e-02 0.75363399 1.585e-17 [1044,] 4.995245e-02 0.75221804 -6.918e-19 [1045,] 5.014126e-02 0.75080320 -1.627e-16 [1046,] 5.033078e-02 0.74938949 -1.851e-16 [1047,] 5.052101e-02 0.74797691 2.93e-17 [1048,] 5.071197e-02 0.74656544 1.236e-16 [1049,] 5.090364e-02 0.74515511 1.939e-16 [1050,] 5.109604e-02 0.74374590 1.912e-16 [1051,] 5.128917e-02 0.74233783 4.982e-17 [1052,] 5.148303e-02 0.74093089 8.152e-17 [1053,] 5.167762e-02 0.73952509 9.513e-17 [1054,] 5.187294e-02 0.73812042 3.885e-16 [1055,] 5.206901e-02 0.73671689 -3.598e-16 [1056,] 5.226581e-02 0.73531451 -2.818e-17 [1057,] 5.246336e-02 0.73391326 -7.462e-17 [1058,] 5.266166e-02 0.73251317 -8.524e-17 [1059,] 5.286070e-02 0.73111421 1.743e-16 [1060,] 5.306050e-02 0.72971641 -6.557e-18 [1061,] 5.326105e-02 0.72831976 2.242e-16 [1062,] 5.346236e-02 0.72692426 -1.775e-16 [1063,] 5.366443e-02 0.72552992 -8.488e-17 [1064,] 5.386727e-02 0.72413674 -1.443e-16 [1065,] 5.407087e-02 0.72274471 -7.736e-18 [1066,] 5.427524e-02 0.72135384 1.582e-16 [1067,] 5.448038e-02 0.71996414 9.875e-17 [1068,] 5.468630e-02 0.71857560 -3.399e-16 [1069,] 5.489300e-02 0.71718822 -4.858e-16 [1070,] 5.510048e-02 0.71580202 -1.397e-18 [1071,] 5.530874e-02 0.71441699 -9.825e-17 [1072,] 5.551779e-02 0.71303312 1.013e-16 [1073,] 5.572763e-02 0.71165044 1.431e-17 [1074,] 5.593827e-02 0.71026892 3.364e-16 [1075,] 5.614970e-02 0.70888859 -1.191e-16 [1076,] 5.636192e-02 0.70750944 -1.337e-16 [1077,] 5.657495e-02 0.70613146 3.001e-17 [1078,] 5.678879e-02 0.70475468 2.713e-17 [1079,] 5.700343e-02 0.70337907 2.624e-16 [1080,] 5.721889e-02 0.70200466 1.905e-16 [1081,] 5.743516e-02 0.70063144 -2.462e-16 [1082,] 5.765225e-02 0.69925940 -4.048e-16 [1083,] 5.787016e-02 0.69788856 -3.308e-16 [1084,] 5.808889e-02 0.69651892 -1.71e-16 [1085,] 5.830844e-02 0.69515047 -1.253e-16 [1086,] 5.852883e-02 0.69378322 -1.712e-16 [1087,] 5.875005e-02 0.69241718 -6.146e-17 [1088,] 5.897211e-02 0.69105233 -1.575e-16 [1089,] 5.919501e-02 0.68968870 -2.219e-16 [1090,] 5.941875e-02 0.68832626 -1.287e-16 [1091,] 5.964333e-02 0.68696504 -2.788e-17 [1092,] 5.986876e-02 0.68560503 -1.237e-16 [1093,] 6.009505e-02 0.68424623 -3.028e-17 [1094,] 6.032219e-02 0.68288865 -4.954e-16 [1095,] 6.055019e-02 0.68153228 -1.23e-16 [1096,] 6.077905e-02 0.68017713 2.12e-16 [1097,] 6.100878e-02 0.67882320 1.73e-17 [1098,] 6.123937e-02 0.67747050 -2.211e-17 [1099,] 6.147084e-02 0.67611901 -5.129e-17 [1100,] 6.170318e-02 0.67476876 -3.067e-16 [1101,] 6.193640e-02 0.67341973 -2.323e-16 [1102,] 6.217050e-02 0.67207193 -2.402e-16 [1103,] 6.240548e-02 0.67072537 -5.503e-17 [1104,] 6.264136e-02 0.66938003 -7.772e-17 [1105,] 6.287812e-02 0.66803594 2.519e-16 [1106,] 6.311578e-02 0.66669308 1.54e-16 [1107,] 6.335434e-02 0.66535146 1.369e-16 [1108,] 6.359380e-02 0.66401108 -4.458e-16 [1109,] 6.383417e-02 0.66267195 1.202e-16 [1110,] 6.407544e-02 0.66133406 -1.001e-16 [1111,] 6.431762e-02 0.65999741 4.817e-17 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0.25992438 1.757e-16 [1488,] 2.666913e-01 0.25915810 -3.679e-16 [1489,] 2.676993e-01 0.25839345 -4.611e-18 [1490,] 2.687111e-01 0.25763042 -2.337e-16 [1491,] 2.697268e-01 0.25686901 -8.232e-17 [1492,] 2.707462e-01 0.25610922 -4.374e-17 [1493,] 2.717696e-01 0.25535105 -5.281e-16 [1494,] 2.727968e-01 0.25459451 1.284e-16 [1495,] 2.738279e-01 0.25383958 1.544e-16 [1496,] 2.748629e-01 0.25308627 -2.509e-16 [1497,] 2.759018e-01 0.25233458 -7.287e-16 [1498,] 2.769446e-01 0.25158450 -8.101e-16 [1499,] 2.779913e-01 0.25083605 -3.884e-16 [1500,] 2.790421e-01 0.25008921 -2.452e-16 [1501,] 2.800968e-01 0.24934398 4.504e-16 [1502,] 2.811554e-01 0.24860037 -7.957e-16 [1503,] 2.822181e-01 0.24785838 -5.271e-16 [1504,] 2.832848e-01 0.24711800 -2.021e-16 [1505,] 2.843555e-01 0.24637923 -1.339e-15 [1506,] 2.854303e-01 0.24564207 -5.214e-16 [1507,] 2.865092e-01 0.24490653 2.946e-17 [1508,] 2.875921e-01 0.24417259 5.027e-16 [1509,] 2.886791e-01 0.24344027 1.136e-16 [1510,] 2.897702e-01 0.24270955 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3.787743e-01 0.19488541 -5.004e-17 [1582,] 3.802060e-01 0.19426800 5.028e-16 [1583,] 3.816430e-01 0.19365212 2.126e-16 [1584,] 3.830855e-01 0.19303777 5.853e-16 [1585,] 3.845335e-01 0.19242495 4.849e-16 [1586,] 3.859869e-01 0.19181364 5.855e-16 [1587,] 3.874458e-01 0.19120387 -4.346e-16 [1588,] 3.889102e-01 0.19059561 -1.255e-16 [1589,] 3.903802e-01 0.18998888 8.808e-17 [1590,] 3.918557e-01 0.18938366 -7.575e-17 [1591,] 3.933368e-01 0.18877996 1.043e-15 [1592,] 3.948235e-01 0.18817778 2.023e-16 [1593,] 3.963158e-01 0.18757711 -2.089e-16 [1594,] 3.978137e-01 0.18697795 -1.615e-15 [1595,] 3.993174e-01 0.18638030 -5.111e-16 [1596,] 4.008267e-01 0.18578417 -2.194e-16 [1597,] 4.023417e-01 0.18518954 1.559e-16 [1598,] 4.038624e-01 0.18459642 -1.157e-15 [1599,] 4.053889e-01 0.18400480 6.487e-16 [1600,] 4.069211e-01 0.18341469 -1.516e-15 [1601,] 4.084591e-01 0.18282608 -1.421e-16 [1602,] 4.100030e-01 0.18223896 -1.141e-15 [1603,] 4.115527e-01 0.18165335 -1.281e-15 [1604,] 4.131082e-01 0.18106923 4.074e-16 [1605,] 4.146696e-01 0.18048661 4.134e-17 [1606,] 4.162370e-01 0.17990549 -1.03e-15 [1607,] 4.178102e-01 0.17932585 5.916e-18 [1608,] 4.193894e-01 0.17874770 -2.425e-16 [1609,] 4.209746e-01 0.17817105 2.063e-16 [1610,] 4.225657e-01 0.17759588 5.442e-16 [1611,] 4.241629e-01 0.17702219 -3.431e-16 [1612,] 4.257661e-01 0.17644999 -5.287e-17 [1613,] 4.273754e-01 0.17587927 -8.467e-16 [1614,] 4.289907e-01 0.17531003 -8.31e-16 [1615,] 4.306122e-01 0.17474227 -6.29e-16 [1616,] 4.322398e-01 0.17417599 -6.413e-16 [1617,] 4.338735e-01 0.17361118 -1.068e-15 [1618,] 4.355134e-01 0.17304784 1.413e-16 [1619,] 4.371595e-01 0.17248598 -5.43e-17 [1620,] 4.388118e-01 0.17192558 -4.358e-16 [1621,] 4.404704e-01 0.17136666 -1.251e-15 [1622,] 4.421352e-01 0.17080919 -4.14e-16 [1623,] 4.438064e-01 0.17025320 4.618e-16 [1624,] 4.454838e-01 0.16969866 -1.025e-15 [1625,] 4.471676e-01 0.16914559 -6.728e-16 [1626,] 4.488578e-01 0.16859398 3.475e-16 [1627,] 4.505543e-01 0.16804382 -4.854e-16 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0.15527234 -7e-16 [1652,] 4.951164e-01 0.15475800 -3.931e-16 [1653,] 4.969878e-01 0.15424507 -8.736e-16 [1654,] 4.988662e-01 0.15373354 -2.568e-16 [1655,] 5.007518e-01 0.15322342 -9.073e-16 [1656,] 5.026445e-01 0.15271469 -1.324e-15 [1657,] 5.045443e-01 0.15220737 1.857e-16 [1658,] 5.064513e-01 0.15170144 -1.315e-15 [1659,] 5.083656e-01 0.15119691 -2.262e-15 [1660,] 5.102870e-01 0.15069378 9.427e-16 [1661,] 5.122158e-01 0.15019203 1.72e-16 [1662,] 5.141518e-01 0.14969167 1.832e-17 [1663,] 5.160951e-01 0.14919270 1.137e-16 [1664,] 5.180458e-01 0.14869512 -6.444e-17 [1665,] 5.200039e-01 0.14819892 2.359e-16 [1666,] 5.219693e-01 0.14770409 -1.585e-16 [1667,] 5.239422e-01 