R Under development (unstable) (2023-11-02 r85465 ucrt) -- "Unsuffered Consequences" Copyright (C) 2023 The R Foundation for Statistical Computing Platform: x86_64-w64-mingw32/x64 R is free software and comes with ABSOLUTELY NO WARRANTY. You are welcome to redistribute it under certain conditions. Type 'license()' or 'licence()' for distribution details. R is a collaborative project with many contributors. Type 'contributors()' for more information and 'citation()' on how to cite R or R packages in publications. Type 'demo()' for some demos, 'help()' for on-line help, or 'help.start()' for an HTML browser interface to help. Type 'q()' to quit R. > #### Examples for non-central t > #### ========================== > ### > ### Johnson, Kotz, ...; 2nd ed, Vol 2. Chapter 31, around p.520 > > c({}## This stems from the two files + , "/u/maechler/R/MM/NUMERICS/dpq-functions/pnt-ex.R" + , "/u/maechler/R/MM/NUMERICS/dpq-functions/t-nonc-approx.R" + , "SEE ALSO ./pnt-prec.R (still current problems!)" + , " ==========" + ) [1] "/u/maechler/R/MM/NUMERICS/dpq-functions/pnt-ex.R" [2] "/u/maechler/R/MM/NUMERICS/dpq-functions/t-nonc-approx.R" [3] "SEE ALSO ./pnt-prec.R (still current problems!)" [4] " ==========" > ## and originally, had > ## source("/u/maechler/R/MM/NUMERICS/dpq-functions/t-nonc-fn.R") > ## - - - - - - - - - - - - - - - - - - - - - - - - - - > library(DPQ) ## source("/u/maechler/R/Pkgs/DPQ/R/t-nonc-fn.R") > > stopifnot(exprs = { + require(graphics) + require(sfsmisc) + }) Loading required package: sfsmisc > > source(system.file(package="DPQ", "test-tools.R", mustWork=TRUE)) > ## => showProc.time(), ... list_() , loadList() , readRDS_() , save2RDS() > relErrV <- sfsmisc::relErrV > > if(!dev.interactive(orNone=TRUE)) pdf("t-nonc_P-1.pdf") > > ### Part I -- former pnt-ex.R > ### --------------------------- > > ### Find value of delta after which pt(*,*, ncp=delta) === 0 : > ### MM{2014}: This is no longer true: pt(ncp -> Inf) now "goes all the way to underflow" > ### and pt(*, log.p=TRUE) "is ok" > > ## This would be from theory : > (dmax <- sqrt(-2*log(2* .Machine$double.xmin))) # 37.62189 [1] 37.62189 > .5*exp(-dmax^2/2) # 2.225074e-308 [1] 2.225074e-308 > > pt.pos <- function(delta,q,df) (pt(q=q,df=df,ncp=delta) > 0) - 1/2 > > ## see 'MM{2014}' above: this no longer works > ## d1 <- uniroot(pt.pos, q=10,df=2, lower=30, upper= 40, tol=1e-10)$root > ## d2 <- uniroot(pt.pos, q= 1,df=2, lower=30, upper= 40, tol=1e-10)$root > ## d1 == d2 # TRUE > d3 <- uniroot(pt.pos, q=.1, df=20, lower=30, upper= 40, tol=1e-10)$root > ## d4 <- uniroot(pt.pos, q=100,df=20, lower=30, upper= 40, tol=1e-10)$root > ## d3 != d4 # T !! > ## d1 == d4 # T > ## c(d1,d3) > ## 38.57550 38.56226 > d5 <- uniroot(pt.pos, q=.01,df= 2, lower=30, upper= 40, tol=1e-10)$root > d6 <- uniroot(pt.pos, q=.01,df=20, lower=30, upper= 40, tol=1e-10)$root > c(d3, d5, d6) [1] 37.62274 37.62189 37.62189 > ## unique(sort(c(d1,d2,d3,d4,d5,d6))) > ##[1] 38.45745 38.46036 38.56226 38.57550 > > > ###--- When DEBUG_pnt is compiled in: > pt(20,20,20) [1] 0.4607831 > pt(20,20,25) [1] 0.05962046 > pt( 1,20,25) [1] 5.728954e-126 > pt(30, 1,25) [1] 0.4049176 > pt(100, 1,25) [1] 0.802597 > ## p #{iter} result > pt(30,30,30) # 1.84694e-196 600 .4672569 [1] 0.4672569 > pt(35,35,35) # 4.93855e-267 786 .4694869 [1] 0.4694869 > pt(36,36,36) # 1.88862e-282 827 .4698806 [1] 0.4698806 > d <- 37 ;pt(d,d,d)# 2.65703e-298 868 .4702595 [1] 0.4702595 > d <- 38 ;pt(d,d,d)# 1.37516e-314 902 .4706245 -- no longer [1] 0.4777101 > d <- 38.5;pt(d,d,d)# 6.91692e-323 (799) s<0 => error-bound < 0 (!) wrong convergence => .4702131 [1] 0.4778479 > ##--- > pt(39,39,39)# p = 0 ==> result=0 --- no longer [1] 0.4779833 > > ## FIXME: > pntR1(40,40, 38.5, verbose=2) ## and then edit ---> gives the *.out file large 'df' or "large" 'ncp' ---> return()ing pnorm(*) = 0.6074858972535062 [1] 0.6074859 > ##--------- FIXME --- work via sink() or capture.output -- to get rr data.frame() > if(FALSE) { + t.file <- "/u/maechler/R/MM/NUMERICS/dpq-functions/pnt-40-40-38.5.out" + nr <- names(rr <- read.table(t.file, header= TRUE)) + summary(rr) + matplot(rr[,"it"], rr[,-1], type='l', log="y") + ##--> Graphic axis "buglet" + + par(mfrow=c(4,2)); f <- FF ~ it + for(i in 2:ncol(rr)) {f[[2]] <- as.name(nr[i]); plot(f, data=rr, type='l')} + + par(mfrow=c(4,2)); f <- FF ~ it + for(i in 2:ncol(rr)) {f[[2]] <- as.name(nr[i]); plot(f, data=rr, type='l', log='xy')} + } > ##----------- end{FIXME}--------------------------------------------- > > ###------------------------ dcdflib ptnc() ---- computations and plots > ## ======= ~~~~~ > ##==> ~/R/Pkgs/dcdflib/tests/ --------- > > ## How does density look in these extremes ? > ## ------- > curve(dt(x, df=10,ncp=10000), 0, 85000, col=2, log="y") # log-density Warning message: In xy.coords(x, y, xlabel, ylabel, log) : 2 y values <= 0 omitted from logarithmic plot > curve(dt(x, df=10,ncp=10000, log=TRUE), 1e3, 1e9, col=2, ylim = c(-30,-8),log="x") > ## is it *not* log concave -- or is that a numeric error in the tails? > ## ditto ? > curve(dt(x, df=10,ncp= 1e7, log=TRUE), 1e5, 1e11, col=2, ylim = c(-35,-15),log="x") > curve(dt(x, df=10,ncp= 1e7), 5e6, 3e7, col=2) > integrate(function(x) dt(x, df=10,ncp= 1e7), 5e6, 3e7) 1.004937 with absolute error < 7.9e-08 > ## 1.004937 with absolute error < 8e-08 hm... --> seems error in tails > > > yl <- c(-80,0) > curve(dt(x, df=10,ncp= 10, log=TRUE), 0, 600, col=2, n=5001, ylim=yl) There were 50 or more warnings (use warnings() to see the first 50) > curve(dt(x, df=10,ncp= 15, log=TRUE), 0, 800, col=2, n=5001, ylim=yl) There were 50 or more warnings (use warnings() to see the first 50) > > ## Log-log > curve(exp(dt(x, df=10,ncp= 15, log=TRUE)), 1, 1e7, log="xy", + col=2, n=5001, axes=FALSE, ylab = "", + main="dt(x, df=10,ncp= 15) -- Log-Log scale") There were 50 or more warnings (use warnings() to see the first 50) > eaxis(1) > op <- par(las=2) > eaxis(2, at = 10^pretty(log10(axTicks(2, log=TRUE)), 10), + at.small =FALSE) > par(op) > mtext(R.version.string, cex = .6, adj=1) > ## NB: Mathematica (e.g) ./pnt-ex.nb -- shows linear {in log-log} tail > > ## From: Jerry Lewis > ## To: Martin Maechler > ## Subject: R pnt.c function > ## Date: Wed, 6 Nov 2013 21:36:16 +0000 > > ## I noticed your comment in the code > > ## /*----------- DEBUGGING ------------- > ## * > ## * make CFLAGS='-DDEBUG_pnt -g' > > ## * -- Feb.3, 1999; M.Maechler: > ## - For 't > ncp > 20' (or so) the result is completely WRONG! > ## */ > > ## while chasing down accuracy problems for calculating 1-sided normal > ## tolerance limits. For example trying to reproduce values in Odeh & > ## Owen's Table 1.1 by > n <- c(102,104); round(qt(.995,n-1,-qnorm(.0001)*sqrt(n))/sqrt(n),3) [1] 4.577 4.592 Warning message: In qt(0.995, n - 1, -qnorm(1e-04) * sqrt(n)) : full precision may not have been achieved in 'pnt{final}' > ## the value R calculates for n==104 is clearly wrong, since the > ## k-factor fails to be decreasing in n (inaccurate for all larger n > ## as well). > > ## MM: graphically convincing is e.g. > n <- seq(80,120, by=0.5) > summary(ncp.n <- -qnorm(.0001)*sqrt(n)) Min. 1st Qu. Median Mean 3rd Qu. Max. 33.26 35.28 37.19 37.13 39.01 40.74 > qt995 <- qt(.995, n-1, ncp=ncp.n) / sqrt(n) There were 45 warnings (use warnings() to see them) > summary(warnings())# 45 identical warnings: 45 identical warnings: In qt(0.995, n - 1, ncp = ncp.n) : full precision may not have been achieved in 'pnt{final}' > ## full precision may not have been achieved in 'pnt{final}' > plot(n, qt995, type = "l", col=2, lwd=2) ## => visual kink in {102, 103}: > abline(v = 102:103, lty=3, col="gray") > > qtSc <- function(n, p = 0.995, delta = .0001) { + stopifnot(length(p) == 1, length(delta) == 1, n > 1, delta > 0, + 0 < p, p < 1) + ncp.n <- -qnorm(delta) * sqrt(n) + qt(p, n-1, ncp=ncp.n) / sqrt(n) + } > > n <- seq(80,120, by=.5); plot(n, qtSc(n), type = "l", col=2, lwd=1.5) There were 45 warnings (use warnings() to see them) > summary(warnings()) # --> "full precision may not have been achieved in 'pnt{final}'" 45 identical warnings: In qt(p, n - 1, ncp = ncp.n) : full precision may not have been achieved in 'pnt{final}' > ## It's the *smaller* n's which give the warning: > assign("last.warning", NULL, envir=as.environment(find("last.warning"))) # (non-API) > n <- seq(80,102, by=.5); plot(n, qtSc(n), type = "l", col=2, lwd=1.5) There were 45 warnings (use warnings() to see them) > try(# works "locally" but nut with the whole script + stopifnot(length(n) == length(warnings())) ## + ) > > ## These give *no* warnings [and we know they are inaccurate!]: > n <- seq(104,120, by=.5); plot(n, qtSc(n), type = "l", col=2, lwd=1.5) > > ## MM: the bug really is in pt(*,df, ncp), i.e., pnt.c: > n <- seq(80,120, by=0.25) > summary(ncp.n <- -qnorm(.0001)*sqrt(n)) Min. 1st Qu. Median Mean 3rd Qu. Max. 33.26 35.28 37.19 37.13 39.01 40.74 > qq <- 4.6*sqrt(n) > plot(qq, pt(qq, n-1, ncp=ncp.n), type = "l") ## - yes! > cbind(n, qq, ncp.n, pt = pt(qq, n-1, ncp=ncp.n))## it is clearly at n qq ncp.n pt [1,] 80.00 41.14365 33.26389 0.9898506 [2,] 80.25 41.20789 33.31583 0.9899534 [3,] 80.50 41.27202 33.36768 0.9900551 [4,] 80.75 41.33606 33.41946 0.9901557 [5,] 81.00 41.40000 33.47115 0.9902553 [6,] 81.25 41.46384 33.52276 0.9903539 [7,] 81.50 41.52758 33.57430 0.9904514 [8,] 81.75 41.59123 33.62575 0.9905479 [9,] 82.00 41.65477 33.67713 0.9906434 [10,] 82.25 41.71822 33.72842 0.9907379 [11,] 82.50 41.78157 33.77964 0.9908315 [12,] 82.75 41.84483 33.83079 0.9909240 [13,] 83.00 41.90799 33.88185 0.9910156 [14,] 83.25 41.97106 33.93284 0.9911062 [15,] 83.50 42.03403 33.98375 0.9911959 [16,] 83.75 42.09691 34.03459 0.9912847 [17,] 84.00 42.15970 34.08535 