0.14721065 1.222e-15 [1668,] 5.259225e-01 0.14671858 3.188e-16 [1669,] 5.279104e-01 0.14622789 -7.751e-17 [1670,] 5.299057e-01 0.14573857 -8.038e-16 [1671,] 5.319086e-01 0.14525062 -5.876e-16 [1672,] 5.339190e-01 0.14476404 -1.059e-15 [1673,] 5.359371e-01 0.14427883 -1.882e-15 [1674,] 5.379628e-01 0.14379497 5.84e-16 [1675,] 5.399961e-01 0.14331248 -3.253e-16 [1676,] 5.420371e-01 0.14283135 1.764e-16 [1677,] 5.440859e-01 0.14235158 -1.551e-15 [1678,] 5.461423e-01 0.14187316 2.928e-16 [1679,] 5.482066e-01 0.14139610 -1.068e-15 [1680,] 5.502786e-01 0.14092039 -2.062e-15 [1681,] 5.523585e-01 0.14044602 -5.525e-16 [1682,] 5.544463e-01 0.13997300 9.956e-16 [1683,] 5.565419e-01 0.13950133 -1.363e-15 [1684,] 5.586455e-01 0.13903100 -2.88e-16 [1685,] 5.607570e-01 0.13856201 -5.278e-16 [1686,] 5.628765e-01 0.13809435 -9.562e-16 [1687,] 5.650040e-01 0.13762803 5.985e-16 [1688,] 5.671395e-01 0.13716305 -1.207e-15 [1689,] 5.692831e-01 0.13669939 -8.945e-16 [1690,] 5.714348e-01 0.13623707 -4.746e-17 [1691,] 5.735947e-01 0.13577607 -3.856e-16 [1692,] 5.757627e-01 0.13531640 -1.232e-17 [1693,] 5.779389e-01 0.13485805 3.008e-16 [1694,] 5.801233e-01 0.13440101 -5.096e-18 [1695,] 5.823160e-01 0.13394530 -5.266e-16 [1696,] 5.845170e-01 0.13349090 -9.822e-16 [1697,] 5.867263e-01 0.13303782 -2.835e-17 [1698,] 5.889439e-01 0.13258604 1.137e-16 [1699,] 5.911699e-01 0.13213558 -6.022e-16 [1700,] 5.934044e-01 0.13168642 -7.816e-18 [1701,] 5.956473e-01 0.13123856 -1.295e-16 [1702,] 5.978986e-01 0.13079201 -7.699e-16 [1703,] 6.001585e-01 0.13034675 1.986e-16 [1704,] 6.024269e-01 0.12990280 9.597e-17 [1705,] 6.047039e-01 0.12946013 -1.334e-15 [1706,] 6.069895e-01 0.12901877 -5.002e-16 [1707,] 6.092837e-01 0.12857869 -7.604e-16 [1708,] 6.115867e-01 0.12813990 9.921e-16 [1709,] 6.138983e-01 0.12770239 -3.948e-16 [1710,] 6.162186e-01 0.12726617 -8.623e-16 [1711,] 6.185477e-01 0.12683123 -3.212e-16 [1712,] 6.208856e-01 0.12639757 -6.225e-16 [1713,] 6.232324e-01 0.12596518 2.979e-16 [1714,] 6.255880e-01 0.12553407 2.84e-17 [1715,] 6.279526e-01 0.12510423 1.025e-15 [1716,] 6.303260e-01 0.12467566 -1.388e-18 [1717,] 6.327085e-01 0.12424836 6.73e-16 [1718,] 6.350999e-01 0.12382232 1.104e-15 [1719,] 6.375004e-01 0.12339754 -5.357e-16 [1720,] 6.399099e-01 0.12297403 -1.313e-16 [1721,] 6.423286e-01 0.12255177 -2.243e-16 [1722,] 6.447564e-01 0.12213076 5.531e-16 [1723,] 6.471934e-01 0.12171101 -8.83e-16 [1724,] 6.496396e-01 0.12129251 -6.394e-16 [1725,] 6.520950e-01 0.12087526 -1.733e-15 [1726,] 6.545597e-01 0.12045926 3.46e-16 [1727,] 6.570338e-01 0.12004449 -8.612e-16 [1728,] 6.595172e-01 0.11963097 -8.904e-16 [1729,] 6.620099e-01 0.11921869 -1.33e-16 [1730,] 6.645121e-01 0.11880764 -1.115e-16 [1731,] 6.670238e-01 0.11839783 -1.151e-15 [1732,] 6.695449e-01 0.11798924 -1.458e-15 [1733,] 6.720756e-01 0.11758189 -2.222e-15 [1734,] 6.746158e-01 0.11717576 -1.044e-15 [1735,] 6.771657e-01 0.11677086 -1.044e-15 [1736,] 6.797252e-01 0.11636718 -1.998e-15 [1737,] 6.822943e-01 0.11596472 3.001e-18 [1738,] 6.848732e-01 0.11556348 2.869e-16 [1739,] 6.874618e-01 0.11516345 -1.274e-15 [1740,] 6.900602e-01 0.11476463 -6.112e-16 [1741,] 6.926684e-01 0.11436702 -1.544e-15 [1742,] 6.952865e-01 0.11397062 2.232e-17 [1743,] 6.979144e-01 0.11357542 -1.518e-15 [1744,] 7.005523e-01 0.11318143 -1.89e-15 [1745,] 7.032002e-01 0.11278863 1.37e-16 [1746,] 7.058581e-01 0.11239704 -1.57e-15 [1747,] 7.085260e-01 0.11200663 -4.838e-16 [1748,] 7.112040e-01 0.11161742 -2.345e-16 [1749,] 7.138922e-01 0.11122940 1.059e-15 [1750,] 7.165905e-01 0.11084257 3.715e-17 [1751,] 7.192990e-01 0.11045692 -1.808e-16 [1752,] 7.220177e-01 0.11007246 -1.191e-15 [1753,] 7.247467e-01 0.10968918 -1.643e-15 [1754,] 7.274860e-01 0.10930707 -1.244e-15 [1755,] 7.302357e-01 0.10892614 -2.028e-15 [1756,] 7.329957e-01 0.10854638 1.307e-15 [1757,] 7.357662e-01 0.10816779 -1.078e-15 [1758,] 7.385472e-01 0.10779037 3.187e-16 [1759,] 7.413387e-01 0.10741412 1.588e-16 [1760,] 7.441407e-01 0.10703903 -1.054e-15 [1761,] 7.469534e-01 0.10666510 1.241e-15 [1762,] 7.497766e-01 0.10629232 3.308e-16 [1763,] 7.526105e-01 0.10592071 9.225e-16 [1764,] 7.554552e-01 0.10555024 -7.354e-16 [1765,] 7.583106e-01 0.10518093 -4.656e-16 [1766,] 7.611767e-01 0.10481276 2.157e-16 [1767,] 7.640538e-01 0.10444574 8.45e-16 [1768,] 7.669416e-01 0.10407986 -1.974e-15 [1769,] 7.698404e-01 0.10371512 -8.008e-16 [1770,] 7.727502e-01 0.10335153 7.169e-16 [1771,] 7.756710e-01 0.10298906 -1.049e-15 [1772,] 7.786028e-01 0.10262773 -2.062e-16 [1773,] 7.815456e-01 0.10226753 -1.546e-15 [1774,] 7.844996e-01 0.10190846 -6.437e-16 [1775,] 7.874648e-01 0.10155051 -9.851e-16 [1776,] 7.904412e-01 0.10119369 -8.18e-16 [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 [2121,] 2.904821 0.02857815 1.729e-15 [2122,] 2.915801 0.02847134 -4.923e-15 [2123,] 2.926821 0.02836492 -1.957e-15 [2124,] 2.937884 0.02825889 -9.779e-15 [2125,] 2.948988 0.02815326 1.284e-15 [2126,] 2.960134 0.02804801 -1.668e-14 [2127,] 2.971323 0.02794315 -6.288e-15 [2128,] 2.982554 0.02783868 1.341e-14 [2129,] 2.993827 0.02773460 -1.762e-15 [2130,] 3.005142 0.02763089 2.047e-14 [2131,] 3.016501 0.02752757 -3.802e-15 [2132,] 3.027902 0.02742464 9.456e-15 [2133,] 3.039347 0.02732208 -8.935e-15 [2134,] 3.050835 0.02721990 2.3e-14 [2135,] 3.062366 0.02711809 8.832e-15 [2136,] 3.073941 0.02701667 -1.739e-14 [2137,] 3.085559 0.02691561 3.67e-14 [2138,] 3.097222 0.02681493 -1.292e-14 [2139,] 3.108928 0.02671463 1.277e-14 [2140,] 3.120679 0.02661469 -4.227e-14 [2141,] 3.132474 0.02651512 1.998e-15 [2142,] 3.144314 0.02641592 3.547e-14 [2143,] 3.156199 0.02631709 -3.097e-16 [2144,] 3.168128 0.02621862 -4.147e-14 [2145,] 3.180103 0.02612051 3.88e-15 [2146,] 3.192122 0.02602277 -2.751e-14 [2147,] 3.204188 0.02592539 -4.672e-15 [2148,] 3.216299 0.02582837 2.133e-14 [2149,] 3.228455 0.02573171 3.707e-15 [2150,] 3.240658 0.02563540 -2.907e-14 [2151,] 3.252906 0.02553946 5.333e-15 [2152,] 3.265201 0.02544386 -2.067e-14 [2153,] 3.277543 0.02534862 -1.225e-14 [2154,] 3.289931 0.02525374 -1.29e-15 [2155,] 3.302366 0.02515920 -1.76e-14 [2156,] 3.314848 0.02506501 -8.576e-15 [2157,] 3.327377 0.02497118 1.201e-14 [2158,] 3.339953 0.02487769 2.425e-15 [2159,] 3.352577 0.02478454 3.987e-15 [2160,] 3.365249 0.02469175 -1.137e-14 [2161,] 3.377969 0.02459929 -2.833e-14 [2162,] 3.390736 0.02450718 1.043e-14 [2163,] 3.403552 0.02441541 -1.381e-14 [2164,] 3.416417 0.02432397 2.764e-15 [2165,] 3.429330 0.02423288 -2.095e-14 [2166,] 3.442292 0.02414213 -2.777e-14 [2167,] 3.455302 0.02405171 -8.692e-15 [2168,] 3.468362 0.02396162 2.8e-14 [2169,] 3.481472 0.02387187 5.828e-15 [2170,] 3.494631 0.02378246 -2.806e-14 [2171,] 3.507839 0.02369337 -6.228e-15 [2172,] 3.521098 0.02360462 -1.116e-14 [2173,] 3.534406 0.02351619 