0.9913725 [18,] 84.25 42.22239 34.13603 0.9914594 [19,] 84.50 42.28499 34.18664 0.9915454 [20,] 84.75 42.34749 34.23718 0.9916305 [21,] 85.00 42.40990 34.28764 0.9917148 [22,] 85.25 42.47223 34.33802 0.9917981 [23,] 85.50 42.53446 34.38834 0.9918806 [24,] 85.75 42.59660 34.43857 0.9919623 [25,] 86.00 42.65865 34.48874 0.9920431 [26,] 86.25 42.72060 34.53883 0.9921230 [27,] 86.50 42.78247 34.58885 0.9922022 [28,] 86.75 42.84425 34.63880 0.9922805 [29,] 87.00 42.90594 34.68868 0.9923580 [30,] 87.25 42.96755 34.73848 0.9924347 [31,] 87.50 43.02906 34.78821 0.9925106 [32,] 87.75 43.09049 34.83788 0.9925857 [33,] 88.00 43.15182 34.88747 0.9926601 [34,] 88.25 43.21308 34.93699 0.9927337 [35,] 88.50 43.27424 34.98644 0.9928065 [36,] 88.75 43.33532 35.03582 0.9928786 [37,] 89.00 43.39631 35.08513 0.9929499 [38,] 89.25 43.45722 35.13437 0.9930205 [39,] 89.50 43.51804 35.18355 0.9930904 [40,] 89.75 43.57878 35.23265 0.9931595 [41,] 90.00 43.63943 35.28169 0.9932279 [42,] 90.25 43.70000 35.33066 0.9932957 [43,] 90.50 43.76048 35.37956 0.9933627 [44,] 90.75 43.82089 35.42839 0.9934291 [45,] 91.00 43.88120 35.47716 0.9934947 [46,] 91.25 43.94144 35.52586 0.9935597 [47,] 91.50 44.00159 35.57449 0.9936241 [48,] 91.75 44.06166 35.62305 0.9936877 [49,] 92.00 44.12165 35.67155 0.9937507 [50,] 92.25 44.18156 35.71999 0.9938131 [51,] 92.50 44.24138 35.76836 0.9938748 [52,] 92.75 44.30113 35.81666 0.9939359 [53,] 93.00 44.36079 35.86490 0.9939964 [54,] 93.25 44.42038 35.91307 0.9940562 [55,] 93.50 44.47988 35.96118 0.9941155 [56,] 93.75 44.53931 36.00922 0.9941741 [57,] 94.00 44.59865 36.05720 0.9942322 [58,] 94.25 44.65792 36.10512 0.9942896 [59,] 94.50 44.71711 36.15297 0.9943464 [60,] 94.75 44.77622 36.20076 0.9944027 [61,] 95.00 44.83525 36.24849 0.9944584 [62,] 95.25 44.89421 36.29615 0.9945135 [63,] 95.50 44.95309 36.34375 0.9945681 [64,] 95.75 45.01189 36.39129 0.9946221 [65,] 96.00 45.07061 36.43877 0.9946756 [66,] 96.25 45.12926 36.48619 0.9947285 [67,] 96.50 45.18783 36.53354 0.9947809 [68,] 96.75 45.24633 36.58083 0.9948327 [69,] 97.00 45.30475 36.62806 0.9948840 [70,] 97.25 45.36309 36.67524 0.9949348 [71,] 97.50 45.42136 36.72235 0.9949851 [72,] 97.75 45.47956 36.76940 0.9950349 [73,] 98.00 45.53768 36.81638 0.9950841 [74,] 98.25 45.59572 36.86331 0.9951329 [75,] 98.50 45.65370 36.91018 0.9951811 [76,] 98.75 45.71160 36.95700 0.9952289 [77,] 99.00 45.76942 37.00375 0.9952762 [78,] 99.25 45.82718 37.05044 0.9953230 [79,] 99.50 45.88486 37.09707 0.9953693 [80,] 99.75 45.94246 37.14365 0.9954152 [81,] 100.00 46.00000 37.19016 0.9954606 [82,] 100.25 46.05746 37.23662 0.9955055 [83,] 100.50 46.11486 37.28302 0.9955500 [84,] 100.75 46.17218 37.32937 0.9955941 [85,] 101.00 46.22943 37.37565 0.9956376 [86,] 101.25 46.28661 37.42188 0.9956808 [87,] 101.50 46.34372 37.46805 0.9957235 [88,] 101.75 46.40075 37.51417 0.9957658 [89,] 102.00 46.45772 37.56023 0.9958076 [90,] 102.25 46.51462 37.60623 0.9958490 [91,] 102.50 46.57145 37.65217 0.9949992 [92,] 102.75 46.62821 37.69806 0.9950460 [93,] 103.00 46.68490 37.74390 0.9950924 [94,] 103.25 46.74152 37.78967 0.9951383 [95,] 103.50 46.79808 37.83540 0.9951838 [96,] 103.75 46.85456 37.88106 0.9952288 [97,] 104.00 46.91098 37.92668 0.9952734 [98,] 104.25 46.96733 37.97223 0.9953176 [99,] 104.50 47.02361 38.01774 0.9953613 [100,] 104.75 47.07983 38.06318 0.9954047 [101,] 105.00 47.13597 38.10858 0.9954476 [102,] 105.25 47.19205 38.15392 0.9954901 [103,] 105.50 47.24807 38.19921 0.9955322 [104,] 105.75 47.30402 38.24444 0.9955739 [105,] 106.00 47.35990 38.28962 0.9956152 [106,] 106.25 47.41571 38.33474 0.9956561 [107,] 106.50 47.47147 38.37982 0.9956967 [108,] 106.75 47.52715 38.42484 0.9957368 [109,] 107.00 47.58277 38.46981 0.9957765 [110,] 107.25 47.63832 38.51472 0.9958159 [111,] 107.50 47.69382 38.55958 0.9958549 [112,] 107.75 47.74924 38.60439 0.9958935 [113,] 108.00 47.80460 38.64915 0.9959318 [114,] 108.25 47.85990 38.69386 0.9959697 [115,] 108.50 47.91513 38.73852 0.9960072 [116,] 108.75 47.97030 38.78312 0.9960444 [117,] 109.00 48.02541 38.82767 0.9960812 [118,] 109.25 48.08045 38.87217 0.9961177 [119,] 109.50 48.13543 38.91662 0.9961538 [120,] 109.75 48.19035 38.96102 0.9961896 [121,] 110.00 48.24521 39.00537 0.9962250 [122,] 110.25 48.30000 39.04967 0.9962601 [123,] 110.50 48.35473 39.09392 0.9962949 [124,] 110.75 48.40940 39.13812 0.9963293 [125,] 111.00 48.46401 39.18227 0.9963634 [126,] 111.25 48.51855 39.22637 0.9963972 [127,] 111.50 48.57304 39.27042 0.9964307 [128,] 111.75 48.62746 39.31442 0.9964638 [129,] 112.00 48.68182 39.35837 0.9964967 [130,] 112.25 48.73613 39.40227 0.9965292 [131,] 112.50 48.79037 39.44613 0.9965614 [132,] 112.75 48.84455 39.48993 0.9965933 [133,] 113.00 48.89867 39.53369 0.9966249 [134,] 113.25 48.95273 39.57740 0.9966562 [135,] 113.50 49.00673 39.62105 0.9966872 [136,] 113.75 49.06068 39.66467 0.9967180 [137,] 114.00 49.11456 39.70823 0.9967484 [138,] 114.25 49.16838 39.75175 0.9967785 [139,] 114.50 49.22215 39.79521 0.9968084 [140,] 114.75 49.27586 39.83864 0.9968380 [141,] 115.00 49.32950 39.88201 0.9968672 [142,] 115.25 49.38309 39.92534 0.9968963 [143,] 115.50 49.43663 39.96861 0.9969250 [144,] 115.75 49.49010 40.01185 0.9969535 [145,] 116.00 49.54352 40.05503 0.9969817 [146,] 116.25 49.59687 40.09817 0.9970096 [147,] 116.50 49.65018 40.14127 0.9970373 [148,] 116.75 49.70342 40.18431 0.9970647 [149,] 117.00 49.75661 40.22731 0.9970918 [150,] 117.25 49.80974 40.27027 0.9971187 [151,] 117.50 49.86281 40.31318 0.9971454 [152,] 117.75 49.91583 40.35604 0.9971718 [153,] 118.00 49.96879 40.39886 0.9971979 [154,] 118.25 50.02170 40.44163 0.9972238 [155,] 118.50 50.07454 40.48436 0.9972494 [156,] 118.75 50.12734 40.52704 0.9972749 [157,] 119.00 50.18008 40.56968 0.9973000 [158,] 119.25 50.23276 40.61227 0.9973250 [159,] 119.50 50.28539 40.65482 0.9973497 [160,] 119.75 50.33796 40.69733 0.9973741 [161,] 120.00 50.39048 40.73978 0.9973984 > ## ncp = > ## even simpler -- > curve(pt(37,37,x), 37, 38) > > ## When x>0 and ncp>0, a cheap fix would substitute pf(x^2,1,df,ncp^2) > ## for pt(x,df,ncp) whenever pnorm(-ncp) is negligible > ## (as it should be when ncp>20). > ## MM: According to > ## https://en.wikipedia.org/wiki/Noncentral_t-distribution#Related_distributions > ## If T is noncentral t-distributed with ν degrees of freedom and > ## noncentrality parameter μ and F = T^2, then F has a noncentral > ## F-distribution with 1 numerator degree of freedom, ν denominator degrees of freedom, > ## and noncentrality parameter μ^2. > ## MM: If t >= 0, > ## pf(t^2, df1=1, df2=nu, ncp=mu^2) = P(T^2 <= t^2) = P(|T| <= t) = > ## ================================ > ## = P(-t <= T <= t) = pt(t, df, ncp) - pt(-t, df, ncp) > ## ================================ > ## and now, indeed, if pt(-t, df,ncp) << pt(t, df, ncp) is negligble, we can use pf(...) > ## HOWEVER, 1) this not only depends on ncp !! > ## 2) Code below suggests that noncentral pf() is not quite ok for small t^2 > > ## MM: let's see: > ptq <- pt(qq, df= n-1, ncp = ncp.n) > pfq2 <- pf(qq^2, df1=1, df2= n-1, ncp = ncp.n^2) > cbind(n, qq, ncp.n, pt = ptq, "pf.q^2" = pfq2, rel.E = 1 - ptq/pfq2) n qq ncp.n pt pf.q^2 rel.E [1,] 80.00 41.14365 33.26389 0.9898506 0.9898506 -9.752095e-10 [2,] 80.25 41.20789 33.31583 0.9899534 0.9899534 -9.643182e-10 [3,] 80.50 41.27202 33.36768 0.9900551 0.9900551 -9.533394e-10 [4,] 80.75 41.33606 33.41946 0.9901557 0.9901557 -9.419696e-10 [5,] 81.00 41.40000 33.47115 0.9902553 0.9902553 -9.310510e-10 [6,] 81.25 41.46384 33.52276 0.9903539 0.9903539 -9.199472e-10 [7,] 81.50 41.52758 33.57430 0.9904514 0.9904514 -9.092975e-10 [8,] 81.75 41.59123 33.62575 0.9905479 0.9905479 -8.987620e-10 [9,] 82.00 41.65477 33.67713 0.9906434 0.9906434 -8.875003e-10 [10,] 82.25 41.71822 33.72842 0.9907379 0.9907379 -8.775953e-10 [11,] 82.50 41.78157 33.77964 0.9908315 0.9908315 -8.668604e-10 [12,] 82.75 41.84483 33.83079 0.9909240 0.9909240 -8.562548e-10 [13,] 83.00 41.90799 33.88185 0.9910156 0.9910156 -8.457022e-10 [14,] 83.25 41.97106 33.93284 0.9911062 0.9911062 -8.360073e-10 [15,] 83.50 42.03403 33.98375 0.9911959 0.9911959 -8.260310e-10 [16,] 83.75 42.09691 34.03459 0.9912847 0.9912847 -8.153072e-10 [17,] 84.00 42.15970 34.08535 0.9913725 0.9913725 -8.057364e-10 [18,] 84.25 42.22239 34.13603 0.9914594 0.9914594 -7.959433e-10 [19,] 84.50 42.28499 34.18664 0.9915454 0.9915454 -9.963537e-10 [20,] 84.75 42.34749 34.23718 0.9916305 0.9916305 -9.838161e-10 [21,] 85.00 42.40990 34.28764 0.9917148 0.9917148 -9.714383e-10 [22,] 85.25 42.47223 34.33802 0.9917981 0.9917981 -9.596544e-10 [23,] 85.50 42.53446 34.38834 0.9918806 0.9918806 -9.470369e-10 [24,] 85.75 42.59660 34.43857 0.9919623 0.9919623 -9.343006e-10 [25,] 86.00 42.65865 34.48874 0.9920431 0.9920431 -9.230285e-10 [26,] 86.25 42.72060 34.53883 0.9921230 0.9921230 -9.102628e-10 [27,] 86.50 42.78247 34.58885 0.9922022 0.9922022 -8.988223e-10 [28,] 86.75 42.84425 34.63880 0.9922805 0.9922805 -8.867860e-10 [29,] 87.00 42.90594 34.68868 0.9923580 0.9923580 -8.752421e-10 [30,] 87.25 42.96755 34.73848 0.9924347 0.9924347 -8.637333e-10 [31,] 87.50 43.02906 34.78821 0.9925106 0.9925106 -8.520205e-10 [32,] 87.75 43.09049 34.83788 0.9925857 0.9925857 -8.417740e-10 [33,] 88.00 43.15182 34.88747 0.9926601 0.9926601 -8.301733e-10 [34,] 88.25 43.21308 34.93699 0.9927337 0.9927337 -8.198513e-10 [35,] 88.50 43.27424 34.98644 0.9928065 0.9928065 -8.080541e-10 [36,] 88.75 43.33532 35.03582 0.9928786 0.9928786 -7.981547e-10 [37,] 89.00 43.39631 35.08513 0.9929499 0.9929499 -9.924599e-10 [38,] 89.25 43.45722 35.13437 0.9930205 0.9930205 -9.784080e-10 [39,] 89.50 43.51804 35.18355 0.9930904 0.9930904 -9.656693e-10 [40,] 89.75 43.57878 35.23265 0.9931595 0.9931595 -9.520009e-10 [41,] 90.00 