6.554e-15 [2174,] 3.547765 0.02342809 5.027e-15 [2175,] 3.561175 0.02334032 -1.557e-14 [2176,] 3.574635 0.02325288 1.773e-14 [2177,] 3.588146 0.02316576 -3.626e-15 [2178,] 3.601708 0.02307896 5.965e-15 [2179,] 3.615321 0.02299248 5.202e-15 [2180,] 3.628986 0.02290633 -1.359e-14 [2181,] 3.642703 0.02282049 3.184e-15 [2182,] 3.656471 0.02273498 2.588e-15 [2183,] 3.670291 0.02264978 1.591e-14 [2184,] 3.684164 0.02256489 2.367e-15 [2185,] 3.698089 0.02248033 -1.583e-14 [2186,] 3.712067 0.02239607 -2.267e-15 [2187,] 3.726097 0.02231213 2.653e-14 [2188,] 3.740181 0.02222850 -7.235e-15 [2189,] 3.754317 0.02214519 -1.133e-14 [2190,] 3.768507 0.02206218 -1.341e-15 [2191,] 3.782751 0.02197948 2.246e-14 [2192,] 3.797049 0.02189709 -3.414e-15 [2193,] 3.811401 0.02181500 3.123e-14 [2194,] 3.825807 0.02173322 -4.081e-14 [2195,] 3.840267 0.02165174 2.553e-14 [2196,] 3.854782 0.02157056 4.162e-15 [2197,] 3.869352 0.02148969 -2.068e-16 [2198,] 3.883977 0.02140912 5.858e-15 [2199,] 3.898657 0.02132885 1.111e-14 [2200,] 3.913393 0.02124887 1.784e-14 [2201,] 3.928184 0.02116920 2.004e-14 [2202,] 3.943032 0.02108982 2.115e-14 [2203,] 3.957935 0.02101073 -2.166e-14 [2204,] 3.972895 0.02093194 -1.834e-14 [2205,] 3.987911 0.02085344 -1.337e-14 [2206,] 4.002984 0.02077524 -1.782e-14 [2207,] 4.018114 0.02069732 3.492e-14 [2208,] 4.033301 0.02061970 -1.066e-14 [2209,] 4.048546 0.02054236 -1.643e-14 [2210,] 4.063848 0.02046531 2.873e-14 [2211,] 4.079208 0.02038855 -2.834e-15 [2212,] 4.094627 0.02031207 2.428e-14 [2213,] 4.110103 0.02023588 7.048e-15 [2214,] 4.125638 0.02015998 3.205e-16 [2215,] 4.141232 0.02008435 2.216e-14 [2216,] 4.156884 0.02000901 -1.197e-14 [2217,] 4.172596 0.01993395 8.496e-15 [2218,] 4.188367 0.01985916 -1.817e-14 [2219,] 4.204198 0.01978466 2.803e-14 [2220,] 4.220088 0.01971043 4.22e-14 [2221,] 4.236039 0.01963648 -1.142e-14 [2222,] 4.252050 0.01956281 -4.042e-15 [2223,] 4.268121 0.01948941 -3.662e-15 [2224,] 4.284254 0.01941628 7.838e-15 [2225,] 4.300447 0.01934342 -1.058e-14 [2226,] 4.316701 0.01927084 3.638e-14 [2227,] 4.333017 0.01919853 -3.42e-14 [2228,] 4.349394 0.01912649 1.33e-14 [2229,] 4.365834 0.01905471 2.927e-15 [2230,] 4.382335 0.01898320 -2.53e-15 [2231,] 4.398899 0.01891196 2.953e-14 [2232,] 4.415526 0.01884099 1.204e-18 [2233,] 4.432215 0.01877028 1.656e-14 [2234,] 4.448967 0.01869983 -3.022e-14 [2235,] 4.465783 0.01862965 9.523e-15 [2236,] 4.482662 0.01855973 2.432e-14 [2237,] 4.499605 0.01849007 -2.592e-14 [2238,] 4.516613 0.01842067 -2.806e-14 [2239,] 4.533684 0.01835152 -1.235e-14 [2240,] 4.550820 0.01828264 -2.499e-14 [2241,] 4.568021 0.01821401 -3.187e-14 [2242,] 4.585286 0.01814564 1.066e-14 [2243,] 4.602617 0.01807752 -1.584e-14 [2244,] 4.620014 0.01800966 -2.676e-14 [2245,] 4.637476 0.01794205 5.412e-14 [2246,] 4.655004 0.01787469 -4.056e-14 [2247,] 4.672599 0.01780759 5.187e-14 [2248,] 4.690260 0.01774073 -4.522e-14 [2249,] 4.707987 0.01767413 2.076e-14 [2250,] 4.725782 0.01760777 5.359e-15 [2251,] 4.743644 0.01754166 -1.136e-14 [2252,] 4.761574 0.01747580 5.553e-14 [2253,] 4.779571 0.01741018 1.348e-15 [2254,] 4.797636 0.01734481 2.047e-14 [2255,] 4.815770 0.01727969 -3.709e-14 [2256,] 4.833972 0.01721480 -3.184e-14 [2257,] 4.852243 0.01715016 -2.711e-14 [2258,] 4.870583 0.01708576 -1.427e-14 [2259,] 4.888992 0.01702160 1.085e-14 [2260,] 4.907471 0.01695768 -1.001e-13 [2261,] 4.926020 0.01689400 5.44e-15 [2262,] 4.944639 0.01683055 -6.957e-15 [2263,] 4.963328 0.01676735 -4.092e-14 [2264,] 4.982088 0.01670438 8.318e-14 [2265,] 5.000919 0.01664164 5.794e-14 [2266,] 5.019821 0.01657914 1.396e-14 [2267,] 5.038794 0.01651687 -5.058e-15 [2268,] 5.057839 0.01645484 8.617e-14 [2269,] 5.076956 0.01639304 -5.755e-15 [2270,] 5.096145 0.01633146 3.28e-14 [2271,] 5.115407 0.01627012 -5.38e-14 [2272,] 5.134742 0.01620901 5.241e-15 [2273,] 5.154150 0.01614813 -9.658e-15 [2274,] 5.173631 0.01608747 1.193e-14 [2275,] 5.193186 0.01602704 3.445e-14 [2276,] 5.212814 0.01596684 4.872e-14 [2277,] 5.232517 0.01590686 2.591e-14 [2278,] 5.252294 0.01584710 3.649e-14 [2279,] 5.272146 0.01578757 -7.908e-14 [2280,] 5.292073 0.01572827 3.314e-14 [2281,] 5.312076 0.01566918 1.114e-13 [2282,] 5.332154 0.01561031 1.814e-14 [2283,] 5.352308 0.01555167 3.136e-14 [2284,] 5.372538 0.01549324 -6.247e-14 [2285,] 5.392844 0.01543503 -5.404e-14 [2286,] 5.413228 0.01537704 3.201e-15 [2287,] 5.433688 0.01531927 2.81e-14 [2288,] 5.454226 0.01526171 -4.178e-14 [2289,] 5.474841 0.01520437 -9.667e-15 [2290,] 5.495534 0.01514725 -5.403e-15 [2291,] 5.516306 0.01509033 -8.227e-14 [2292,] 5.537156 0.01503363 1.659e-14 [2293,] 5.558084 0.01497714 2.683e-14 [2294,] 5.579092 0.01492087 4.75e-14 [2295,] 5.600179 0.01486480 5.193e-14 [2296,] 5.621346 0.01480894 4.224e-15 [2297,] 5.642593 0.01475330 -3.066e-14 [2298,] 5.663921 0.01469786 -1.7e-14 [2299,] 5.685329 0.01464263 -2.026e-14 [2300,] 5.706817 0.01458760 2.051e-14 [2301,] 5.728387 0.01453278 4.409e-14 [2302,] 5.750039 0.01447817 2.303e-14 [2303,] 5.771772 0.01442376 8.009e-14 [2304,] 5.793588 0.01436955 -2.318e-14 [2305,] 5.815486 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-1.369e-16 [3160,] 1.463601e+02 0.00056937 -8.7e-17 [3161,] 1.469133e+02 0.00056723 -1.721e-16 [3162,] 1.474685e+02 0.00056509 6.221e-18 [3163,] 1.480259e+02 0.00056296 -1.26e-16 [3164,] 1.485854e+02 0.00056084 -4.474e-18 [3165,] 1.491470e+02 0.00055873 -7.337e-17 [3166,] 1.497108e+02 0.00055663 3.247e-17 [3167,] 1.502766e+02 0.00055453 -1.71e-16 [3168,] 1.508446e+02 0.00055244 -1.339e-16 [3169,] 1.514148e+02 0.00055036 -2.206e-16 [3170,] 1.519871e+02 0.00054829 1.098e-17 [3171,] 1.525615e+02 0.00054623 -1.076e-16 [3172,] 1.531382e+02 0.00054417 2.344e-17 [3173,] 1.537170e+02 0.00054212 -1.098e-16 [3174,] 1.542980e+02 0.00054008 6.063e-17 [3175,] 1.548812e+02 0.00053805 3.361e-17 [3176,] 1.554666e+02 0.00053602 -1.589e-16 [3177,] 1.560542e+02 0.00053400 2.344e-17 [3178,] 1.566440e+02 0.00053199 -6.861e-17 [3179,] 1.572361e+02 0.00052999 2.321e-17 [3180,] 1.578304e+02 0.00052799 -8.577e-17 [3181,] 1.584270e+02 0.00052600 8.257e-17 [3182,] 1.590258e+02 0.00052402 -8.943e-17 [3183,] 1.596268e+02 0.00052205 2.366e-17 [3184,] 1.602302e+02 0.00052008 2.577e-18 [3185,] 1.608358e+02 0.00051813 -2.093e-17 [3186,] 1.614437e+02 0.00051618 -1.413e-16 [3187,] 1.620539e+02 0.00051423 -1.563e-16 [3188,] 1.626664e+02 0.00051230 2.488e-17 [3189,] 1.632813e+02 0.00051037 4.435e-18 [3190,] 1.638984e+02 0.00050844 6.55e-18 [3191,] 1.645179e+02 0.00050653 -1.446e-16 [3192,] 1.651397e+02 0.00050462 -1.07e-16 [3193,] 1.657639e+02 0.00050272 -4.43e-18 [3194,] 1.663904e+02 0.00050083 -1.033e-16 [3195,] 1.670193e+02 0.00049894 -1.21e-16 [3196,] 1.676506e+02 0.00049706 3.098e-17 [3197,] 1.682843e+02 0.00049519 -6.2e-18 [3198,] 1.689204e+02 0.00049333 -7.671e-17 [3199,] 1.695588e+02 0.00049147 -1.656e-18 [3200,] 1.701997e+02 0.00048962 -1.128e-16 [3201,] 1.708430e+02 0.00048778 -7.794e-17 [3202,] 1.714887e+02 0.00048594 1.611e-17 [3203,] 1.721369e+02 0.00048411 -7.969e-17 [3204,] 1.727875e+02 0.00048229 -7.004e-17 [3205,] 1.734406e+02 0.00048047 5.569e-17 [3206,] 1.740962e+02 0.00047866 -9.314e-17 [3207,] 1.747542e+02 0.00047686 1.348e-17 [3208,] 1.754147e+02 0.00047506 -2.984e-17 [3209,] 1.760777e+02 0.00047328 -4.942e-17 [3210,] 1.767433e+02 0.00047149 -9.103e-17 [3211,] 1.774113e+02 0.00046972 -1.174e-16 [3212,] 1.780819e+02 0.00046795 -9.147e-17 [3213,] 1.787549e+02 0.00046619 -5.984e-17 [3214,] 1.794306e+02 0.00046443 -1.516e-17 [3215,] 1.801088e+02 0.00046268 1.003e-17 [3216,] 1.807895e+02 0.00046094 4.784e-17 [3217,] 1.814729e+02 0.00045921 -3.962e-17 [3218,] 1.821588e+02 0.00045748 -3.255e-17 [3219,] 1.828473e+02 0.00045575 -4.219e-17 [3220,] 1.835384e+02 0.00045404 -9.778e-17 [3221,] 1.842321e+02 0.00045233 -5.377e-18 [3222,] 1.849284e+02 0.00045062 -1.18e-16 [3223,] 1.856274e+02 0.00044893 -7.965e-17 [3224,] 1.863290e+02 0.00044724 -1.545e-16 [3225,] 1.870333e+02 0.00044555 -3.558e-17 [3226,] 1.877402e+02 0.00044388 -1.32e-16 [3227,] 1.884498e+02 0.00044220 -5.104e-17 [3228,] 1.891621e+02 0.00044054 -6.386e-17 [3229,] 1.898771e+02 0.00043888 -2.725e-17 [3230,] 1.905948e+02 0.00043723 -1.218e-16 [3231,] 1.913152e+02 0.00043558 2.998e-17 [3232,] 1.920383e+02 0.00043394 -1.031e-16 [3233,] 1.927641e+02 0.00043231 -5.198e-17 [3234,] 1.934927e+02 0.00043068 -5.665e-17 [3235,] 1.942240e+02 0.00042906 -1.209e-16 [3236,] 1.949581e+02 0.00042744 -4.562e-18 [3237,] 1.956950e+02 0.00042583 6.689e-17 [3238,] 1.964347e+02 0.00042423 4.069e-17 [3239,] 1.971772e+02 0.00042263 -1.41e-16 [3240,] 1.979224e+02 0.00042104 -1.328e-16 [3241,] 1.986705e+02 0.00041945 -4.777e-17 [3242,] 1.994214e+02 0.00041788 -7.158e-17 [3243,] 2.001752e+02 0.00041630 8.632e-17 [3244,] 2.009318e+02 0.00041473 1.718e-17 [3245,] 2.016912e+02 0.00041317 -4.299e-17 [3246,] 2.024536e+02 0.00041162 3.896e-17 [3247,] 2.032188e+02 0.00041007 9.571e-17 [3248,] 2.039869e+02 0.00040852 5.922e-17 [3249,] 2.047579e+02 0.00040698 1.224e-16 [3250,] 2.055318e+02 0.00040545 1.279e-16 [3251,] 2.063087e+02 0.00040393 4.033e-17 [3252,] 2.070885e+02 0.00040240 5.443e-17 [3253,] 2.078712e+02 0.00040089 6.93e-18 [3254,] 2.086569e+02 0.00039938 1.145e-16 [3255,] 2.094455e+02 0.00039788 2.768e-17 [3256,] 2.102372e+02 0.00039638 2.032e-17 [3257,] 2.110318e+02 0.00039488 -6.663e-17 [3258,] 2.118294e+02 0.00039340 -7.207e-19 [3259,] 2.126301e+02 0.00039192 -1.677e-17 [3260,] 2.134338e+02 0.00039044 -2.098e-17 [3261,] 2.142405e+02 0.00038897 -1.667e-17 [3262,] 2.150502e+02 0.00038751 6.843e-17 [3263,] 2.158631e+02 0.00038605 -1.002e-16 [3264,] 2.166790e+02 0.00038459 -3.397e-17 [3265,] 2.174979e+02 0.00038315 -1.579e-17 [3266,] 2.183200e+02 0.00038170 -2.195e-17 [3267,] 2.191452e+02 0.00038027 2.978e-17 [3268,] 2.199735e+02 0.00037883 3.916e-17 [3269,] 2.208049e+02 0.00037741 -1.359e-18 [3270,] 2.216395e+02 0.00037599 1.514e-16 [3271,] 2.224772e+02 0.00037457 -8.836e-18 [3272,] 2.233181e+02 0.00037316 -1.277e-17 [3273,] 2.241622e+02 0.00037175 -5.196e-17 [3274,] 2.250095e+02 0.00037035 5.013e-17 [3275,] 2.258599e+02 0.00036896 -1.151e-16 [3276,] 2.267136e+02 0.00036757 4.035e-18 [3277,] 2.275705e+02 0.00036619 -1.395e-17 [3278,] 2.284307e+02 0.00036481 -2.672e-17 [3279,] 2.292941e+02 0.00036343 6.357e-18 [3280,] 2.301607e+02 0.00036207 -2.806e-17 [3281,] 2.310307e+02 0.00036070 -7.831e-17 [3282,] 2.319039e+02 0.00035934 -6.061e-17 [3283,] 2.327804e+02 0.00035799 -3.55e-17 [3284,] 2.336603e+02 0.00035664 -2.664e-17 [3285,] 2.345434e+02 0.00035530 -8.669e-17 [3286,] 2.354299e+02 0.00035396 -5.3e-18 [3287,] 2.363198e+02 0.00035263 -6.212e-17 [3288,] 2.372130e+02 0.00035130 -6.332e-17 [3289,] 2.381096e+02 0.00034998 -6.671e-18 [3290,] 2.390096e+02 0.00034866 -4.82e-17 [3291,] 2.399129e+02 0.00034735 4.382e-17 [3292,] 2.408197e+02 0.00034604 -1.851e-17 [3293,] 2.417300e+02 0.00034474 1.131e-16 [3294,] 2.426436e+02 0.00034344 2.42e-17 [3295,] 2.435608e+02 0.00034215 5.737e-18 [3296,] 2.444813e+02 0.00034086 -5.028e-17 [3297,] 2.454054e+02 0.00033957 -7.983e-17 [3298,] 2.463330e+02 0.00033830 3.818e-18 [3299,] 2.472640e+02 0.00033702 -9.877e-18 [3300,] 2.481986e+02 0.00033575 -3.224e-17 [3301,] 2.491367e+02 0.00033449 -6.783e-17 [3302,] 2.500784e+02 0.00033323 -6.814e-17 [3303,] 2.510236e+02 0.00033197 4.484e-17 [3304,] 2.519724e+02 0.00033072 3.122e-17 [3305,] 2.529248e+02 0.00032948 -7.562e-17 [3306,] 2.538807e+02 0.00032824 9.513e-17 [3307,] 2.548403e+02 0.00032700 6.378e-17 [3308,] 2.558036e+02 0.00032577 -1.632e-18 [3309,] 2.567704e+02 0.00032454 -1.955e-17 [3310,] 2.577409e+02 0.00032332 2.177e-17 [3311,] 2.587151e+02 0.00032210 -1.385e-16 [3312,] 2.596930e+02 0.00032089 -1.755e-16 [3313,] 2.606745e+02 0.00031968 2.97e-17 [3314,] 2.616598e+02 0.00031848 -1.029e-16 [3315,] 2.626488e+02 0.00031728 5.912e-17 [3316,] 2.636415e+02 0.00031609 -1.389e-16 [3317,] 2.646380e+02 0.00031490 -8.888e-17 [3318,] 2.656383e+02 0.00031371 5.294e-17 [3319,] 2.666423e+02 0.00031253 2.386e-17 [3320,] 2.676501e+02 0.00031135 -2.752e-18 [3321,] 2.686618e+02 0.00031018 2.439e-17 [3322,] 2.696772e+02 0.00030901 -1.071e-16 [3323,] 2.706965e+02 0.00030785 -6.552e-17 [3324,] 2.717197e+02 0.00030669 -6.817e-17 [3325,] 2.727467e+02 0.00030553 4.124e-17 [3326,] 2.737776e+02 0.00030438 -6.375e-18 [3327,] 2.748124e+02 0.00030324 -6.67e-17 [3328,] 2.758511e+02 0.00030210 -1.088e-16 [3329,] 2.768937e+02 0.00030096 2.735e-17 [3330,] 2.779403e+02 0.00029982 -3.585e-17 [3331,] 2.789908e+02 0.00029870 -1.741e-17 [3332,] 2.800453e+02 0.00029757 -2.116e-17 [3333,] 2.811038e+02 0.00029645 -6.155e-17 [3334,] 2.821663e+02 0.00029533 -6.778e-17 [3335,] 2.832328e+02 0.00029422 -5.701e-17 [3336,] 2.843033e+02 0.00029311 -8.057e-17 [3337,] 2.853779e+02 0.00029201 -1.084e-16 [3338,] 2.864565e+02 0.00029091 -1.651e-17 [3339,] 2.875393e+02 0.00028982 -1.483e-16 [3340,] 2.886261e+02 0.00028872 -1.405e-16 [3341,] 2.897170e+02 0.00028764 -4.299e-17 [3342,] 2.908120e+02 0.00028655 3.878e-18 [3343,] 2.919112e+02 0.00028547 -3.419e-17 [3344,] 2.930145e+02 0.00028440 3.893e-17 [3345,] 2.941220e+02 0.00028333 3.149e-17 [3346,] 2.952337e+02 0.00028226 -1.008e-16 [3347,] 2.963496e+02 0.00028120 -3.58e-17 [3348,] 2.974697e+02 0.00028014 -1.201e-16 [3349,] 2.985941e+02 0.00027909 -1.057e-16 [3350,] 2.997227e+02 0.00027803 -1.381e-16 [3351,] 3.008555e+02 0.00027699 -4.397e-17 [3352,] 3.019927e+02 0.00027594 6.891e-17 [3353,] 3.031341e+02 0.00027491 -1.42e-17 [3354,] 3.042799e+02 0.00027387 -6.778e-17 [3355,] 3.054300e+02 0.00027284 -2.81e-17 [3356,] 3.065844e+02 