43.63943 35.28169 0.9932279 0.9932279 -9.386070e-10 [42,] 90.25 43.70000 35.33066 0.9932957 0.9932957 -9.261794e-10 [43,] 90.50 43.76048 35.37956 0.9933627 0.9933627 -9.137004e-10 [44,] 90.75 43.82089 35.42839 0.9934291 0.9934291 -9.010899e-10 [45,] 91.00 43.88120 35.47716 0.9934947 0.9934947 -8.879895e-10 [46,] 91.25 43.94144 35.52586 0.9935597 0.9935597 -8.759125e-10 [47,] 91.50 44.00159 35.57449 0.9936241 0.9936241 -8.632199e-10 [48,] 91.75 44.06166 35.62305 0.9936877 0.9936877 -8.513157e-10 [49,] 92.00 44.12165 35.67155 0.9937507 0.9937507 -8.385141e-10 [50,] 92.25 44.18156 35.71999 0.9938131 0.9938131 -8.277177e-10 [51,] 92.50 44.24138 35.76836 0.9938748 0.9938748 -8.158045e-10 [52,] 92.75 44.30113 35.81666 0.9939359 0.9939359 -8.037269e-10 [53,] 93.00 44.36079 35.86490 0.9939964 0.9939964 -9.945353e-10 [54,] 93.25 44.42038 35.91307 0.9940562 0.9940562 -9.806602e-10 [55,] 93.50 44.47988 35.96118 0.9941155 0.9941155 -9.663741e-10 [56,] 93.75 44.53931 36.00922 0.9941741 0.9941741 -9.529899e-10 [57,] 94.00 44.59865 36.05720 0.9942322 0.9942322 -9.383374e-10 [58,] 94.25 44.65792 36.10512 0.9942896 0.9942896 -9.250216e-10 [59,] 94.50 44.71711 36.15297 0.9943464 0.9943464 -9.113803e-10 [60,] 94.75 44.77622 36.20076 0.9944027 0.9944027 -8.985728e-10 [61,] 95.00 44.83525 36.24849 0.9944584 0.9944584 -8.854284e-10 [62,] 95.25 44.89421 36.29615 0.9945135 0.9945135 -8.726022e-10 [63,] 95.50 44.95309 36.34375 0.9945681 0.9945681 -8.590253e-10 [64,] 95.75 45.01189 36.39129 0.9946221 0.9946221 -8.464482e-10 [65,] 96.00 45.07061 36.43877 0.9946756 0.9946756 -8.342309e-10 [66,] 96.25 45.12926 36.48619 0.9947285 0.9947285 -8.217378e-10 [67,] 96.50 45.18783 36.53354 0.9947809 0.9947809 -8.092742e-10 [68,] 96.75 45.24633 36.58083 0.9948327 0.9948327 -9.965657e-10 [69,] 97.00 45.30475 36.62806 0.9948840 0.9948840 -9.811714e-10 [70,] 97.25 45.36309 36.67524 0.9949348 0.9949348 -9.663432e-10 [71,] 97.50 45.42136 36.72235 0.9949851 0.9949851 -9.520003e-10 [72,] 97.75 45.47956 36.76940 0.9950349 0.9950349 -9.374992e-10 [73,] 98.00 45.53768 36.81638 0.9950841 0.9950841 -9.235404e-10 [74,] 98.25 45.59572 36.86331 0.9951329 0.9951329 -9.094012e-10 [75,] 98.50 45.65370 36.91018 0.9951811 0.9951811 -8.952707e-10 [76,] 98.75 45.71160 36.95700 0.9952289 0.9952289 -8.817342e-10 [77,] 99.00 45.76942 37.00375 0.9952762 0.9952762 -8.681562e-10 [78,] 99.25 45.82718 37.05044 0.9953230 0.9953230 -8.550898e-10 [79,] 99.50 45.88486 37.09707 0.9953693 0.9953693 -8.419550e-10 [80,] 99.75 45.94246 37.14365 0.9954152 0.9954152 -8.288359e-10 [81,] 100.00 46.00000 37.19016 0.9954606 0.9954606 -8.162051e-10 [82,] 100.25 46.05746 37.23662 0.9955055 0.9955055 -8.035057e-10 [83,] 100.50 46.11486 37.28302 0.9955500 0.9955500 -9.850514e-10 [84,] 100.75 46.17218 37.32937 0.9955941 0.9955941 -9.694627e-10 [85,] 101.00 46.22943 37.37565 0.9956376 0.9956376 -9.543191e-10 [86,] 101.25 46.28661 37.42188 0.9956808 0.9956808 -9.394150e-10 [87,] 101.50 46.34372 37.46805 0.9957235 0.9957235 -9.246874e-10 [88,] 101.75 46.40075 37.51417 0.9957658 0.9957658 -9.098120e-10 [89,] 102.00 46.45772 37.56023 0.9958076 0.9958076 -8.962489e-10 [90,] 102.25 46.51462 37.60623 0.9958490 0.9958490 -8.816146e-10 [91,] 102.50 46.57145 37.65217 0.9949992 0.9958901 8.945090e-04 [92,] 102.75 46.62821 37.69806 0.9950460 0.9959306 8.882341e-04 [93,] 103.00 46.68490 37.74390 0.9950924 0.9959708 8.819991e-04 [94,] 103.25 46.74152 37.78967 0.9951383 0.9960106 8.758038e-04 [95,] 103.50 46.79808 37.83540 0.9951838 0.9960500 8.696480e-04 [96,] 103.75 46.85456 37.88106 0.9952288 0.9960890 8.635314e-04 [97,] 104.00 46.91098 37.92668 0.9952734 0.9961275 8.574540e-04 [98,] 104.25 46.96733 37.97223 0.9953176 0.9961657 8.514155e-04 [99,] 104.50 47.02361 38.01774 0.9953613 0.9962036 8.454157e-04 [100,] 104.75 47.07983 38.06318 0.9954047 0.9962410 8.394545e-04 [101,] 105.00 47.13597 38.10858 0.9954476 0.9962780 8.335315e-04 [102,] 105.25 47.19205 38.15392 0.9954901 0.9963147 8.276467e-04 [103,] 105.50 47.24807 38.19921 0.9955322 0.9963510 8.217998e-04 [104,] 105.75 47.30402 38.24444 0.9955739 0.9963870 8.159906e-04 [105,] 106.00 47.35990 38.28962 0.9956152 0.9964226 8.102189e-04 [106,] 106.25 47.41571 38.33474 0.9956561 0.9964578 8.044846e-04 [107,] 106.50 47.47147 38.37982 0.9956967 0.9964926 7.987874e-04 [108,] 106.75 47.52715 38.42484 0.9957368 0.9965272 7.931271e-04 [109,] 107.00 47.58277 38.46981 0.9957765 0.9965613 7.875034e-04 [110,] 107.25 47.63832 38.51472 0.9958159 0.9965952 7.819165e-04 [111,] 107.50 47.69382 38.55958 0.9958549 0.9966287 7.763659e-04 [112,] 107.75 47.74924 38.60439 0.9958935 0.9966618 7.708514e-04 [113,] 108.00 47.80460 38.64915 0.9959318 0.9966946 7.653730e-04 [114,] 108.25 47.85990 38.69386 0.9959697 0.9967271 7.599303e-04 [115,] 108.50 47.91513 38.73852 0.9960072 0.9967593 7.545232e-04 [116,] 108.75 47.97030 38.78312 0.9960444 0.9967911 7.491515e-04 [117,] 109.00 48.02541 38.82767 0.9960812 0.9968227 7.438150e-04 [118,] 109.25 48.08045 38.87217 0.9961177 0.9968539 7.385136e-04 [119,] 109.50 48.13543 38.91662 0.9961538 0.9968848 7.332469e-04 [120,] 109.75 48.19035 38.96102 0.9961896 0.9969153 7.280149e-04 [121,] 110.00 48.24521 39.00537 0.9962250 0.9969456 7.228173e-04 [122,] 110.25 48.30000 39.04967 0.9962601 0.9969756 7.176538e-04 [123,] 110.50 48.35473 39.09392 0.9962949 0.9970053 7.125245e-04 [124,] 110.75 48.40940 39.13812 0.9963293 0.9970347 7.074292e-04 [125,] 111.00 48.46401 39.18227 0.9963634 0.9970637 7.023675e-04 [126,] 111.25 48.51855 39.22637 0.9963972 0.9970925 6.973393e-04 [127,] 111.50 48.57304 39.27042 0.9964307 0.9971210 6.923444e-04 [128,] 111.75 48.62746 39.31442 0.9964638 0.9971493 6.873826e-04 [129,] 112.00 48.68182 39.35837 0.9964967 0.9971772 6.824538e-04 [130,] 112.25 48.73613 39.40227 0.9965292 0.9972049 6.775578e-04 [131,] 112.50 48.79037 39.44613 0.9965614 0.9972322 6.726943e-04 [132,] 112.75 48.84455 39.48993 0.9965933 0.9972594 6.678633e-04 [133,] 113.00 48.89867 39.53369 0.9966249 0.9972862 6.630644e-04 [134,] 113.25 48.95273 39.57740 0.9966562 0.9973128 6.582973e-04 [135,] 113.50 49.00673 39.62105 0.9966872 0.9973391 6.535623e-04 [136,] 113.75 49.06068 39.66467 0.9967180 0.9973651 6.488590e-04 [137,] 114.00 49.11456 39.70823 0.9967484 0.9973909 6.441871e-04 [138,] 114.25 49.16838 39.75175 0.9967785 0.9974164 6.395466e-04 [139,] 114.50 49.22215 39.79521 0.9968084 0.9974417 6.349371e-04 [140,] 114.75 49.27586 39.83864 0.9968380 0.9974667 6.303586e-04 [141,] 115.00 49.32950 39.88201 0.9968672 0.9974915 6.258109e-04 [142,] 115.25 49.38309 39.92534 0.9968963 0.9975160 6.212937e-04 [143,] 115.50 49.43663 39.96861 0.9969250 0.9975403 6.168070e-04 [144,] 115.75 49.49010 40.01185 0.9969535 0.9975643 6.123505e-04 [145,] 116.00 49.54352 40.05503 0.9969817 0.9975881 6.079240e-04 [146,] 116.25 49.59687 40.09817 0.9970096 0.9976117 6.035272e-04 [147,] 116.50 49.65018 40.14127 0.9970373 0.9976350 5.991603e-04 [148,] 116.75 49.70342 40.18431 0.9970647 0.9976581 5.948230e-04 [149,] 117.00 49.75661 40.22731 0.9970918 0.9976810 5.905150e-04 [150,] 117.25 49.80974 40.27027 0.9971187 0.9977036 5.862362e-04 [151,] 117.50 49.86281 40.31318 0.9971454 0.9977260 5.819864e-04 [152,] 117.75 49.91583 40.35604 0.9971718 0.9977482 5.777654e-04 [153,] 118.00 49.96879 40.39886 0.9971979 0.9977702 5.735731e-04 [154,] 118.25 50.02170 40.44163 0.9972238 0.9977919 5.694093e-04 [155,] 118.50 50.07454 40.48436 0.9972494 0.9978135 5.652739e-04 [156,] 118.75 50.12734 40.52704 0.9972749 0.9978348 5.611665e-04 [157,] 119.00 50.18008 40.56968 0.9973000 0.9978559 5.570870e-04 [158,] 119.25 50.23276 40.61227 0.9973250 0.9978768 5.530356e-04 [159,] 119.50 50.28539 40.65482 0.9973497 0.9978975 5.490117e-04 [160,] 119.75 50.33796 40.69733 0.9973741 0.9979180 5.450154e-04 [161,] 120.00 50.39048 40.73978 0.9973984 0.9979383 5.410464e-04 > lines(qq, pfq2, col=adjustcolor("blue", 0.4), lwd=3) > ##==> he is right that pf() is good *here* > > chk.t.F <- function(x, df, ncp, tolerance = 1e-9) { + pt. <- pt(x, df= df, ncp = ncp) + pf. <- pf(x^2, df1=1, df2= df, ncp = ncp^2) + re <- relErrV(pf., pt.) + if(any(abs(re) > tolerance)) + warning("rel.error too large: ", max(abs(re))) + invisible(re) + } > chk.t.F(x=qq, df=n-1, ncp=ncp.n) ## too large: 0.00894 Warning message: In chk.t.F(x = qq, df = n - 1, ncp = ncp.n) : rel.error too large: 0.000894508963078855 > > ## many are not so good.. > ## these look good: > cols <- c(palette()[3], adjustcolor(2, 0.4)) > curve(pt(x, df=11.282, ncp=30), 0, 50, n=512, col=3) > curve(pf(x^2, df1=1, df2=11.282, ncp=30^2), n=512, col=cols[2], lwd=4, add=TRUE) > legend("top", c("pt()", "pf()"), col=cols, lwd=c(1,4), bty="n") > ## but not if we look at the left tail in log scale: > ## ========== > curve(pt(x, df=11.282, ncp=30), 0, 30, n=512, col=3, log="y",yaxt="n"); eaxis(2) > curve(pf(x^2, df1=1, df2=11.282, ncp=30^2), n=512, col=adjustcolor(2, 0.4), lwd=4, add=TRUE) > legend("topleft", c("pt()", "pf()"), col=cols, lwd=c(1,4), bty="n") > ## pf() and pt() very much differ when x gets small even for this > ## large ncp > > ## The difference should *only* be "the other" pt() part : pt(-x, df, ncp) > ## but that looks quite bad : > ## Note that it should be monotone !!! > pxy <- curve(pt(x, df=11.282, ncp=30, log=TRUE), -25, 30, n=512, col=3) > curve(pt(x, df=5, ncp=10, log=TRUE), -20, 20, n=512, col=2, lwd=2, type="o") > > ## If we look at the corresponding *density* function, it's not better: > curve(dt(x, df=11.282, ncp=30, log=TRUE), -25, 30, n=512, col=3) > > ## What about negative ncp ? [the same phenomenon} > ## all fine "regular scale": > curve(dt(x, df=11, ncp=-10), -25, 30, n=512, col=2, lwd=2) There were 50 or more warnings (use warnings() to see the first 50) > ## "all bad" on log scale: > curve(dt(x, df=11, ncp=-10, log=TRUE), -40, 30, n=512, col=2, lwd=2) There were 50 or more warnings (use warnings() to see the first 50) > > > > > ## [Jerry, from his e-mail:] ---------------------------------------------------- > > ## That alone would improve the minimum accuracy from 2 to 6 significant > ## figures when calculating tolerance factors for the cases that Odeh and > ## Owen tabulated. > > ## The series expansion used by Lenth (basis of R's pnt.c function) can > ## be implemented in a way that can accurately handle these cases, but I > ## have not digested R's c code to see where the problem lies. The > ## boundary where Lenth's p0 crosses from a normalized to a denormal binary representation is > ## between the cases of n==102 and n==104, so I suspect that p0 is not > ## factored out of the summation, which would accumulate reduced accuracy > ## terms. > > ## Jerry > > if(!dev.interactive(orNone=TRUE)) { dev.off(); pdf("t-nonc_P-2.pdf") } > > ### Part II -- former t-nonc-fn.R > ### ------------------------------ > > ### b_nu ----------------------------------- > mult.fig(2, main = "b(nu) = E[ Chi_nu ] / sqrt(nu)")$old.par -> opar > plot(b_chi, n=1024, col=2) > > ## Unfortunately, the above switch gives a small kink at nu = 300 > ## 2015-01-02: b_chi() improved: no kink anymore > ## ================================= ! > > curve(b_chi, 200,1200) > curve(b_chiAsymp, add=TRUE, col='red') > > plot(b_chi,1,10, col='red'); curve(b_chiAsymp, add=TRUE, col='blue') > > ## New cutoff > plot(b_chi,340.9, 341.1, n=1001)# no jump/kink visible > plot(b_chi,999.9, 1000.1, n=1001)# no jump/kink visible > > nu <- seq(200, 1000, length=1001) > plot(nu, b_chi(nu) - b_chiAsymp(nu), type='l')# b_chi(nu) > b_chiAsymp() ALWAYS > ## with 'one.minus=TRUE' the reverse: b_chi(*, one.minus=TRUE) < b_chiAsymp(*, ..TRUE) > plot(nu, b_chi(nu,one.minus=TRUE) - b_chiAsymp(nu,one.minus=TRUE), type='l') > > par(opar)# reset > > ### ----------- pntJW39() and pntR() ---------------------- > > pntR(30,30,30, verbose=2)# 600 iter pnt(t= 30 , df= 30 , delta= 30 ) ==> x= 0.9677419 : p= 1.846942e-196 it 1e5*(godd, geven) p q s pnt(*) D(pnt) errbd 1 1.817e-16 4.789e-16|8.311e-194 1.326e-192 0.5000 1.41399952298972e-192 1.409e-192 1.000 2 1.160e-15 2.626e-15|1.870e-191 2.387e-190 0.5000 2.58844956407531e-190 2.574e-190 1.000 3 5.615e-15 1.144e-14|2.805e-189 3.069e-188 0.5000 3.37578311922961e-188 3.350e-188 1.000 4 2.234e-14 4.206e-14|3.156e-187 3.069e-186 0.5000 3.41871948547031e-186 3.385e-186 1.000 5 7.665e-14 1.357e-13|2.840e-185 2.511e-184 0.5000 2.82952041640782e-184 2.795e-184 1.000 6 2.339e-13 3.939e-13|2.130e-183 1.739e-182 0.5000 1.97991111996521e-182 1.952e-182 1.000 7 6.490e-13 1.048e-12|1.369e-181 1.043e-180 0.5000 1.19989757117047e-180 1.180e-180 1.000 8 1.663e-12 2.593e-12|7.703e-180 5.523e-179 0.5000 6.41287877005700e-179 6.293e-179 1.000 9 3.980e-12 6.021e-12|3.851e-178 2.616e-177 0.5000 3.06524073653017e-177 3.001e-177 1.000 10 8.987e-12 1.324e-11|1.733e-176 1.121e-175 0.5000 1.32509594067733e-175 1.294e-175 1.000 11 1.929e-11 2.777e-11|7.090e-175 4.387e-174 0.5000 5.22854871534360e-174 5.096e-174 1.000 12 3.957e-11 5.581e-11|2.659e-173 1.579e-172 0.5000 1.89749429768168e-172 1.845e-172 1.000 13 7.800e-11 1.080e-10|9.203e-172 5.264e-171 0.5000 6.37453112790375e-171 6.185e-171 1.000 14 1.484e-10 2.021e-10|2.958e-170 1.634e-169 0.5000 1.99336083350716e-169 1.930e-169 1.000 15 2.733e-10 3.667e-10|8.874e-169 4.743e-168 0.5000 5.83007290866416e-168 5.631e-168 1.000 16 4.888e-10 6.472e-10|2.496e-167 1.294e-166 0.5000 1.60151967506007e-166 1.543e-166 1.000 17 8.515e-10 1.113e-09|6.607e-166 3.326e-165 0.5000 4.14730609835969e-165 3.987e-165 1.000 18 1.448e-09 1.871e-09|1.652e-164 8.091e-164 0.5000 1.01578548093631e-163 9.743e-164 1.000 19 2.407e-09 3.079e-09|3.912e-163 1.867e-162 0.5000 2.36002253506348e-162 2.258e-162 1.000 20 3.920e-09 4.966e-09|8.802e-162 4.099e-161 0.5000 5.21503039115181e-161 4.979e-161 1.000 21 6.263e-09 7.864e-09|1.886e-160 8.579e-160 0.5000 1.09865882516043e-159 1.047e-159 1.000 22 9.832e-09 1.224e-08|3.858e-159 1.716e-158 0.5000 2.21145625114537e-158 2.102e-158 1.000 23 1.518e-08 1.876e-08|7.548e-158 3.286e-157 0.5000 4.26152772361481e-157 4.040e-157 1.000 24 2.309e-08 2.832e-08|1.415e-156 6.035e-156 0.5000 7.87614637622144e-156 7.450e-156 1.000 25 3.461e-08 4.216e-08|2.548e-155 1.065e-154 0.5000 1.39846138582248e-154 1.320e-154 1.000 26 5.119e-08 6.196e-08|4.409e-154 1.808e-153 0.5000 2.38916524539328e-153 2.249e-153 1.000 27 7.476e-08 8.994e-08|7.349e-153 2.959e-152 0.5000 3.93298171979182e-152 3.694e-152 1.000 28 1.079e-07 1.291e-07|1.181e-151 4.672e-151 0.5000 6.24675260036192e-151 5.853e-151 1.000 29 1.540e-07 1.832e-07|1.833e-150 7.127e-150 0.5000 9.58473524022436e-150 8.960e-150 1.000 30 2.174e-07 2.573e-07|2.749e-149 1.052e-148 0.5000 1.42232990753580e-148 1.326e-148 1.000 31 3.039e-07 3.580e-07|3.990e-148 1.502e-147 0.5000 2.04354207968611e-147 1.901e-147 1.000 32 4.208e-07 4.934e-07|5.612e-147 2.080e-146 0.5000 2.84556915746517e-146 2.641e-146 1.000 33 5.774e-07 6.741e-07|7.652e-146 2.794e-145 0.5000 3.84387863311175e-145 3.559e-145 1.000 34 7.855e-07 9.133e-07|1.013e-144 3.644e-144 0.5000 5.04165934136426e-144 4.657e-144 1.000 35 1.060e-06 1.228e-06|1.302e-143 4.620e-143 0.5000 6.42608953166429e-143 5.922e-143 1.000 36 1.419e-06 1.637e-06|1.628e-142 5.696e-142 0.5000 7.96590760762411e-142 7.323e-142 1.000 37 1.886e-06 2.168e-06|1.980e-141 6.835e-141 0.5000 9.61094392441889e-141 8.814e-141 1.000 38 2.489e-06 2.852e-06|2.344e-140 7.989e-140 0.5000 1.12940302080014e-139 1.033e-139 1.000 39 3.262e-06 3.726e-06|2.705e-139 9.101e-139 0.5000 1.29353416772310e-138 1.181e-138 1.000 40 4.248e-06 4.837e-06|3.043e-138 1.011e-137 0.5000 1.44488122644401e-137 1.316e-137 1.000 41 5.498e-06 6.241e-06|3.340e-137 1.097e-136 0.5000 1.57498807211009e-136 1.430e-136 1.000 42 7.074e-06 8.006e-06|3.578e-136 1.161e-135 0.5000 1.67635580748714e-135 1.519e-135 1.000 43 9.049e-06 1.021e-05|3.745e-135 1.201e-134 0.5000 1.74317148603647e-134 1.576e-134 1.000 44 1.151e-05 1.296e-05|3.830e-134 1.215e-133 0.5000 1.77185913095305e-133 1.598e-133 1.000 45 1.457e-05 1.636e-05|3.830e-133 1.201e-132 0.5000 1.76138133463989e-132 1.584e-132 1.000 46 1.834e-05 2.054e-05|3.747e-132 1.162e-131 0.5000 1.71325928065278e-131 1.537e-131 1.000 47 2.298e-05 2.568e-05|3.587e-131 1.101e-130 0.5000 1.63132211979109e-130 1.460e-130 1.000 48 2.866e-05 3.195e-05|3.363e-130 1.022e-129 0.5000 1.52123500867858e-129 1.358e-129 1.000 49 3.558e-05 3.958e-05|3.089e-129 9.289e-129 0.5000 1.38988220302377e-128 1.238e-128 1.000 50 4.398e-05 4.882e-05|2.780e-128 8.277e-128 0.5000 1.24469373529882e-127 1.106e-127 1.000 51 5.413e-05 5.996e-05|2.453e-127 7.233e-127 0.5000 1.09300109545521e-126 9.685e-127 1.000 52 6.636e-05 7.336e-05|2.123e-126 6.199e-126 0.5000 9.41491621170714e-126 8.322e-126 1.000 53 8.102e-05 8.940e-05|1.802e-125 5.214e-125 0.5000 7.95807587179735e-125 7.017e-125 1.000 54 9.855e-05 0.0001085|1.502e-124 4.305e-124 0.5000 6.60309545641147e-124 5.807e-124 1.000 55 0.0001194 0.0001313|1.229e-123 3.491e-123 0.5000 5.37999107398021e-123 4.720e-123 1.000 56 0.0001442 0.0001583|9.874e-123 2.780e-122 0.5000 4.30577538123675e-122 3.768e-122 1.000 57 0.0001735 0.0001901|7.795e-122 2.176e-121 0.5000 3.38605037443983e-121 2.955e-121 1.000 58 0.0002081 0.0002277|6.048e-121 1.674e-120 0.5000 2.61721485461610e-120 2.279e-120 1.000 59 0.0002488 0.0002717|4.613e-120 1.266e-119 0.5000 1.98891616211367e-119 1.727e-119 1.000 60 0.0002965 0.0003233|3.460e-119 9.416e-119 0.5000 1.48644106298172e-118 1.288e-118 1.000 61 0.0003523 0.0003835|2.552e-118 6.890e-118 0.5000 1.09282840007157e-117 9.442e-118 1.000 62 0.0004173 0.0004536|1.852e-117 4.961e-117 0.5000 7.90577441626963e-117 6.813e-117 1.000 63 0.0004928 0.0005350|1.323e-116 3.515e-116 0.5000 5.62906779618244e-116 4.838e-116 1.000 64 0.0005804 0.0006293|9.303e-116 2.453e-115 0.5000 3.94580956900457e-115 3.383e-115 1.000 65 0.0006818 0.0007382|6.441e-115 1.685e-114 0.5000 2.72362923110002e-114 2.329e-114 1.000 66 0.0007987 0.0008636|4.391e-114 1.140e-113 0.5000 1.85171243465609e-113 1.579e-113 1.000 67 0.0009332 0.001008|2.949e-113 7.601e-113 0.5000 1.24025569434351e-112 1.055e-112 1.000 68 0.001088 0.001173|1.952e-112 4.994e-112 0.5000 8.18571441796497e-112 6.945e-112 1.000 69 0.001265 0.001362|1.273e-111 3.233e-111 0.5000 5.32478887161532e-111 4.506e-111 1.000 70 0.001467 0.001578|8.183e-111 2.064e-110 0.5000 3.41459154615803e-110 2.882e-110 1.000 71 0.001698 0.001825|5.187e-110 1.299e-109 0.5000 2.15900313962047e-109 1.818e-109 1.000 72 0.001960 0.002104|3.242e-109 8.062e-109 0.5000 1.34626662305498e-108 1.130e-108 1.000 73 0.002258 0.002422|1.998e-108 4.936e-108 0.5000 8.28047001178962e-108 6.934e-108 1.000 74 0.002596 0.002781|1.215e-107 2.981e-107 0.5000 5.02464724722950e-107 4.197e-107 1.000 75 0.002978 0.003187|7.291e-107 1.777e-106 0.5000 3.00857796790790e-106 2.506e-106 1.000 76 0.003409 0.003645|4.317e-106 1.045e-105 0.5000 1.77786355883123e-105 1.477e-105 1.000 77 0.003895 0.004161|2.523e-105 6.070e-105 0.5000 1.03703025351955e-104 8.592e-105 1.000 78 0.004442 0.004740|1.456e-104 3.479e-104 0.5000 5.97190284732231e-104 4.935e-104 1.000 79 0.005056 0.005390|8.291e-104 1.969e-103 0.5000 3.39572869317608e-103 2.799e-103 1.000 80 0.005744 0.006118|4.664e-103 1.101e-102 0.5000 1.90686981886150e-102 1.567e-102 1.000 81 0.006513 0.006931|2.591e-102 6.079e-102 0.5000 1.05765513455147e-101 8.670e-102 1.000 82 0.007373 0.007839|1.422e-101 3.316e-101 0.5000 5.79518925564547e-101 4.738e-101 1.000 83 0.008331 0.008850|7.709e-101 1.787e-100 0.5000 3.13729687890309e-100 2.558e-100 1.000 84 0.009398 0.009975|4.130e-100 9.516e-100 0.5000 1.67830345862489e-99 