0.00027181 6.559e-17 [3357,] 3.077432e+02 0.00027079 3.428e-17 [3358,] 3.089064e+02 0.00026977 7.481e-17 [3359,] 3.100739e+02 0.00026875 -2.332e-17 [3360,] 3.112459e+02 0.00026774 -9.819e-17 [3361,] 3.124223e+02 0.00026673 3.822e-17 [3362,] 3.136032e+02 0.00026573 -8.975e-18 [3363,] 3.147885e+02 0.00026473 8.011e-17 [3364,] 3.159783e+02 0.00026373 8.958e-17 [3365,] 3.171726e+02 0.00026274 -1.459e-16 [3366,] 3.183714e+02 0.00026175 -1.192e-16 [3367,] 3.195748e+02 0.00026076 1.217e-16 [3368,] 3.207827e+02 0.00025978 2.076e-17 [3369,] 3.219951e+02 0.00025880 -1.432e-16 [3370,] 3.232122e+02 0.00025783 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0.00007509 -6.123e-17 [3698,] 1.113999e+03 0.00007481 2.319e-17 [3699,] 1.118209e+03 0.00007452 -8.463e-17 [3700,] 1.122436e+03 0.00007424 -2.001e-17 [3701,] 1.126678e+03 0.00007396 -7.427e-17 [3702,] 1.130937e+03 0.00007369 -1.065e-16 [3703,] 1.135211e+03 0.00007341 -3.109e-17 [3704,] 1.139502e+03 0.00007313 -1.998e-16 [3705,] 1.143809e+03 0.00007286 -1.164e-16 [3706,] 1.148132e+03 0.00007258 9.744e-18 [3707,] 1.152472e+03 0.00007231 -6.285e-17 [3708,] 1.156828e+03 0.00007204 -6.569e-17 [3709,] 1.161200e+03 0.00007176 2.067e-17 [3710,] 1.165589e+03 0.00007149 1.095e-16 [3711,] 1.169995e+03 0.00007123 1.818e-17 [3712,] 1.174417e+03 0.00007096 -2.973e-17 [3713,] 1.178856e+03 0.00007069 -8.103e-17 [3714,] 1.183312e+03 0.00007042 -7.133e-17 [3715,] 1.187784e+03 0.00007016 9.324e-17 [3716,] 1.192274e+03 0.00006989 -8.996e-17 [3717,] 1.196780e+03 0.00006963 9.583e-18 [3718,] 1.201304e+03 0.00006937 -1.118e-16 [3719,] 1.205844e+03 0.00006911 -5.527e-18 [3720,] 1.210402e+03 0.00006885 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1.325108e+03 0.00006289 -7.974e-17 [3745,] 1.330117e+03 0.00006265 4.273e-17 [3746,] 1.335144e+03 0.00006242 9.701e-18 [3747,] 1.340190e+03 0.00006218 -5.181e-17 [3748,] 1.345256e+03 0.00006195 -1.345e-16 [3749,] 1.350341e+03 0.00006171 -5.686e-17 [3750,] 1.355445e+03 0.00006148 1.28e-16 [3751,] 1.360568e+03 0.00006125 5.62e-17 [3752,] 1.365710e+03 0.00006102 -1.227e-17 [3753,] 1.370872e+03 0.00006079 -4.489e-17 [3754,] 1.376054e+03 0.00006056 -1.069e-16 [3755,] 1.381255e+03 0.00006033 -5.961e-17 [3756,] 1.386475e+03 0.00006010 -6.772e-18 [3757,] 1.391716e+03 0.00005988 -8.757e-17 [3758,] 1.396976e+03 0.00005965 -8.111e-17 [3759,] 1.402256e+03 0.00005943 -3.615e-17 [3760,] 1.407556e+03 0.00005920 -6.775e-17 [3761,] 1.412877e+03 0.00005898 -5.229e-17 [3762,] 1.418217e+03 0.00005876 -6.296e-17 [3763,] 1.423577e+03 0.00005854 -1.392e-16 [3764,] 1.428958e+03 0.00005832 -7.402e-17 [3765,] 1.434359e+03 0.00005810 -3.792e-17 [3766,] 1.439780e+03 0.00005788 -5.223e-17 [3767,] 1.445222e+03 0.00005766 -1.004e-16 [3768,] 1.450685e+03 0.00005744 -1.547e-16 [3769,] 1.456168e+03 0.00005723 -5.411e-17 [3770,] 1.461672e+03 0.00005701 -4.989e-17 [3771,] 1.467196e+03 0.00005680 -1.374e-16 [3772,] 1.472742e+03 0.00005658 1.805e-17 [3773,] 1.478308e+03 0.00005637 5.789e-17 [3774,] 1.483896e+03 0.00005616 -1.609e-16 [3775,] 1.489505e+03 0.00005595 -1.037e-16 [3776,] 1.495135e+03 0.00005574 -4.44e-17 [3777,] 1.500786e+03 0.00005553 -1.087e-16 [3778,] 1.506458e+03 0.00005532 -1.006e-16 [3779,] 1.512152e+03 0.00005511 5.25e-17 [3780,] 1.517868e+03 0.00005490 -1.426e-16 [3781,] 1.523605e+03 0.00005469 -1.033e-16 [3782,] 1.529363e+03 0.00005449 -5.479e-17 [3783,] 1.535144e+03 0.00005428 -1.953e-16 [3784,] 1.540946e+03 0.00005408 5.668e-17 [3785,] 1.546771e+03 0.00005388 1.57e-17 [3786,] 1.552617e+03 0.00005367 -4.915e-17 [3787,] 1.558485e+03 0.00005347 -3.96e-17 [3788,] 1.564376e+03 0.00005327 1.038e-17 [3789,] 1.570289e+03 0.00005307 -1.458e-16 [3790,] 1.576224e+03 0.00005287 -5.595e-17 [3791,] 1.582182e+03 0.00005267 -5.961e-17 [3792,] 1.588162e+03 0.00005247 -3.442e-17 [3793,] 1.594165e+03 0.00005227 -4.612e-18 [3794,] 1.600190e+03 0.00005208 -1.294e-16 [3795,] 1.606238e+03 0.00005188 -1.226e-18 [3796,] 1.612309e+03 0.00005169 -1.782e-16 [3797,] 1.618403e+03 0.00005149 -8.842e-17 [3798,] 1.624521e+03 0.00005130 -3.705e-17 [3799,] 1.630661e+03 0.00005110 1.596e-17 [3800,] 1.636824e+03 0.00005091 -1.264e-16 [3801,] 1.643011e+03 0.00005072 -6.847e-18 [3802,] 1.649221e+03 0.00005053 -1.033e-16 [3803,] 1.655454e+03 0.00005034 -1.439e-16 [3804,] 1.661712e+03 0.00005015 -1.646e-16 [3805,] 1.667992e+03 0.00004996 3.464e-17 [3806,] 1.674297e+03 0.00004977 -2.304e-17 [3807,] 1.680625e+03 0.00004958 -7.741e-17 [3808,] 1.686977e+03 0.00004940 -7.616e-17 [3809,] 1.693354e+03 0.00004921 -5.708e-17 [3810,] 1.699754e+03 0.00004903 -8.392e-18 [3811,] 1.706179e+03 0.00004884 -8.698e-17 [3812,] 1.712627e+03 0.00004866 -1.48e-16 [3813,] 1.719101e+03 0.00004847 -1.259e-16 [3814,] 1.725598e+03 0.00004829 2.155e-17 [3815,] 1.732120e+03 0.00004811 -1.966e-16 [3816,] 1.738667e+03 0.00004793 -4.277e-17 [3817,] 1.745239e+03 0.00004775 -2.423e-17 [3818,] 1.751835e+03 0.00004757 -9.572e-17 [3819,] 1.758457e+03 0.00004739 -2.509e-17 [3820,] 1.765103e+03 0.00004721 -4.249e-17 [3821,] 1.771775e+03 0.00004703 -1.306e-16 [3822,] 1.778472e+03 0.00004686 1.476e-17 [3823,] 1.785194e+03 0.00004668 -8.15e-17 [3824,] 1.791941e+03 0.00004650 -6.134e-17 [3825,] 1.798714e+03 0.00004633 -7.264e-17 [3826,] 1.805513e+03 0.00004615 -1.66e-16 [3827,] 1.812337e+03 0.00004598 5.507e-17 [3828,] 1.819187e+03 0.00004581 6.645e-17 [3829,] 1.826063e+03 0.00004564 -6.336e-17 [3830,] 1.832965e+03 0.00004546 -6.526e-18 [3831,] 1.839893e+03 0.00004529 -1.448e-17 [3832,] 1.846847e+03 0.00004512 -5.838e-17 [3833,] 1.853828e+03 0.00004495 -1.658e-16 [3834,] 1.860835e+03 0.00004478 3.953e-17 [3835,] 1.867868e+03 0.00004461 -4.179e-17 [3836,] 1.874928e+03 0.00004445 -7.784e-17 [3837,] 1.882015e+03 0.00004428 -1.665e-16 [3838,] 1.889128e+03 0.00004411 -1.432e-16 [3839,] 1.896268e+03 0.00004395 -7.699e-17 [3840,] 1.903436e+03 0.00004378 -1.16e-16 [3841,] 1.910630e+03 0.00004362 -4.908e-17 [3842,] 1.917852e+03 0.00004345 -1.317e-16 [3843,] 1.925101e+03 0.00004329 -1.224e-18 [3844,] 1.932377e+03 0.00004312 -1.275e-16 [3845,] 1.939681e+03 0.00004296 -1.655e-16 [3846,] 1.947012e+03 0.00004280 -3.166e-17 [3847,] 1.954371e+03 0.00004264 -1.508e-16 [3848,] 1.961758e+03 0.00004248 -6.698e-17 [3849,] 1.969173e+03 0.00004232 -8.921e-17 [3850,] 1.976616e+03 0.00004216 -1.003e-17 [3851,] 1.984087e+03 0.00004200 -2.066e-18 [3852,] 1.991586e+03 0.00004184 1.466e-17 [3853,] 1.999114e+03 0.00004169 -7.229e-18 [3854,] 2.006670e+03 0.00004153 -1.506e-16 [3855,] 2.014254e+03 0.00004137 1.892e-17 [3856,] 2.021868e+03 0.00004122 -1.225e-17 [3857,] 2.029510e+03 0.00004106 -1.623e-16 [3858,] 2.037181e+03 0.00004091 -1.359e-17 [3859,] 2.044880e+03 0.00004075 -1.139e-16 [3860,] 2.052610e+03 0.00004060 -1.677e-16 [3861,] 2.060368e+03 0.00004045 -2.856e-17 [3862,] 2.068155e+03 0.00004029 -1.258e-16 [3863,] 2.075972e+03 0.00004014 -5.412e-17 [3864,] 2.083819e+03 0.00003999 3.602e-17 [3865,] 2.091695e+03 0.00003984 4.158e-17 [3866,] 2.099601e+03 0.00003969 -1.855e-16 [3867,] 2.107537e+03 0.00003954 -2.019e-17 [3868,] 2.115503e+03 0.00003939 -9.64e-17 [3869,] 2.123499e+03 0.00003924 -4.828e-17 [3870,] 2.131525e+03 0.00003910 -1.153e-16 [3871,] 2.139581e+03 0.00003895 -1.877e-16 [3872,] 2.147668e+03 0.00003880 -3.797e-17 [3873,] 2.155786e+03 0.00003866 5.904e-17 [3874,] 2.163934e+03 0.00003851 -3.025e-17 [3875,] 2.172113e+03 0.00003837 -6.818e-17 [3876,] 2.180323e+03 0.00003822 -7.739e-17 [3877,] 2.188564e+03 0.00003808 -8.337e-17 [3878,] 2.196836e+03 0.00003793 -2.083e-16 [3879,] 2.205139e+03 0.00003779 -5.766e-17 [3880,] 2.213474e+03 0.00003765 -1.581e-16 [3881,] 2.221840e+03 0.00003751 -1.519e-16 [3882,] 2.230238e+03 0.00003737 -6.108e-17 [3883,] 2.238668e+03 0.00003722 -1.372e-17 [3884,] 2.247129e+03 0.00003708 -4.304e-17 [3885,] 2.255623e+03 0.00003694 -1.585e-16 [3886,] 2.264148e+03 0.00003681 -8.411e-18 [3887,] 2.272706e+03 0.00003667 -9.629e-17 [3888,] 2.281296e+03 0.00003653 -4.34e-17 [3889,] 2.289919e+03 0.00003639 -1.581e-17 [3890,] 2.298574e+03 0.00003625 -8.862e-17 [3891,] 2.307262e+03 0.00003612 -1.123e-17 [3892,] 2.315983e+03 0.00003598 -9.665e-17 [3893,] 2.324736e+03 0.00003585 5.788e-18 [3894,] 2.333523e+03 0.00003571 -1.359e-16 [3895,] 2.342343e+03 0.00003558 -9.103e-17 [3896,] 2.351197e+03 0.00003544 -1.608e-18 [3897,] 2.360083e+03 0.00003531 -3.048e-17 [3898,] 2.369004e+03 0.00003518 -2.469e-18 [3899,] 2.377958e+03 0.00003504 4.293e-17 [3900,] 2.386946e+03 0.00003491 -3.656e-17 [3901,] 2.395968e+03 0.00003478 -2.614e-17 [3902,] 2.405024e+03 0.00003465 3.539e-17 [3903,] 2.414114e+03 0.00003452 -1.593e-16 [3904,] 2.423239e+03 0.00003439 2.538e-17 [3905,] 2.432398e+03 0.00003426 -3.882e-17 [3906,] 2.441591e+03 0.00003413 -6.605e-17 [3907,] 2.450820e+03 0.00003400 3.208e-18 [3908,] 2.460083e+03 0.00003387 -2.111e-16 [3909,] 2.469382e+03 0.00003375 -4.001e-17 [3910,] 2.478715e+03 0.00003362 -1.838e-18 [3911,] 2.488084e+03 0.00003349 6.961e-18 [3912,] 2.497488e+03 0.00003337 -1.104e-16 [3913,] 2.506928e+03 0.00003324 2.253e-17 [3914,] 2.516403e+03 0.00003312 -7.563e-17 [3915,] 2.525914e+03 0.00003299 -1.963e-18 [3916,] 2.535462e+03 0.00003287 1.729e-17 [3917,] 2.545045e+03 0.00003274 -5.984e-17 [3918,] 2.554664e+03 0.00003262 -1.02e-16 [3919,] 2.564320e+03 0.00003250 8.589e-17 [3920,] 2.574013e+03 0.00003237 -4.521e-17 [3921,] 2.583742e+03 0.00003225 -1.254e-16 [3922,] 2.593507e+03 0.00003213 -1.993e-17 [3923,] 2.603310e+03 0.00003201 -1.859e-17 [3924,] 2.613150e+03 0.00003189 -6.338e-17 [3925,] 2.623027e+03 0.00003177 -3.025e-17 [3926,] 2.632941e+03 0.00003165 2.458e-17 [3927,] 2.642892e+03 0.00003153 1.116e-16 [3928,] 2.652882e+03 0.00003141 -1.083e-17 [3929,] 2.662909e+03 0.00003129 -4.546e-17 [3930,] 2.672974e+03 0.00003118 4.106e-17 [3931,] 2.683077e+03 0.00003106 -1.305e-16 [3932,] 2.693218e+03 0.00003094 -1.709e-16 [3933,] 2.703398e+03 0.00003083 -2.226e-16 [3934,] 2.713616e+03 0.00003071 1.913e-17 [3935,] 2.723872e+03 0.00003059 -8.502e-17 [3936,] 2.734168e+03 0.00003048 -4.9e-17 [3937,] 2.744502e+03 0.00003036 -4.135e-17 [3938,] 2.754875e+03 0.00003025 -1.701e-16 [3939,] 2.765288e+03 0.00003014 1.576e-17 [3940,] 2.775740e+03 0.00003002 -1.103e-16 [3941,] 2.786231e+03 0.00002991 -1.002e-16 [3942,] 2.796762e+03 0.00002980 -4.343e-17 [3943,] 2.807333e+03 0.00002968 -1.025e-16 [3944,] 2.817944e+03 0.00002957 4.219e-18 [3945,] 2.828595e+03 0.00002946 2.981e-17 [3946,] 2.839286e+03 0.00002935 -7.496e-17 [3947,] 2.850018e+03 0.00002924 -1.202e-16 [3948,] 2.860790e+03 0.00002913 -9.194e-17 [3949,] 2.871603e+03 0.00002902 -3.186e-17 [3950,] 2.882457e+03 0.00002891 -1.235e-16 [3951,] 2.893352e+03 0.00002880 -2.866e-17 [3952,] 2.904288e+03 0.00002869 -2.027e-17 [3953,] 2.915265e+03 0.00002859 -1.252e-16 [3954,] 2.926284e+03 0.00002848 -6.191e-17 [3955,] 2.937344e+03 0.00002837 -8.739e-18 [3956,] 2.948447e+03 0.00002826 -5.252e-17 [3957,] 2.959591e+03 0.00002816 -1.803e-16 [3958,] 2.970777e+03 0.00002805 -6.675e-17 [3959,] 2.982006e+03 0.00002795 -1.465e-16 [3960,] 2.993277e+03 0.00002784 -1.537e-16 [3961,] 3.004590e+03 0.00002774 -1.082e-16 [3962,] 3.015947e+03 0.00002763 -1.089e-16 [3963,] 3.027346e+03 0.00002753 -1.22e-17 [3964,] 3.038789e+03 0.00002742 -1.714e-16 [3965,] 3.050274e+03 0.00002732 -2.695e-17 [3966,] 3.061803e+03 0.00002722 -3.94e-17 [3967,] 3.073376e+03 0.00002711 -8.237e-17 [3968,] 3.084993e+03 0.00002701 -3.034e-17 [3969,] 3.096653e+03 0.00002691 -5.874e-17 [3970,] 3.108357e+03 0.00002681 4.489e-17 [3971,] 3.120106e+03 0.00002671 2.906e-17 [3972,] 3.131899e+03 0.00002661 -3.297e-17 [3973,] 3.143737e+03 0.00002651 -1.375e-17 [3974,] 3.155619e+03 0.00002641 3.7e-18 [3975,] 3.167546e+03 0.00002631 -1.17e-16 [3976,] 3.179519e+03 0.00002621 -8.258e-17 [3977,] 3.191536e+03 0.00002611 -9.097e-17 [3978,] 3.203599e+03 0.00002601 -1.062e-16 [3979,] 3.215708e+03 0.00002591 -1.205e-16 [3980,] 3.227862e+03 0.00002582 -6.906e-17 [3981,] 3.240062e+03 0.00002572 -1.735e-16 [3982,] 3.252309e+03 0.00002562 -3.353e-17 [3983,] 3.264602e+03 0.00002553 4.112e-17 [3984,] 3.276941e+03 0.00002543 -3.137e-17 [3985,] 3.289327e+03 0.00002533 -8.655e-17 [3986,] 3.301759e+03 0.00002524 -1.175e-16 [3987,] 3.314239e+03 0.00002514 -2.415e-16 [3988,] 3.326766e+03 0.00002505 -1.664e-16 [3989,] 3.339340e+03 0.00002496 -4.848e-17 [3990,] 3.351962e+03 0.00002486 -1.629e-16 [3991,] 3.364631e+03 0.00002477 -1.066e-16 [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): 2.33 0 2.33 > > > ##-- 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 ; `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): 2.7 0.02 2.72 > > > ## ========== 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): 3.11 0.04 3.16 > > ##--- 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): 6.64 0 6.64 > > > 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): 6.38 0.08 6.45 > > > 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-_87038__windows_WS2022x64_20_.