1.365e-99 1.000 85 0.01058 0.01123| 2.186e-99 5.008e-99 0.5000 8.87305546225975e-99 7.195e-99 1.000 86 0.01190 0.01261| 1.144e-98 2.606e-98 0.5000 4.63685244263684e-98 3.750e-98 1.000 87 0.01336 0.01415| 5.917e-98 1.340e-97 0.5000 2.39540470185845e-97 1.932e-97 1.000 88 0.01497 0.01584| 3.026e-97 6.814e-97 0.5000 1.22348009868480e-96 9.839e-97 1.000 89 0.01676 0.01772| 1.530e-96 3.426e-96 0.5000 6.17920894197970e-96 4.956e-96 1.000 90 0.01872 0.01978| 7.650e-96 1.703e-95 0.5000 3.08632136759401e-95 2.468e-95 1.000 91 0.02089 0.02206| 3.783e-95 8.378e-95 0.5000 1.52466645336620e-94 1.216e-94 1.000 92 0.02328 0.02456| 1.850e-94 4.076e-94 0.5000 7.45050933137625e-94 5.926e-94 1.000 93 0.02590 0.02731| 8.953e-94 1.962e-93 0.5000 3.60184788030238e-93 2.857e-93 1.000 94 0.02878 0.03032| 4.286e-93 9.340e-93 0.5000 1.72283164843876e-92 1.363e-92 1.000 95 0.03193 0.03362| 2.030e-92 4.401e-92 0.5000 8.15430819574882e-92 6.431e-92 1.000 96 0.03539 0.03723| 9.517e-92 2.052e-91 0.5000 3.81949485756384e-91 3.004e-91 1.000 97 0.03916 0.04118| 4.415e-91 9.473e-91 0.5000 1.77070413536735e-90 1.389e-90 1.000 98 0.04329 0.04549| 2.027e-90 4.328e-90 0.5000 8.12555661209967e-90 6.355e-90 1.000 99 0.04779 0.05018| 9.215e-90 1.957e-89 0.5000 3.69123881779387e-89 2.879e-89 1.000 100 0.05269 0.05530| 4.147e-89 8.764e-89 0.5000 1.66014886933081e-88 1.291e-88 1.000 101 0.05802 0.06086| 1.848e-88 3.885e-88 0.5000 7.39300295195676e-88 5.733e-88 1.000 102 0.06382 0.06690| 8.151e-88 1.706e-87 0.5000 3.26013756955835e-87 2.521e-87 1.000 103 0.07011 0.07346| 3.561e-87 7.416e-87 0.5000 1.42374955023985e-86 1.098e-86 1.000 104 0.07694 0.08057| 1.541e-86 3.194e-86 0.5000 6.15821047193283e-86 4.734e-86 1.000 105 0.08434 0.08826| 6.604e-86 1.362e-85 0.5000 2.63838740488722e-85 2.023e-85 1.000 106 0.09235 0.09659| 2.803e-85 5.756e-85 0.5000 1.11975965018355e-84 8.559e-85 1.000 107 0.1010 0.1056| 1.179e-84 2.409e-84 0.5000 4.70816469146770e-84 3.588e-84 1.000 108 0.1104 0.1153| 4.913e-84 9.993e-84 0.5000 1.96135732834831e-83 1.491e-83 1.000 109 0.1205 0.1258| 2.028e-83 4.107e-83 0.5000 8.09612142770548e-83 6.135e-83 1.000 110 0.1313 0.1371| 8.297e-83 1.672e-82 0.5000 3.31168343737185e-82 2.502e-82 1.000 111 0.1431 0.1493| 3.364e-82 6.750e-82 0.5000 1.34247984668257e-81 1.011e-81 1.000 112 0.1557 0.1623| 1.351e-81 2.700e-81 0.5000 5.39372620013171e-81 4.051e-81 1.000 113 0.1692 0.1764| 5.382e-81 1.070e-80 0.5000 2.14796187963917e-80 1.609e-80 1.000 114 0.1838 0.1915| 2.124e-80 4.207e-80 0.5000 8.47919411446916e-80 6.331e-80 1.000 115 0.1994 0.2077| 8.313e-80 1.639e-79 0.5000 3.31822719193952e-79 2.470e-79 1.000 116 0.2162 0.2250| 3.225e-79 6.331e-79 0.5000 1.28740091867507e-78 9.556e-79 1.000 117 0.2341 0.2436| 1.240e-78 2.425e-78 0.5000 4.95233180682573e-78 3.665e-78 1.000 118 0.2534 0.2635| 4.730e-78 9.208e-78 0.5000 1.88897148180418e-77 1.394e-77 1.000 119 0.2739 0.2847| 1.789e-77 3.467e-77 0.5000 7.14482848448998e-77 5.256e-77 1.000 120 0.2959 0.3074| 6.708e-77 1.295e-76 0.5000 2.68002861370252e-76 1.966e-76 1.000 121 0.3193 0.3316| 2.495e-76 4.796e-76 0.5000 9.97007082771253e-76 7.290e-76 1.000 122 0.3443 0.3575| 9.201e-76 1.762e-75 0.5000 3.67872841979143e-75 2.682e-75 1.000 123 0.3710 0.3850| 3.366e-75 6.419e-75 0.5000 1.34637762190678e-74 9.785e-75 1.000 124 0.3994 0.4143| 1.222e-74 2.320e-74 0.5000 4.88803435218240e-74 3.542e-74 1.000 125 0.4297 0.4455| 4.398e-74 8.319e-74 0.5000 1.76046704342550e-73 1.272e-73 0.9999 126 0.4618 0.4786| 1.571e-73 2.959e-73 0.5000 6.29035949780922e-73 4.530e-73 0.9999 127 0.4960 0.5139| 5.565e-73 1.045e-72 0.5000 2.22999632338926e-72 1.601e-72 0.9999 128 0.5323 0.5513| 1.957e-72 3.658e-72 0.5000 7.84405649513297e-72 5.614e-72 0.9999 129 0.5708 0.5909| 6.825e-72 1.271e-71 0.5000 2.73786178780913e-71 1.953e-71 0.9999 130 0.6116 0.6330| 2.363e-71 4.383e-71 0.5000 9.48292946291098e-71 6.745e-71 0.9999 131 0.6549 0.6775| 8.116e-71 1.500e-70 0.5000 3.25956215344469e-70 2.311e-70 0.9999 132 0.7008 0.7247| 2.767e-70 5.094e-70 0.5000 1.11195426962193e-69 7.860e-70 0.9999 133 0.7493 0.7746| 9.361e-70 1.717e-69 0.5000 3.76487263815026e-69 2.653e-69 0.9999 134 0.8006 0.8273| 3.144e-69 5.745e-69 0.5000 1.26524250015668e-68 8.888e-69 0.9999 135 0.8548 0.8831| 1.048e-68 1.908e-68 0.5000 4.22067032518793e-68 2.955e-68 0.9999 136 0.9121 0.9419| 3.467e-68 6.290e-68 0.5000 1.39764530215204e-67 9.756e-68 0.9999 137 0.9725 1.004| 1.139e-67 2.058e-67 0.5000 4.59455804132474e-67 3.197e-67 0.9999 138 1.036 1.069| 3.714e-67 6.688e-67 0.5000 1.49949412144624e-66 1.040e-66 0.9999 139 1.104 1.138| 1.202e-66 2.157e-66 0.5000 4.85872960801228e-66 3.359e-66 0.9998 140 1.174 1.211| 3.865e-66 6.910e-66 0.5000 1.56314762062852e-65 1.077e-65 0.9998 141 1.249 1.288| 1.233e-65 2.198e-65 0.5000 4.99342579499814e-65 3.430e-65 0.9998 142 1.327 1.368| 3.909e-65 6.940e-65 0.5000 1.58394567978780e-64 1.085e-64 0.9998 143 1.410 1.453| 1.230e-64 2.176e-64 0.5000 4.98937874636304e-64 3.405e-64 0.9998 144 1.497 1.542| 3.844e-64 6.777e-64 0.5000 1.56076780638901e-63 1.062e-63 0.9998 145 1.588 1.635| 1.193e-63 2.096e-63 0.5000 4.84882600131870e-63 3.288e-63 0.9998 146 1.683 1.733| 3.677e-63 6.438e-63 0.5000 1.49610450432197e-62 1.011e-62 0.9997 147 1.784 1.836| 1.125e-62 1.964e-62 0.5000 4.58494957277087e-62 3.089e-62 0.9997 148 1.889 1.943| 3.422e-62 5.952e-62 0.5000 1.39564350173897e-61 9.371e-62 0.9997 149 1.999 2.056| 1.034e-61 1.792e-61 0.5000 4.21989323713688e-61 2.824e-61 0.9997 150 2.114 2.174| 3.101e-61 5.357e-61 0.5000 1.26746194037986e-60 8.455e-61 0.9997 151 2.235 2.298| 9.240e-61 1.591e-60 0.5000 3.78176115775169e-60 2.514e-60 0.9996 152 2.362 2.427| 2.736e-60 4.695e-60 0.5000 1.12098016464135e-59 7.428e-60 0.9996 153 2.494 2.562| 8.046e-60 1.376e-59 0.5000 3.30114900050893e-59 2.180e-59 0.9996 154 2.632 2.704| 2.351e-59 4.009e-59 0.5000 9.65859948715239e-59 6.357e-59 0.9996 155 2.777 2.851| 6.826e-59 1.160e-58 0.5000 2.80778108525725e-58 1.842e-58 0.9995 156 2.928 3.005| 1.969e-58 3.336e-58 0.5000 8.11017783021479e-58 5.302e-58 0.9995 157 3.085 3.166| 5.643e-58 9.531e-58 0.5000 2.32773382276976e-57 1.517e-57 0.9995 158 3.249 3.334| 1.607e-57 2.706e-57 0.5000 6.63880021884768e-57 4.311e-57 0.9994 159 3.420 3.509| 4.549e-57 7.635e-57 0.5000 1.88155335842244e-56 1.218e-56 0.9994 160 3.599 3.691| 1.279e-56 2.141e-56 0.5000 5.29945528323633e-56 3.418e-56 0.9994 161 3.785 3.880| 3.576e-56 5.964e-56 0.5000 1.48337365996304e-55 9.534e-56 0.9993 162 3.978 4.078| 9.933e-56 1.652e-55 0.5000 4.12658891599893e-55 2.643e-55 0.9993 163 4.179 4.283| 2.742e-55 4.546e-55 0.5000 1.14095793923966e-54 7.283e-55 0.9993 164 4.389 4.497| 7.524e-55 1.244e-54 0.5000 3.13546612831464e-54 1.995e-54 0.9992 165 4.606 4.719| 2.052e-54 3.381e-54 0.5000 8.56455326100101e-54 5.429e-54 0.9992 166 4.833 4.949| 5.563e-54 9.139e-54 0.5000 2.32537668504198e-53 1.469e-53 0.9991 167 5.068 5.189| 1.499e-53 2.455e-53 0.5000 6.27600956650867e-53 3.951e-53 0.9991 168 5.312 5.437| 4.015e-53 6.557e-53 0.5000 1.68380215863509e-52 1.056e-52 0.9990 169 5.565 5.695| 1.069e-52 1.741e-52 0.5000 4.49087557782998e-52 2.807e-52 0.9990 170 5.828 5.963| 2.830e-52 4.594e-52 0.5000 1.19074457826861e-51 7.417e-52 0.9989 171 6.100 6.240| 7.448e-52 1.206e-51 0.5000 3.13883717088709e-51 1.948e-51 0.9988 172 6.382 6.527| 1.948e-51 3.145e-51 0.5000 8.22614435474002e-51 5.087e-51 0.9988 173 6.675 6.825| 5.068e-51 8.157e-51 0.5000 2.14346225384754e-50 1.321e-50 0.9987 174 6.978 7.133| 1.311e-50 2.103e-50 0.5000 5.55318022557425e-50 3.410e-50 0.9986 175 7.291 7.452| 3.371e-50 5.393e-50 0.5000 1.43050172717999e-49 8.752e-50 0.9986 176 7.616 7.782| 8.618e-50 1.375e-49 0.5000 3.66412099076941e-49 2.234e-49 0.9985 177 7.951 8.124| 2.191e-49 3.486e-49 0.5000 9.33254253490499e-49 5.668e-49 0.9984 178 8.299 8.476| 5.539e-49 8.789e-49 0.5000 2.36370186066287e-48 1.430e-48 0.9983 179 8.657 8.841| 1.392e-48 2.203e-48 0.5000 5.95335255735877e-48 3.590e-48 0.9982 180 9.028 9.218| 3.481e-48 5.493e-48 0.5000 1.49114604267898e-47 8.958e-48 0.9982 181 9.410 9.606| 8.655e-48 1.362e-47 0.5000 3.71434096681652e-47 2.223e-47 0.9981 182 9.805 10.01| 2.140e-47 3.358e-47 0.5000 9.20151961145897e-47 5.487e-47 0.9980 183 10.21 10.42| 5.262e-47 8.235e-47 0.5000 2.26707865683414e-46 1.347e-46 0.9979 184 10.63 10.85| 1.287e-46 2.009e-46 0.5000 5.55540553720619e-46 3.288e-46 0.9978 185 11.07 11.29| 3.130e-46 4.872e-46 0.5000 1.35400371247698e-45 7.985e-46 0.9976 186 11.51 11.74| 7.574e-46 1.176e-45 0.5000 3.28239978603939e-45 1.928e-45 0.9975 187 11.97 12.21| 1.823e-45 2.822e-45 0.5000 7.91485780948235e-45 4.632e-45 0.9974 188 12.45 12.69| 4.362e-45 6.736e-45 0.5000 1.89839741688758e-44 1.107e-44 0.9973 189 12.94 13.19| 1.039e-44 1.600e-44 0.5000 4.52934862905042e-44 2.631e-44 0.9972 190 13.44 13.70| 2.460e-44 3.778e-44 0.5000 1.07498166850720e-43 6.220e-44 0.9970 191 13.96 14.22| 5.796e-44 8.879e-44 0.5000 2.53801991402374e-43 1.463e-43 0.9969 192 14.49 14.76| 1.358e-43 2.076e-43 0.5000 5.96114234070403e-43 3.423e-43 0.9967 193 15.04 15.32| 3.167e-43 4.827e-43 0.5000 1.39288793303486e-42 7.968e-43 0.9966 194 15.60 15.89| 7.347e-43 1.117e-42 0.5000 3.23792416592779e-42 1.845e-42 0.9964 195 16.18 16.47| 1.695e-42 2.571e-42 0.5000 7.48845905933859e-42 4.251e-42 0.9963 196 16.77 17.07| 3.893e-42 5.887e-42 0.5000 1.72307782968975e-41 9.742e-42 0.9961 197 17.38 17.69| 8.892e-42 1.341e-41 0.5000 3.94471126156959e-41 2.222e-41 0.9959 198 18.01 18.32| 2.021e-41 3.041e-41 0.5000 8.98534101582250e-41 5.041e-41 0.9957 199 18.65 18.98| 4.570e-41 6.859e-41 0.5000 2.03645089985789e-40 1.138e-40 0.9956 200 19.31 19.64| 1.028e-40 1.539e-40 0.5000 4.59244815660310e-40 2.556e-40 0.9954 201 19.98 20.33| 2.302e-40 3.438e-40 0.5000 1.03051974378254e-39 5.713e-40 0.9952 202 20.67 