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$c1), 4))) ks c1. c1.2 relD [1,] 1 3.326e-02 3.331e-02 -0.0017 [2,] 2 9.532e-03 9.511e-03 0.0022 [3,] 3 7.042e-03 7.049e-03 -0.0009 [4,] 4 9.845e-03 9.811e-03 0.0035 [5,] 5 2.172e-02 2.169e-02 0.0015 [6,] 6 7.028e-02 6.967e-02 0.0088 [7,] 7 3.061e-01 3.041e-01 0.0066 [8,] 8 1.771e+00 1.744e+00 0.0156 [9,] 9 1.275e+01 1.259e+01 0.0128 [10,] 10 1.156e+02 1.129e+02 0.0235 [11,] 11 1.242e+03 1.224e+03 0.0152 [12,] 12 1.638e+04 1.592e+04 0.0294 [13,] 13 2.466e+05 2.427e+05 0.0162 [14,] 14 4.434e+06 4.331e+06 0.0238 [15,] 15 8.994e+07 8.846e+07 0.0167 [16,] 16 2.134e+09 2.082e+09 0.0253 [17,] 17 5.608e+10 5.515e+10 0.0169 [18,] 18 1.704e+12 1.655e+12 0.0292 > > c1.. <- c1.r2[c("ks", "c1")] # just the smallest part is needed here: > (s.c1.fil <- paste0("stirlerr-c1r2-", myPlatform(), ".rds")) [1] "stirlerr-c1r2-R-_87038__windows_WS2022x64_20_.rds" > saveRDS(c1.., file = s.c1.fil) # was "stirlerr-c1.rds" > > ## 2. Now, find the n(k) +/- se.n(k) intervals > ## Use these c1 from above > > ##' Given c1-results relErr(n,k); |relE(n,k) ~= c1 * n^{-2k} , find n such that > ##' |relE(n,k)| line {in log-log scale} cuts y = eps, i.e., n* such that |relE(n,k)| <= eps for all n >= n* > n.ep <- function(eps, c1Res, ks = c1Res$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() 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 Warning in cobs(n, Y, tau = tau, nknots = 6, lambda = -1, print.warn = FALSE, : The algorithm has not converged after 100 iterations for at least one lambda user system elapsed 14.05 0.38 14.42 Warning messages: 1: In chol(, *): Replaced 5 tiny diagonal entries by 'Large' 2: In chol(, *): Replaced 7 tiny diagonal entries by 'Large' 3: In cobs(n, Y, tau = tau, nknots = 6, lambda = -1, print.warn = FALSE, : drqssbc2(): Not all flags are normal (== 1), ifl : 111111118118181818181818181818181818181818 4: In chol(, *): Replaced 5 tiny diagonal entries by 'Large' 5: In chol(, *): Replaced 7 tiny diagonal entries by 'Large' 6: In cobs(n, Y, tau = tau, nknots = 6, lambda = -1, print.warn = FALSE, : drqssbc2(): Not all flags are normal (== 1), ifl : 11111111111118181818181818181818181818 > ## needs 12 sec (!!) user system elapsed = > (s.find15.fil <- paste0("stirlerr-find1_1-15_", myPlatform(), ".rds")) [1] "stirlerr-find1_1-15_R-_87038__windows_WS2022x64_20_.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..$c1)) > > ok1cutsLst <- function(res) { + stopifnot(is.list(res), sapply(res, is.list)) # must be list of lists + cobsL <- lapply(lapply(res, `[[`, "smooths"), `[[`, "cobs") + vapply(cobsL, is.array, NA) + } > (resLok <- ok1cutsLst(resL)) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE > ## 1 2 3 4 5 6 ...... 15 > ## FALSE TRUE TRUE TRUE TRUE TRUE ...... TRUE > > > if(FALSE) { + ## e.g. in R-devel-no-ldouble: + (r1 <- find1cuts(k=1, c1=c1..$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: num [1:1500, 1:2] 5.32e-14 5.28e-14 5.25e-14 5.22e-14 5.19e-14 ... .. ..- attr(*, "dimnames")=List of 2 .. .. ..$ : NULL .. .. ..$ : chr [1:2] "cs1" "cs2" $ i.n : int [1:2, 1:3] 1 NA 1 NA 1 NA ..- attr(*, "dimnames")=List of 2 .. ..$ : chr [1:2] "i" "n." .. ..$ : chr [1:3] "spl" "low" "cobs" > plot1cuts(r17) # no-ldouble is *very* different than normal: k *much better* than k+1 > > (s.find17.fil <- paste0("stirlerr-find1_17_", myPlatform(), ".rds")) [1] "stirlerr-find1_17_R-_87038__windows_WS2022x64_20_.rds" > saveRDS(r17, file = s.find17.fil) # was "stirlerr-find1_17.rds" > ## } > > ## ditto for tau = 0.8 {quantile for cobs()}: > system.time( + resL.8 <- lapply(setNames(,k.), function(k) find1cuts(k=k, c1=c1..$c1, tau = 0.80)) + ) # warnings from cobs {again 12.1 sec} Warning in cobs(n, Y, tau = tau, nknots = 6, lambda = -1, print.warn = FALSE, : The algorithm has not converged after 100 iterations for at least one lambda Warning in cobs(n, Y, tau = tau, nknots = 6, lambda = -1, print.warn = FALSE, : The algorithm has not converged after 100 iterations for at least one lambda user system elapsed 12.45 0.36 12.81 Warning messages: 1: In chol(, *): Replaced 5 tiny diagonal entries by 'Large' 2: In chol(, *): Replaced 7 tiny diagonal entries by 'Large' 3: In cobs(n, Y, tau = tau, nknots = 6, lambda = -1, print.warn = FALSE, : drqssbc2(): Not all flags are normal (== 1), ifl : 111111111111818181818181818181818181818 4: In chol(, *): Replaced 5 tiny diagonal entries by 'Large' 5: In chol(, *): Replaced 7 tiny diagonal entries by 'Large' 6: In cobs(n, Y, tau = tau, nknots = 6, lambda = -1, print.warn = FALSE, : drqssbc2(): Not all flags are normal (== 1), ifl : 11111111118181818181818181818181818181818 > (resL8ok <- ok1cutsLst(resL.8)) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE > > mult.fig(15, main = "stirlerr(n, order=k) vs order = k+1 -- tau = 0.8") > invisible(lapply(resL.8[resL8ok], plot1cuts)) > ## plus "last" one > str(r17.8 <- find1cuts(k=17, n = seq(5.1, 6.5, length.out = 1500), c1=c1..$c1, tau = 0.80)) 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: num [1:1500, 1:2] 5.31e-14 5.28e-14 5.25e-14 5.22e-14 5.19e-14 ... .. ..- attr(*, "dimnames")=List of 2 .. .. ..$ : NULL .. .. ..$ : chr [1:2] "cs1" "cs2" $ i.n : int [1:2, 1:3] 1 NA 1 NA 1 NA ..- attr(*, "dimnames")=List of 2 .. ..$ : chr [1:2] "i" "n." .. ..$ : chr [1:3] "spl" "low" "cobs" > plot1cuts(r17.8) > par(opar) > > n.mn <- sapply(resL, `[[`, "i.n", simplify = "array") > n.mn8 <- sapply(resL.8, `[[`, "i.n", simplify = "array") > ## of course, only the cobs part differs: > ## ... when I later see errors here ... what's going on?? > str(n.mn) # all NULL for "no-double" !! num [1:2, 1:3, 1:15] 390 15637376 402 15821059 404 ... - attr(*, "dimnames")=List of 3 ..$ : chr [1:2] "i" "n." ..$ : chr [1:3] "spl" "low" "cobs" ..$ : chr [1:15] "1" "2" "3" "4" ... > dim(n.mn["n.",,]) [1] 3 15 > str(n.mn8["n.", "cobs",]) Named num [1:15] 1.57e+07 NA 2.05e+02 8.62e+01 2.71e+01 ... - attr(*, "names")= chr [1:15] "1" "2" "3" "4" ... > > sessionInfo() R Under development (unstable) (2024-08-21 r87038 ucrt) Platform: x86_64-w64-mingw32/x64 Running under: Windows Server 2022 x64 (build 20348) Matrix products: default locale: [1] LC_COLLATE=C LC_CTYPE=German_Germany.utf8 LC_MONETARY=C [4] LC_NUMERIC=C LC_TIME=C time zone: Europe/Berlin tzcode source: internal attached base packages: [1] stats graphics grDevices utils datasets methods base other attached packages: [1] cobs_1.3-8 sfsmisc_1.1-19 Rmpfr_0.9-5 gmp_0.7-5 DPQ_0.5-9 loaded via a namespace (and not attached): [1] MASS_7.3-61 DPQmpfr_0.3-3 compiler_4.5.0 Matrix_1.7-0 quantreg_5.98 [6] tools_4.5.0 SparseM_1.84-2 survival_3.7-0 MatrixModels_0.5-3 splines_4.5.0 [11] grid_4.5.0 lattice_0.22-6 > > form(data.frame(k = k., cutoffs = rev(.stirl.cutoffs("R4.4_0")[-1])[k.], + t(n.mn ["n.",,]), cobs.80 = n.mn8["n.", "cobs",])) k cutoffs spl low cobs cobs.80 1 1 17400000.00 15600000.00 15800000.00 15800000.00 15700000.00 2 2 3700.00 NA NA NA NA 3 3 200.00 200.00 201.00 206.00 205.00 4 4 81.00 NA NA 86.60 86.20 5 5 36.00 27.20 27.10 27.00 27.10 6 6 25.00 23.50 NA NA NA 7 7 19.00 12.80 12.80 12.80 12.80 8 8 14.00 NA NA 12.30 12.80 9 9 11.00 8.91 8.95 8.89 8.86 10 10 9.50 9.69 NA 9.72 NA 11 11 8.80 7.26 7.31 7.32 7.29 12 12 8.25 8.16 NA 8.10 7.76 13 13 7.60 6.51 6.53 6.51 6.52 14 14 7.10 7.43 NA 7.47 7.43 15 15 6.50 6.10 6.10 6.08 6.09 > ## k cutoffs spl low cobs cobs.80 > ## 1 1 17400000.00 15600000.00 15800000.00 15800000.00 15700000.00 > ## 2 2 3700.00 NA NA NA NA == from tau=0.90 I'd use ~ 5000 > ## 3 3 200.00 200.00 201.00 206.00 205.00 205 > ## 4 4 81.00 NA NA 86.60 86.20 86 > ## 5 5 36.00 27.20 27.10 27.00 27.10 27 > ## 6 6 25.00 23.50 NA NA NA 23.5 > ## 7 7 19.00 12.80 12.80 12.80 12.80 12.8 > ## 8 8 14.00 NA NA 12.30 12.80 12.3 > ## 9 9 11.00 8.91 8.95 8.89 8.86 8.9 > ## 10 10 9.50 9.69 NA 9.72 NA --skip-- > ## 11 11 8.80 7.26 7.31 7.32 7.29 7.3 > ## 12 12 8.25 8.16 NA 8.10 7.76 --skip-- > ## 13 13 7.60 6.51 6.53 6.51 6.52 6.52 > ## 14 14 7.10 7.43 NA 7.47 7.43 --skip-- > ## 15 15 6.50 6.10 6.10 6.08 6.09 6.10 > ## 16 --skip-- > ## 17 5.25 or something all the way dow > > ##---- In the no-ldouble case --- this is very different: > ## k cutoffs spl low cobs cobs.80 > ## 1 17400000.00 8580000.00 8580000.00 8370000.00 8370000.00 > ## 2 3700.00 NA NA 5890.00 6180.00 > ## 3 200.00 159.00 159.00 160.00 159.00 > ## 4 81.00 87.10 87.10 84.80 86.00 > ## 5 36.00 23.50 23.50 23.50 23.50 > ## 6 25.00 23.70 23.70 NA NA > ## 7 19.00 11.50 11.50 11.50 11.50 > ## 8 14.00 NA NA 12.90 13.00 > ## 9 11.00 8.20 8.20 8.21 8.21 > ## 10 9.50 NA NA NA 9.69 > ## 11 8.80 6.84 6.84 6.85 6.83 > ## 12 8.25 NA NA 8.27 7.95 > ## 13 7.60 NA NA NA NA > ## 14 7.10 NA NA 7.43 7.40 > ## 15 6.50 NA NA NA NA > > > ## M1 macOS is similar to no-ldouble : "==" if equaln > ## M1 mac | aarch64-apple-darwin20 | macOS Ventura 13.3.1 | R-devel (2024-01-29 r85841) ; LAPACK version 3.12.0 > ## k cutoffs spl low cobs cobs.80 > ## 1 1 17400000.00 8580000.00 8580000.00 8370000.00 8370000.00 == > ## 2 2 3700.00 NA NA 5900.00 NA ~= > ## 3 3 200.00 159.00 159.00 160.00 159.00 == > ## 4 4 81.00 87.10 87.10 84.80 86.00 == > ## 5 5 36.00 23.50 23.50 23.50 23.50 > ## 6 6 25.00 23.70 23.70 NA NA == > ## 7 7 19.00 11.50 11.50 11.50 11.50 > ## 8 8 14.00 NA NA 12.90 13.00 == > ## 9 9 11.00 8.20 8.20 8.21 8.21 > ## 10 10 9.50 NA NA NA 9.69 == > ## 11 11 8.80 6.84 6.84 6.85 6.83 > ## 12 12 8.25 NA NA 8.27 7.95 == > ## 13 13 7.60 NA NA NA NA == > ## 14 14 7.10 NA NA 7.43 7.40 == > ## 15 15 6.50 NA NA NA NA == > > > > > ## ========== Try a slightly better direct formula ====================================================== > > ## after some trial error: gamma(n+1) = n*gamma(n) ==> lgamma(n+1) = lgamma(n) + log(n) > ## stirlerrD2 <- function(n) lgamma(n) + n*(1-(l.n <- log(n))) + (l.n - log(2*pi))/2 > > > palette("default") > > if(do.pdf) { dev.off(); pdf("stirlerr-tst_others.pdf") } > > n <- n5m # (1/64 ... 5000) -- goes with st.nM > p.stirlerrDev(n=n, stnM=st.nM, cex=1/4, type="o", cutoffs = cuts, abs=TRUE) > axis(1,at= 2:6, col=NA, col.axis=(cola <- "lightblue"), line=-3/4) > abline(v = 2:6, lty=3, col=cola) > i.n <- 1 <= n & n <= 15 ; cr2 <- adjustcolor(2, 0.6) > lines(n[i.n], abs(relErrV(st.nM[i.n], stirlerr_simpl(n[i.n], "MM2" ))), col=cr2, lwd=2) > legend(20, 1e-13, legend = quote( relErr(stirlerr_simpl(n, '"MM2"'))), col=cr2, lwd=3, bty="n") > > > n <- seq(1,6, by= 1/200) > stM <- stirlerr(mpfr(n, 512), use.halves = FALSE) > relE <- asNumeric(relErrV(stM, cbind(stD = stirlerr_simpl(n), stD2 = stirlerr_simpl(n, "MM2")))) > signif(apply(abs(relE), 2, summary), 4) stD stD2 Min. 5.526e-18 5.526e-18 1st Qu. 2.516e-15 1.674e-15 Median 8.959e-15 5.024e-15 Mean 1.651e-14 9.956e-15 3rd Qu. 2.294e-14 1.378e-14 Max. 1.208e-13 8.615e-14 > ## stD stD2 > ## Min. 3.093e-17 7.081e-18 > ## 1st Qu. 2.777e-15 1.936e-15 > ## Median 8.735e-15 5.238e-15 > ## Mean 1.670e-14 1.017e-14 .. well "67% better" > ## 3rd Qu. 2.380e-14 1.408e-14 > ## Max. 1.284e-13 9.382e-14 > > c2 <- adjustcolor(1:2, 0.6) > matplotB(n, abs19(relE), type="o", cex=3/4, log="y", ylim = c(8e-17, 1.3e-13), yaxt="n", col=c2) > eaxis(2) > abline(h = 2^-53, lty=1, col="gray") > smrelE <- apply(abs(relE), 2, \(y) lowess(n, y, f = 0.1)$y) > matlines(n, smrelE, lwd=3, lty=1) > legend("topleft", legend = expression(stirlerr_simpl(n), stirlerr_simple(n, "MM2")), + bty='n', col=c2, lwd=3, lty=1) > drawEps.h(-(53:48)) > > > ## should we e.g., use interpolation spline through sfserr_halves[] for n <= 7.5 > ## -- doing the interpolation on the log(1 - 12*x*stirlerr(x)) vs log2(x) scale -- maybe ? > stirL <- curve(1-12*x*stirlerr(x, cutoffs = cuts, verbose=TRUE), 1/64, 8, log="xy", n=2048) stirlerr(n, cutoffs = 5.22,6.5,7,7.9,8.75,10.5,13,20,26,60,200,3300) : case I (n <= 5.22), using direct formula for n= num [1:1907] 0.0156 0.0157 0.0157 0.0158 0.0158 ... case II (n > 5.22 ), 12 cutoffs: ( 5.22, 6.5, 7, 7.9, 8.75, 10.5, 13, 20, 26, 60, 200, 3300 ): n in cutoff intervals: (5.22,6.5] (6.5,7] (7,7.9] (7.9,8.75] (8.75,10.5] (10.5,13] (13,20] 72 25 39 5 0 0 0 (20,26] (26,60] (60,200] (200,3.3e+03] (3.3e+03,Inf] 0 0 0 0 0 > ## just need "true" values for x = 2^-(6,5,4,3,2) in addition to those we already have at x = 1/2, 1.5, 2, 2.5, ..., 7.5, 8 > xM <- mpfr(stirL$x, 2048) > stilEM <- roundMpfr(1 - 12*xM*stirlerr(xM, verbose=TRUE), 128) stirlerr(n): As 'n' is "mpfr", using "mpfr" & stirlerrM(): > relEsml <- relErrV(stilEM, stirL$y) > plot(stirL$x, relEsml) # again: stirlerr() is "limited.." for ~ [2, 6] > > ## The function to interpolate: > plot(asNumeric( (stilEM)) ~ x, data = stirL, type = "l", log="x") > plot(asNumeric(sqrt(stilEM)) ~ x, data = stirL, type = "l", log="x") > plot(asNumeric( log(stilEM)) ~ x, data = stirL, type = "l", log="x") > > y <- asNumeric( log(stilEM)) > x <- stirL$x > spl <- splinefun(log(x), y) > > plot(asNumeric(log(stilEM)) ~ x, data = stirL, type = "l", log="x") > lines(x, spl(log(x)), col=2) > summary(rE <- relErrV(target = y, current = spl(log(x)))) # all 0 {of course: interpolation at *these* x} Min. 1st Qu. Median Mean 3rd Qu. Max. 0 0 0 0 0 0 > > ssmpl <- smooth.spline(log(x), y) > str(ssmpl$fit)# only 174 knots, 170 coefficients List of 5 $ knot : num [1:174] 0 0 0 0 0.00586 ... $ nk : num 170 $ min : num -4.16 $ range: num 6.24 $ coef : num [1:170] -0.263 -0.265 -0.27 -0.277 -0.285 ... - attr(*, "class")= chr "smooth.spline.fit" > methods(class="smooth.spline.fit") #-> .. [1] predict see '?methods' for accessing help and source code > str(yp <- predict(ssmpl$fit, x=log(x))) List of 2 $ x: num [1:2048] -4.16 -4.16 -4.15 -4.15 -4.15 ... $ y: num [1:2048] -0.263 -0.263 -0.264 -0.264 -0.265 ... > lines(x, yp$y, col=adjustcolor(3, 1/3), lwd=3) # looks fine > ## but > summary(reSpl <- relErrV(target = y, current = yp$y)) Min. 1st Qu. Median Mean 3rd Qu. Max. -6.339e-10 -7.548e-11 1.207e-12 1.000e-16 7.316e-11 5.527e-10 > ## Min. 1st Qu. Median Mean 3rd Qu. Max. > ## -6.35e-10 -7.55e-11 1.10e-12 0.00e+00 7.32e-11 5.53e-10 > ## which is *NOT* good enough, of course .... > > > > showProc.time() Time (user system elapsed): 35.26 0.89 36.16 > > proc.time() user system elapsed 67.51 1.15 68.67