21.03| 5.128e-40 7.640e-40 0.5000 2.30102212955986e-39 1.271e-39 0.9950 203 21.38 21.74| 1.137e-39 1.689e-39 0.5000 5.11267610501515e-39 2.812e-39 0.9947 204 22.11 22.48| 2.508e-39 3.717e-39 0.5000 1.13044451279326e-38 6.192e-39 0.9945 205 22.86 23.23| 5.504e-39 8.140e-39 0.5000 2.48733384755104e-38 1.357e-38 0.9943 206 23.62 24.01| 1.202e-38 1.774e-38 0.5000 5.44644401141379e-38 2.959e-38 0.9941 207 24.40 24.79| 2.614e-38 3.847e-38 0.5000 1.18685131594972e-37 6.422e-38 0.9938 208 25.20 25.60| 5.655e-38 8.303e-38 0.5000 2.57391364509742e-37 1.387e-37 0.9936 209 26.01 26.43| 1.218e-37 1.783e-37 0.5000 5.55540841442813e-37 2.981e-37 0.9933 210 26.85 27.27| 2.609e-37 3.813e-37 0.5000 1.19336215088217e-36 6.378e-37 0.9930 211 27.70 28.14| 5.565e-37 8.112e-37 0.5000 2.55136482275307e-36 1.358e-36 0.9928 212 28.57 29.02| 1.181e-36 1.718e-36 0.5000 5.42908449309285e-36 2.878e-36 0.9925 213 29.47 29.92| 2.495e-36 3.621e-36 0.5000 1.14985737954778e-35 6.069e-36 0.9922 214 30.38 30.84| 5.247e-36 7.596e-36 0.5000 2.42400888279266e-35 1.274e-35 0.9919 215 31.31 31.78| 1.098e-35 1.586e-35 0.5000 5.08635557847970e-35 2.662e-35 0.9916 216 32.26 32.74| 2.288e-35 3.297e-35 0.5000 1.06235789030505e-34 5.537e-35 0.9912 217 33.22 33.72| 4.745e-35 6.821e-35 0.5000 2.20869551823950e-34 1.146e-34 0.9909 218 34.21 34.71| 9.794e-35 1.405e-34 0.5000 4.57099652892044e-34 2.362e-34 0.9906 219 35.22 35.73| 2.013e-34 2.880e-34 0.5000 9.41683706507845e-34 4.846e-34 0.9902 220 36.25 36.77| 4.117e-34 5.877e-34 0.5000 1.93120040836262e-33 9.895e-34 0.9898 221 37.30 37.83| 8.382e-34 1.194e-33 0.5000 3.94263593770438e-33 2.011e-33 0.9895 222 38.36 38.91| 1.699e-33 2.415e-33 0.5000 8.01293998424788e-33 4.070e-33 0.9891 223 39.45 40.00| 3.429e-33 4.862e-33 0.5000 1.62125655584258e-32 8.200e-33 0.9887 224 40.56 41.12| 6.888e-33 9.746e-33 0.5000 3.26568975262206e-32 1.644e-32 0.9883 225 41.69 42.26| 1.378e-32 1.945e-32 0.5000 6.54892518178577e-32 3.283e-32 0.9879 226 42.84 43.42| 2.743e-32 3.864e-32 0.5000 1.30751171746741e-31 6.526e-32 0.9874 227 44.01 44.60| 5.438e-32 7.643e-32 0.5000 2.59902261637817e-31 1.292e-31 0.9870 228 45.20 45.80| 1.073e-31 1.505e-31 0.5000 5.14365480979889e-31 2.545e-31 0.9866 229 46.41 47.02| 2.109e-31 2.951e-31 0.5000 1.01353596927147e-30 4.992e-31 0.9861 230 47.64 48.26| 4.126e-31 5.762e-31 0.5000 1.98847595776604e-30 9.749e-31 0.9856 231 48.89 49.52| 8.038e-31 1.120e-30 0.5000 3.88439614102461e-30 1.896e-30 0.9851 232 50.16 50.80| 1.559e-30 2.168e-30 0.5000 7.55538792798779e-30 3.671e-30 0.9846 233 51.45 52.11| 3.011e-30 4.178e-30 0.5000 1.46328233641070e-29 7.077e-30 0.9841 234 52.77 53.43| 5.791e-30 8.017e-30 0.5000 2.82192558246456e-29 1.359e-29 0.9836 235 54.10 54.77| 1.109e-29 1.532e-29 0.5000 5.41897236418366e-29 2.597e-29 0.9830 236 55.45 56.14| 2.114e-29 2.915e-29 0.5000 1.03621551590960e-28 4.943e-29 0.9825 237 56.83 57.52| 4.015e-29 5.523e-29 0.5000 1.97311672530264e-28 9.369e-29 0.9819 238 58.22 58.93| 7.591e-29 1.042e-28 0.5000 3.74138748706695e-28 1.768e-28 0.9813 239 59.64 60.35| 1.429e-28 1.958e-28 0.5000 7.06476067126582e-28 3.323e-28 0.9807 240 61.07 61.80| 2.680e-28 3.664e-28 0.5000 1.32847882886900e-27 6.220e-28 0.9801 241 62.53 63.27| 5.004e-28 6.827e-28 0.5000 2.48777869402137e-27 1.159e-27 0.9795 242 64.01 64.75| 9.304e-28 1.267e-27 0.5000 4.63955380684426e-27 2.152e-27 0.9789 243 65.50 66.26| 1.723e-27 2.341e-27 0.5000 8.61698694580876e-27 3.977e-27 0.9782 244 67.02 67.79| 3.178e-27 4.309e-27 0.5000 1.59388404914453e-26 7.322e-27 0.9775 245 68.56 69.33| 5.837e-27 7.898e-27 0.5000 2.93621158508602e-26 1.342e-26 0.9768 246 70.11 70.90| 1.068e-26 1.442e-26 0.5000 5.38709347329730e-26 2.451e-26 0.9761 247 71.69 72.49| 1.945e-26 2.621e-26 0.5000 9.84385851506173e-26 4.457e-26 0.9754 248 73.29 74.09| 3.529e-26 4.747e-26 0.5000 1.79154200565973e-25 8.072e-26 0.9747 249 74.90 75.72| 6.379e-26 8.562e-26 0.5000 3.24747969781985e-25 1.456e-25 0.9739 250 76.54 77.36| 1.148e-25 1.538e-25 0.5000 5.86314577678934e-25 2.616e-25 0.9732 251 78.19 79.03| 2.058e-25 2.752e-25 0.5000 1.05435452251433e-24 4.680e-25 0.9724 252 79.86 80.71| 3.676e-25 4.905e-25 0.5000 1.88851847141959e-24 8.342e-25 0.9716 253 81.56 82.41| 6.538e-25 8.706e-25 0.5000 3.36931199539573e-24 1.481e-24 0.9708 254 83.27 84.13| 1.158e-24 1.539e-24 0.5000 5.98760865002649e-24 2.618e-24 0.9700 255 85. 85.87| 2.044e-24 2.711e-24 0.5000 1.05989902056204e-23 4.611e-24 0.9691 256 86.74 87.62| 3.593e-24 4.757e-24 0.5000 1.86887883353758e-23 8.090e-24 0.9682 257 88.51 89.40| 6.291e-24 8.313e-24 0.5000 3.28253969466355e-23 1.414e-23 0.9674 258 90.29 91.19| 1.097e-23 1.447e-23 0.5000 5.74324814417819e-23 2.461e-23 0.9665 259 92.10 93.00| 1.907e-23 2.509e-23 0.5000 1.00099162970122e-22 4.267e-23 0.9655 260 93.91 94.83| 3.300e-23 4.335e-23 0.5000 1.73794108822684e-22 7.369e-23 0.9646 261 95.75 96.67| 5.689e-23 7.460e-23 0.5000 3.00592276509280e-22 1.268e-22 0.9636 262 97.60 98.54| 9.772e-23 1.279e-22 0.5000 5.17922908076025e-22 2.173e-22 0.9627 263 99.47 100.4| 1.672e-22 2.184e-22 0.5000 8.89003126671440e-22 3.711e-22 0.9617 264 101.4 102.3| 2.850e-22 3.716e-22 0.5000 1.52019256322411e-21 6.312e-22 0.9606 265 103.3 104.2| 4.840e-22 6.298e-22 0.5000 2.58974679161077e-21 1.070e-21 0.9596 266 105.2 106.1| 8.187e-22 1.063e-21 0.5000 4.39527022964985e-21 1.806e-21 0.9586 267 107.1 108.1| 1.380e-21 1.789e-21 0.5000 7.43172423037376e-21 3.036e-21 0.9575 268 109.1 110.0| 2.317e-21 2.998e-21 0.5000 1.25191666999739e-20 5.087e-21 0.9564 269 111.0 112.0| 3.876e-21 5.006e-21 0.5000 2.10111175998308e-20 8.492e-21 0.9553 270 113.0 114.0| 6.460e-21 8.328e-21 0.5000 3.51331265261004e-20 1.412e-20 0.9542 271 115.0 116.0| 1.073e-20 1.380e-20 0.5000 5.85307837594742e-20 2.340e-20 0.9530 272 117.0 118.0| 1.775e-20 2.279e-20 0.5000 9.71533343786876e-20 3.862e-20 0.9518 273 119.0 120.0| 2.925e-20 3.750e-20 0.5000 1.60672965449908e-19 6.352e-20 0.9507 274 121.1 122.1| 4.804e-20 6.148e-20 0.5000 2.64755805853089e-19 1.041e-19 0.9494 275 123.1 124.1| 7.861e-20 1.004e-19 0.5000 4.34681920766452e-19 1.699e-19 0.9482 276 125.2 126.2| 1.282e-19 1.634e-19 0.5000 7.11093717976176e-19 2.764e-19 0.9470 277 127.3 128.3| 2.082e-19 2.650e-19 0.5000 1.15908964402459e-18 4.480e-19 0.9457 278 129.3 130.4| 3.371e-19 4.283e-19 0.5000 1.88255567886438e-18 7.235e-19 0.9444 279 131.4 132.5| 5.436e-19 6.895e-19 0.5000 3.04666650957950e-18 1.164e-18 0.9431 280 133.5 134.6| 8.737e-19 1.106e-18 0.5000 4.91308093911293e-18 1.866e-18 0.9417 281 135.7 136.7| 1.399e-18 1.768e-18 0.5000 7.89478706575296e-18 2.982e-18 0.9404 282 137.8 138.9| 2.233e-18 2.817e-18 0.5000 1.26412488055087e-17 4.746e-18 0.9390 283 139.9 141.0| 3.550e-18 4.471e-18 0.5000 2.01701028389022e-17 7.529e-18 0.9376 284 142.1 143.2| 5.625e-18 7.072e-18 0.5000 3.20700992914948e-17 1.190e-17 0.9362 285 144.2 145.3| 8.882e-18 1.115e-17 0.5000 5.08126686580812e-17 1.874e-17 0.9347 286 146.4 147.5| 1.398e-17 1.751e-17 0.5000 8.02284916536076e-17 2.942e-17 0.9333 287 148.6 149.7| 2.191e-17 2.740e-17 0.5000 1.26233770470878e-16 4.601e-17 0.9318 288 150.8 151.9| 3.424e-17 4.274e-17 0.5000 1.97932966569526e-16 7.170e-17 0.9303 289 153. 154.1| 5.331e-17 6.644e-17 0.5000 3.09287010730131e-16 1.114e-16 0.9288 290 155.2 156.3| 8.273e-17 1.029e-16 0.5000 4.81627725213383e-16 1.723e-16 0.9272 291 157.4 158.5| 1.279e-16 1.589e-16 0.5000 7.47433880270897e-16 2.658e-16 0.9256 292 159.6 160.7| 1.972e-16 2.444e-16 0.5000 1.15598128439279e-15 4.085e-16 0.9240 293 161.8 162.9| 3.028e-16 3.748e-16 0.5000 1.78176628491537e-15 6.258e-16 0.9224 294 164.0 165.1| 4.635e-16 5.726e-16 0.5000 2.73701854828988e-15 9.553e-16 0.9208 295 166.3 167.4| 7.070e-16 8.721e-16 0.5000 4.19022112873330e-15 1.453e-15 0.9191 296 168.5 169.6| 1.075e-15 1.324e-15 0.5000 6.39342303117024e-15 2.203e-15 0.9174 297 170.7 171.8| 1.628e-15 2.002e-15 0.5000 9.72237361927662e-15 3.329e-15 0.9157 298 173. 174.1| 2.459e-15 3.018e-15 0.5000 1.47352863244734e-14 5.013e-15 0.9140 299 175.2 176.3| 3.701e-15 4.535e-15 0.5000 2.22585706126337e-14 7.523e-15 0.9122 300 177.5 178.6| 5.551e-15 6.791e-15 0.5000 3.35114614898743e-14 1.125e-14 0.9105 301 179.7 180.8| 8.299e-15 1.014e-14 0.5000 5.02865483614018e-14 1.678e-14 0.9087 302 182. 183.1| 1.237e-14 1.508e-14 0.5000 7.52103442953915e-14 2.492e-14 0.9069 303 184.2 185.3| 1.837e-14 2.235e-14 0.5000 1.12118046303119e-13 3.691e-14 0.9050 304 186.5 187.6| 2.719e-14 3.304e-14 0.5000 1.66590617470761e-13 5.447e-14 0.9031 305 188.7 189.9| 4.011e-14 4.866e-14 0.5000 2.46721773104437e-13 8.013e-14 0.9013 306 191. 192.1| 5.899e-14 7.145e-14 0.5000 3.64209413044695e-13 1.175e-13 0.8993 307 193.2 194.4| 8.647e-14 1.046e-13 0.5000 5.35903266498751e-13 1.717e-13 0.8974 308 195.5 196.6| 1.263e-13 1.525e-13 0.5000 7.85991715133170e-13 2.501e-13 0.8955 309 197.7 198.9| 1.840e-13 2.217e-13 0.5000 1.14908091173404e-12 3.631e-13 0.8935 310 200. 201.1| 2.671e-13 3.214e-13 0.5000 1.67451506235792e-12 5.254e-13 0.8915 311 202.2 203.4| 3.864e-13 4.643e-13 0.5000 2.43241692938000e-12 7.579e-13 0.8895 312 204.5 205.6| 5.573e-13 6.685e-13 0.5000 3.52210394783456e-12 1.090e-12 0.8874 313 206.7 207.8| 8.013e-13 9.596e-13 0.5000 5.08377326539900e-12 1.562e-12 0.8853 314 209. 210.1| 1.148e-12 1.373e-12 0.5000 7.31466878780058e-12 2.231e-12 0.8833 315 211.2 212.3| 1.640e-12 1.958e-12 0.5000 1.04913714388514e-11 3.177e-12 0.8811 316 213.4 214.5| 2.336e-12 2.784e-12 0.5000 1.50004248288258e-11 4.509e-12 0.8790 317 215.6 216.7| 3.316e-12 3.947e-12 0.5000 2.13802670850779e-11 6.380e-12 0.8769 318 217.9 219.| 4.693e-12 5.576e-12 0.5000 3.03784434431789e-11 8.998e-12 0.8747 319 220.1 221.2| 6.620e-12 7.853e-12 0.5000 4.30293974689783e-11 1.265e-11 0.8725 320 222.3 223.4| 9.309e-12 1.103e-11 0.5000 6.07598763176354e-11 1.773e-11 0.8703 321 224.5 225.6| 1.305e-11 1.543e-11 0.5000 8.55312579937132e-11 2.477e-11 0.8680 322 226.7 227.7| 1.824e-11 2.154e-11 0.5000 1.20031069228563e-10 3.450e-11 0.8657 323 228.8 229.9| 2.541e-11 2.996e-11 0.5000 1.67929804480476e-10 4.790e-11 0.8635 324 231.0 232.1| 3.529e-11 4.154e-11 0.5000 2.34224122265910e-10 6.629e-11 0.8611 325 233.2 234.3| 4.886e-11 5.743e-11 0.5000 3.25693887781672e-10 9.147e-11 0.8588 326 235.3 236.4| 6.745e-11 7.916e-11 0.5000 4.51508727841171e-10 1.258e-10 0.8565 327 237.5 238.5| 9.282e-11 1.088e-10 0.5000 6.24030232626706e-10 1.725e-10 0.8541 328 239.6 240.7| 1.273e-10 1.490e-10 0.5000 8.59869254166546e-10 2.358e-10 0.8517 329 241.7 242.8| 1.742e-10 2.035e-10 0.5000 1.18127461933743e-09 3.214e-10 0.8493 330 243.8 244.9| 2.375e-10 2.771e-10 0.5000 1.61795085618058e-09 4.367e-10 0.8468 331 245.9 247.| 3.229e-10 3.761e-10 0.5000 2.20942927072387e-09 5.915e-10 0.8444 332 248.0 249.1| 4.377e-10 5.090e-10 0.5000 3.00815016789438e-09 7.987e-10 0.8419 333 250.1 251.1| 5.915e-10 6.868e-10 0.5000 4.08345569787266e-09 1.075e-09 0.8394 334 252.2 253.2| 7.969e-10 9.239e-10 0.5000 5.52674453524385e-09 1.443e-09 0.8369 335 254.2 255.2| 1.070e-09 1.239e-09 0.5000 7.45810350751681e-09 1.931e-09 0.8343 336 256.2 257.3| 1.434e-09 1.657e-09 0.5000 1.00348099134442e-08 2.577e-09 0.8318 337 258.3 259.3| 1.914e-09 2.210e-09 0.5000 1.34621945795008e-08 3.427e-09 0.8292 338 260.3 261.3| 2.549e-09 2.938e-09 0.5000 1.80074731561535e-08 4.545e-09 0.8266 339 262.3 263.3| 3.383e-09 3.894e-09 0.5000 2.40172957624089e-08 6.010e-09 0.8240 340 264.2 265.2| 4.478e-09 5.146e-09 0.5000 3.19399374573984e-08 7.923e-09 0.8213 341 266.2 267.2| 5.909e-09 6.781e-09 0.5000 4.23532594150631e-08 1.041e-08 0.8187 342 268.1 269.1| 7.775e-09 8.909e-09 0.5000 5.59998190667740e-08 1.365e-08 0.8160 343 270.1 271.0| 1.020e-08 1.167e-08 0.5000 7.38308035608949e-08 1.783e-08 0.8133 344 272. 272.9| 1.334e-08 1.525e-08 0.5000 9.70608121376766e-08 2.323e-08 0.8106 345 273.9 274.8| 1.741e-08 1.986e-08 0.5000 1.27235927677608e-07 3.018e-08 0.8078 346 275.7 276.7| 2.264e-08 2.579e-08 0.5000 1.66318004759228e-07 3.908e-08 0.8051 347 277.6 278.5| 2.936e-08 3.339e-08 0.5000 2.16788670787724e-07 5.047e-08 0.8023 348 279.4 280.3| 3.796e-08 4.312e-08 0.5000 2.81777198503307e-07 6.499e-08 0.7995 349 281.3 282.2| 4.895e-08 5.552e-08 0.5000 3.65217173350375e-07 8.344e-08 0.7967 350 283.1 283.9| 6.293e-08 7.128e-08 0.5000 4.72037759045740e-07 1.068e-07 0.7938 351 284.8 285.7| 8.068e-08 9.125e-08 0.5000 6.08396370619873e-07 1.364e-07 0.7910 352 286.6 287.5| 1.031e-07 1.165e-07 0.5000 7.81960707451966e-07 1.736e-07 0.7881 353 288.3 289.2| 1.315e-07 1.483e-07 0.5000 1.00224939008375e-06 2.203e-07 0.7852 354 290.1 290.9| 1.671e-07 1.882e-07 0.5000 1.28104189361014e-06 2.788e-07 0.7823 355 291.7 292.6| 2.119e-07 2.383e-07 0.5000 1.63287008543005e-06 3.518e-07 0.7794 356 293.4 294.3| 2.678e-07 3.008e-07 0.5000 2.07560546218812e-06 4.427e-07 0.7765 357 295.1 295.9| 3.376e-07 3.786e-07 0.5000 2.63115814308475e-06 5.556e-07 0.7735 358 296.7 297.5| 4.243e-07 4.752e-07 0.5000 3.32630581395408e-06 6.951e-07 0.7706 359 298.3 299.1| 5.319e-07 5.949e-07 0.5000 4.19367312362448e-06 8.674e-07 0.7676 360 299.9 300.7| 6.649e-07 7.426e-07 0.5000 5.27288450056621e-06 1.079e-06 0.7646 361 301.5 302.3| 8.288e-07 9.244e-07 0.5000 6.61191596660625e-06 1.339e-06 0.7616 362 303.0 303.8| 1.030e-06 1.147e-06 0.5000 8.26867425080330e-06 1.657e-06 0.7585 363 304.5 305.3| 1.277e-06 1.421e-06 0.5000 1.03128343169216e-05 2.044e-06 0.7555 364 306.0 306.8| 1.579e-06 1.754e-06 0.5000 1.28279692689682e-05 2.515e-06 0.7524 365 307.5 308.2| 1.947e-06 2.159e-06 0.5000 1.59140094367359e-05 3.086e-06 0.7494 366 309. 309.7| 2.393e-06 2.651e-06 0.5000 1.96900702011814e-05 3.776e-06 0.7463 367 310.4 311.1| 2.935e-06 3.246e-06 0.5000 2.42976907191317e-05 4.608e-06 0.7432 368 311.8 312.5| 3.589e-06 3.964e-06 0.5000 2.99045280564715e-05 5.607e-06 0.7400 369 313.2 313.8| 4.376e-06 4.828e-06 0.5000 3.67085532333775e-05 6.804e-06 0.7369 370 314.5 315.2| 5.322e-06 5.864e-06 0.5000 4.49427972057286e-05 8.234e-06 0.7337 371 315.8 316.5| 6.456e-06 7.103e-06 0.5000 5.48806957219889e-05 9.938e-06 0.7306 372 317.1 317.8| 7.809e-06 8.581e-06 0.5000 6.68420821611529e-05 1.196e-05 0.7274 373 318.4 319.0| 9.422e-06 1.034e-05 0.4999 8.11998767208104e-05 1.436e-05 0.7242 374 319.6 320.3| 1.134e-05 1.242e-05 0.4999 9.83875185227616e-05 1.719e-05 0.7210 375 320.9 321.5| 1.360e-05 1.489e-05 0.4999 0.000118907184169057 2.052e-05 0.7177 376 322.1 322.7| 1.628e-05 1.779e-05 0.4999 0.000143338831843940 2.443e-05 0.7145 377 323.2 323.8| 1.943e-05 2.121e-05 0.4999 0.000172350104049100 2.901e-05 0.7112 378 324.4 324.9| 2.313e-05 2.522e-05 0.4999 0.000206707114320149 3.436e-05 0.7080 379 325.5 326.0| 2.747e-05 2.990e-05 0.4998 0.000247286133652988 4.058e-05 0.7047 380 326.6 327.1| 3.253e-05 3.536e-05 0.4998 0.000295086180740424 4.780e-05 0.7014 381 327.6 328.2| 3.842e-05 4.171e-05 0.4998 0.000351242506378054 5.616e-05 0.6980 382 328.7 329.2| 4.526e-05 4.907e-05 0.4997 0.000417040946472882 6.580e-05 0.6947 383 329.7 330.2| 5.318e-05 5.758e-05 0.4997 0.000493933099947325 7.689e-05 0.6913 384 330.7 331.2| 6.232e-05 6.739e-05 0.4996 0.000583552267491669 8.962e-05 0.6879 385 331.6 332.1| 7.284e-05 7.867e-05 0.4995 0.000687730064651155 0.0001042 0.6845 386 332.6 333.0| 8.491e-05 9.159e-05 0.4994 0.000808513598272830 0.0001208 0.6811 387 333.5 333.9| 9.874e-05 0.0001064 0.4993 0.000948183069083473 0.0001397 0.6776 388 334.3 334.8| 0.0001145 0.0001232 0.4992 0.00110926963539761 0.0001611 0.6741 389 335.2 335.6| 0.0001325 0.0001423 0.4991 0.00129457334401535 0.0001853 0.6706 390 336. 336.4| 0.0001529 0.0001640 0.4989 0.00150718090469447 0.0002126 0.6670 391 336.8 337.2| 0.0001759 0.0001885 0.4988 0.00175048305467970 0.0002433 0.6634 392 337.5 337.9| 0.0002019 0.0002162 0.4986 0.00202819123023319 0.0002777 0.6598 393 338.3 338.6| 0.0002312 0.0002472 0.4983 0.00234435323359413 0.0003162 0.6561 394 339. 339.3| 0.0002641 0.0002820 0.4981 0.00270336755703005 0.0003590 0.6524 395 339.7 340.| 0.0003009 0.0003208 0.4978 0.00310999600141127 0.0004066 0.6486 396 340.3 340.6| 0.0003419 0.0003641 0.4974 0.00356937420586915 0.0004594 0.6448 397 340.9 341.2| 0.0003875 0.0004122 0.4970 0.00408701968844427 0.0005176 0.6409 398 341.5 341.8| 0.0004382 0.0004655 0.4966 0.00466883698605510 0.0005818 0.6370 399 342.1 342.4| 0.0004942 0.0005243 0.4961 0.00532111947647498 0.0006523 0.6329 400 342.6 342.9| 0.0005560 0.0005891 0.4956 0.00605054746611213 0.0007294 0.6288 401 343.1 343.4| 0.0006239 0.0006603 0.4949 0.00686418213600436 0.0008136 0.6246 402 343.6 343.8| 0.0006984 0.0007382 0.4942 0.00776945495523974 0.0009053 0.6204 403 344.1 344.3| 0.0007798 0.0008233 0.4935 0.00877415219655966 0.001005 0.6160 404 344.5 344.7| 0.0008686 0.0009159 0.4926 0.00988639422361704 0.001112 0.6115 405 344.9 345.1| 0.0009652 0.001016 0.4916 0.0111146092635105 0.001228 0.6069 406 345.3 345.4| 0.001070 0.001125 0.4906 0.0124675014318742 0.001353 0.6022 407 345.6 345.8| 0.001183 0.001243 0.4894 0.0139540128408397 0.001487 0.5974 408 345.9 346.1| 0.001305 0.001369 0.4881 0.0155832796922541 0.001629 0.5924 409 346.2 346.3| 0.001435 0.001504 0.4866 0.0173645823390428 0.001781 0.5873 410 346.5 346.6| 0.001575 0.001649 0.4851 0.0193072893857253 0.001943 0.5820 411 346.7 346.8| 0.001725 0.001803 0.4833 0.0214207959937523 0.002114 0.5766 412 346.9 347.| 0.001884 0.001967 0.4814 0.0237144566572243 0.002294 0.5710 413 347.1 347.1| 0.002053 0.002141 0.4794 0.0261975128181470 0.002483 0.5653 414 347.2 347.3| 0.002231 0.002324 0.4772 0.0288790157959354 0.002682 0.5593 415 347.3 347.4| 0.002419 0.002517 0.4747 0.0317677456114870 0.002889 0.5532 416 347.4 347.5| 0.002617 0.002719 0.4721 0.0348721263897415 0.003104 0.5468 417 347.5 347.5| 0.002824 0.002931 0.4693 0.0382001391240640 0.003328 0.5403 418 347.5 347.5| 0.003040 0.003152 0.4663 0.0417592326788157 0.003559 0.5336 419 347.6 347.5| 0.003265 0.003381 0.4630 0.0455562339908622 0.003797 0.5266 420 347.5 347.5| 0.003499 0.003618 0.4595 0.0495972585043393 0.004041 0.5194 421 347.5 347.5| 0.003740 0.003863 0.4558 0.0538876219336481 0.004290 0.5120 422 347.4 347.4| 0.003988 0.004114 0.4518 0.0584317544954551 0.004544 0.5044 423 347.3 347.3| 0.004242 0.004372 0.4475 0.0632331187797018 0.004801 0.4966 424 347.2 347.2| 0.004502 0.004634 0.4430 0.0682941324408249 0.005061 0.4885 425 347.1 347.| 0.004767 0.004901 0.4383 0.0736160968823963 0.005322 0.4802 426 346.9 346.8| 0.005036 0.005171 0.4332 0.0791991330804276 0.005583 0.4717 427 346.7 346.6| 0.005307 0.005443 0.4279 0.0850421256422352 0.005843 0.4629 428 346.5 346.4| 0.005580 0.005716 0.4223 0.0911426761290492 0.006101 0.4540 429 346.2 346.1| 0.005853 0.005989 0.4165 0.0974970665819198 0.006354 0.4448 430 346. 345.8| 0.006125 0.006261 0.4104 0.104100234082813 0.006603 0.4354 431 345.7 345.5| 0.006395 0.006529 0.4040 0.110945757057438 0.006846 0.4258 432 345.3 345.2| 0.006662 0.006793 0.3973 0.118025853885025 0.007080 0.4161 433 345.0 344.8| 0.006923 0.007052 0.3904 0.125331394225170 0.007306 0.4061 434 344.6 344.4| 0.007178 0.007303 0.3832 0.132851923305390 0.007521 0.3960 435 344.2 344.0| 0.007426 0.007546 0.3758 0.140575699238114 0.007724 0.3858 436 343.8 343.6| 0.007664 0.007780 0.3681 0.148489743255370 0.007914 0.3754 437 343.4 343.1| 0.007892 0.008002 0.3602 0.156579902566796 0.008090 0.3648 438 342.9 342.7| 0.008109 0.008212 0.3521 0.164830925365033 0.008251 0.3542 439 342.4 342.2| 0.008312 0.008408 0.3438 0.173226547325496 0.008396 0.3435 440 341.9 341.6| 0.008501 0.008589 0.3353 0.181749588778293 0.008523 0.3327 441 341.4 341.1| 0.008674 0.008755 0.3266 0.190382061571860 0.008632 0.3219 442 340.8 340.5| 0.008831 0.008903 0.3178 0.199105284503776 0.008723 0.3110 443 340.2 339.9| 0.008971 0.009034 0.3088 0.207900006066956 0.008795 0.3001 444 339.6 339.3| 0.009092 0.009145 0.2997 0.216746533151407 0.008847 0.2893 445 339. 338.7| 0.009194 0.009238 0.2905 0.225624864255052 0.008878 0.2784 446 338.3 338.0| 0.009277 0.009310 0.2813 0.234514825693381 0.008890 0.2676 447 337.7 337.3| 0.009339 0.009362 0.2719 0.243396209257979 0.008881 0.2569 448 337. 336.6| 0.009381 0.009394 0.2625 0.252248909759050 0.008853 0.2463 449 336.3 335.9| 0.009401 0.009404 0.2531 0.261053060896917 0.008804 0.2357 450 335.5 335.2| 0.009401 0.009394 0.2437 0.269789167941882 0.008736 0.2254 451 334.8 334.4| 0.009381 0.009362 0.2344 0.278438235759864 0.008649 0.2151 452 334. 333.6| 0.009339 0.009311 0.2250 0.286981890801536 0.008544 0.2050 453 333.2 332.8| 0.009277 0.009239 0.2157 0.295402495773605 0.008421 0.1951 454 332.4 332.| 0.009195 0.009147 0.2065 0.303683255830150 0.008281 0.1855 455 331.6 331.1| 0.009094 0.009037 0.1974 0.311808315257180 0.008125 0.1760 456 330.7 330.3| 0.008975 0.008908 0.1885 0.319762843771995 0.007955 0.1667 457 329.8 329.4| 0.008837 0.008762 0.1796 0.327533111717642 0.007770 0.1577 458 328.9 328.5| 0.008683 0.008600 0.1710 0.335106553598582 0.007573 0.1490 459 328.0 327.6| 0.008513 0.008422 0.1624 0.342471819573656 0.007365 0.1405 460 327.1 326.6| 0.008328 0.008230 0.1541 0.349618814693233 0.007147 0.1323 461 326.1 325.7| 0.008129 0.008025 0.1460 0.356538725836195 0.006920 0.1244 462 325.2 324.7| 0.007918 0.007808 0.1381 0.363224036466132 0.006685 0.1167 463 324.2 323.7| 0.007695 0.007581 0.1304 0.369668529482122 0.006444 0.1094 464 323.2 322.7| 0.007463 0.007344 0.1229 0.375867278585399 0.006199 0.1023 465 322.2 321.6| 0.007223 0.007099 0.1157 0.381816628716781 0.005949 0.09555 466 321.1 320.6| 0.006975 0.006848 0.1087 0.387514166239231 0.005698 0.08909 467 320.1 319.5| 0.006721 0.006592 0.1020 0.392958679643909 0.005445 0.08293 468 319.0 318.5| 0.006462 0.006332 0.09553 0.398150111645350 0.005191 0.07707 469 317.9 317.4| 0.006200 0.006069 0.08933 0.403089503601390 0.004939 0.07150 470 316.8 316.3| 0.005937 0.005804 0.08339 0.407778933245761 0.004689 0.06622 471 315.7 315.1| 0.005672 0.005540 0.07772 0.412221446755885 0.004443 0.06122 472 314.6 314.0| 0.005407 0.005276 0.07231 0.416420986195773 0.004200 0.05651 473 313.4 312.8| 0.005145 0.005014 0.06716 0.420382313374648 0.003961 0.05207 474 312.3 311.7| 0.004884 0.004755 0.06228 0.424110931147013 0.003729 0.04789 475 311.1 310.5| 0.004627 0.004500 0.05765 0.427613003150521 0.003502 0.04397 476 309.9 309.3| 0.004374 0.004250 0.05328 0.430895272935606 0.003282 0.04031 477 308.7 308.1| 0.004127 0.004005 0.04915 0.433964983386956 0.003070 0.03688 478 307.5 306.9| 0.003885 0.003766 0.04527 0.436829797273237 0.002865 0.03369 479 306.2 305.6| 0.003650 0.003535 0.04162 0.439497719689759 0.002668 0.03072 480 305.0 304.4| 0.003422 0.003310 0.03820 0.441977023080778 0.002479 0.02796 481 303.7 303.1| 0.003201 0.003094 0.03500 0.444276175445671 0.002299 0.02540 482 302.5 301.8| 0.002989 0.002885 0.03201 0.446403772247953 0.002128 0.02304 483 301.2 300.6| 0.002784 0.002686 0.02922 0.448368472459748 0.001965 0.02086 484 299.9 299.3| 0.002589 0.002494 0.02663 0.450178939088361 0.001810 0.01885 485 298.6 297.9| 0.002402 0.002312 0.02423 0.451843784447461 0.001665 0.01701 486 297.3 296.6| 0.002224 0.002138 0.02201 0.453371520354339 0.001528 0.01532 487 296. 295.3| 0.002055 0.001974 0.01995 0.454770513357795 0.001399 0.01377 488 294.6 293.9| 0.001895 0.001818 0.01806 0.456048945029423 0.001278 0.01235 489 293.3 292.6| 0.001744 0.001672 0.01631 0.457214777285096 0.001166 0.01106 490 291.9 291.2| 0.001602 0.001534 0.01471 0.458275722643920 0.001061 0.009892 491 290.5 289.8| 0.001468 0.001404 0.01324 0.459239219279122 0.0009635 0.008829 492 289.2 288.5| 0.001343 0.001283 0.01190 0.460112410669695 0.0008732 0.007865 493 287.8 287.1| 0.001225 0.001170 0.01068 0.460902129622966 0.0007897 0.006994 494 286.4 285.7| 0.001116 0.001065 0.009560 0.461614886406767 0.0007128 0.006208 495 285. 284.3| 0.001015 0.0009668 0.008545 0.462256860705204 0.0006420 0.005500 496 283.5 282.8| 0.0009207 0.0008763 0.007625 0.462833897093982 0.0005770 0.004864 497 282.1 281.4| 0.0008336 0.0007926 0.006791 0.463351503719365 0.0005176 0.004294 498 280.7 280.| 0.0007533 0.0007155 0.006038 0.463814853858746 0.0004634 0.003784 499 279.2 278.5| 0.0006793 0.0006446 0.005359 0.464228790039937 0.0004139 0.003328 500 277.8 277.1| 0.0006114 0.0005796 0.004747 0.464597830400095 0.0003690 0.002922 501 276.3 275.6| 0.0005491 0.0005200 0.004198 0.464926176973114 0.0003283 0.002561 502 274.9 274.1| 0.0004922 0.0004657 0.003706 0.465217725605755 0.0002915 0.002240 503 273.4 272.7| 0.0004404 0.0004162 0.003265 0.465476077217159 0.0002584 0.001956 504 271.9 271.2| 0.0003932 0.0003713 0.002872 0.465704550133107 0.0002285 0.001705 505 270.4 269.7| 0.0003504 0.0003305 0.002522 0.465906193244976 0.0002016 0.001483 506 269. 268.2| 0.0003116 0.0002936 0.002210 0.466083799763135 0.0001776 0.001288 507 267.5 266.7| 0.0002766 0.0002604 0.001934 0.466239921355283 0.0001561 0.001117 508 266. 265.2| 0.0002450 0.0002304 0.001689 0.466376882481235 0.0001370 0.0009663 509 264.5 263.7| 0.0002166 0.0002035 0.001472 0.466496794756809 0.0001199 0.0008345 510 262.9 262.2| 0.0001911 0.0001794 0.001281 0.466601571200174 0.0001048 0.0007195 511 261.4 260.7| 0.0001683 0.0001578 0.001113 0.466692940234225 9.137e-05 0.0006191 512 259.9 259.1| 0.0001479 0.0001386 0.0009648 0.466772459337770 7.952e-05 0.0005318 513 258.4 257.6| 0.0001298 0.0001214 0.0008351 0.466841528256630 6.907e-05 0.0004560 514 256.8 256.1| 0.0001136 0.0001062 0.0007215 0.466901401702734 5.987e-05 0.0003902 515 255.3 254.5| 9.926e-05 9.272e-05 0.0006222 0.466953201485077 5.180e-05 0.0003334 516 253.8 253.| 8.656e-05 8.078e-05 0.0005357 0.466997928030752 4.473e-05 0.0002843 517 252.2 251.5| 7.534e-05 7.024e-05 0.0004603 0.467036471267234 3.854e-05 0.0002420 518 250.7 249.9| 6.545e-05 6.096e-05 0.0003949 0.467069620848624 3.315e-05 0.0002056 519 249.1 248.4| 5.675e-05 5.281e-05 0.0003381 0.467098075718722 2.845e-05 0.0001744 520 247.6 246.8| 4.911e-05 4.565e-05 0.0002890 0.467122453012562 2.438e-05 0.0001476 521 246.0 245.2| 4.242e-05 3.939e-05 0.0002466 0.467143296305568 2.084e-05 0.0001247 522 244.5 243.7| 3.657e-05 3.393e-05 0.0002100 0.467161083225730 1.779e-05 0.0001052 523 242.9 242.1| 3.146e-05 2.917e-05 0.0001786 0.467176232449352 1.515e-05 8.857e-05 524 241.3 240.6| 2.702e-05 2.502e-05 0.0001515 0.467189110105001 1.288e-05 7.444e-05 525 239.8 239.0| 2.316e-05 2.143e-05 0.0001284 0.467200035613417 1.093e-05 6.245e-05 526 238.2 237.4| 1.981e-05 1.831e-05 0.0001086 0.467209286993435 9.251e-06 5.229e-05 527 236.7 235.9| 1.692e-05 1.562e-05 9.164e-05 0.467217105665474 7.819e-06 4.371e-05 528 235.1 234.3| 1.442e-05 1.330e-05 7.722e-05 0.467223700785011 6.595e-06 3.647e-05 529 233.5 232.7| 1.227e-05 1.131e-05 6.496e-05 0.467229253138762 5.552e-06 3.037e-05 530 232. 231.2| 1.041e-05 9.590e-06 5.454e-05 0.467233918636062 4.665e-06 2.525e-05 531 230.4 229.6| 8.826e-06 8.119e-06 4.572e-05 0.467237831427375 3.913e-06 2.095e-05 532 228.8 228.0| 7.466e-06 6.861e-06 3.825e-05 0.467241106680897 3.275e-06 1.736e-05 533 227.2 226.5| 6.303e-06 5.788e-06 3.195e-05 0.467243843047062 2.736e-06 1.435e-05 534 225.7 224.9| 5.312e-06 4.873e-06 2.664e-05 0.467246124839347 2.282e-06 1.184e-05 535 224.1 223.3| 4.468e-06 4.095e-06 2.217e-05 0.467248023958252 1.899e-06 9.758e-06 536 222.5 221.7| 3.751e-06 3.434e-06 1.842e-05 0.467249601583699 1.578e-06 8.025e-06 537 221. 220.2| 3.143e-06 2.875e-06 1.527e-05 0.467250909659409 1.308e-06 6.588e-06 538 219.4 218.6| 2.629e-06 2.403e-06 1.265e-05 0.467251992191110 1.083e-06 5.399e-06 539 217.8 217.0| 2.195e-06 2.004e-06 1.045e-05 0.467252886378731 8.942e-07 4.416e-06 540 216.3 215.5| 1.829e-06 1.669e-06 8.621e-06 0.467253623601081 7.372e-07 3.606e-06 541 214.7 213.9| 1.521e-06 1.387e-06 7.100e-06 0.467254230269872 6.067e-07 2.939e-06 542 213.1 212.3| 1.263e-06 1.150e-06 5.837e-06 0.467254728568451 4.983e-07 2.391e-06 543 211.6 210.8| 1.047e-06 9.523e-07 4.790e-06 0.467255137089066 4.085e-07 1.942e-06 544 210.0 209.2| 8.660e-07 7.871e-07 3.924e-06 0.467255471381185 3.343e-07 1.574e-06 545 208.4 207.7| 7.150e-07 6.493e-07 3.209e-06 0.467255744422034 2.730e-07 1.274e-06 546 206.9 206.1| 5.893e-07 5.346e-07 2.619e-06 0.467255967019352 2.226e-07 1.029e-06 547 205.3 204.6| 4.848e-07 4.394e-07 2.135e-06 0.467256148155246 1.811e-07 8.300e-07 548 203.8 203.0| 3.981e-07 3.605e-07 1.736e-06 0.467256295279030 1.471e-07 6.681e-07 549 202.2 201.5| 3.263e-07 2.952e-07 1.410e-06 0.467256414555971 1.193e-07 5.369e-07 550 200.7 199.9| 2.670e-07 2.413e-07 1.143e-06 0.467256511078082 9.652e-08 4.306e-07 551 199.1 198.4| 2.180e-07 1.969e-07 9.251e-07 0.467256589042294 7.796e-08 3.448e-07 552 197.6 196.8| 1.778e-07 1.604e-07 7.474e-07 0.467256651900706 6.286e-08 2.756e-07 553 196.1 195.3| 1.446e-07 1.304e-07 6.027e-07 0.467256702486968 5.059e-08 2.199e-07 554 194.5 193.8| 1.175e-07 1.058e-07 4.852e-07 0.467256743122345 4.064e-08 1.752e-07 555 193.0 192.2| 9.526e-08 8.572e-08 3.900e-07 0.467256775704512 3.258e-08 1.393e-07 556 191.5 190.7| 7.710e-08 6.932e-08 3.129e-07 0.467256801781716 2.608e-08 1.105e-07 557 190. 189.2| 6.229e-08 5.595e-08 2.506e-07 0.467256822614577 2.083e-08 8.757e-08 558 188.4 187.7| 5.023e-08 4.508e-08 2.003e-07 0.467256839227462 1.661e-08 6.926e-08 559 186.9 186.2| 4.044e-08 3.626e-08 1.599e-07 0.467256852451097 1.322e-08 5.468e-08 560 185.4 184.7| 3.250e-08 2.911e-08 1.274e-07 0.467256862957832 1.051e-08 4.310e-08 561 183.9 183.1| 2.607e-08 2.333e-08 1.013e-07 0.467256871290751 8.333e-09 3.391e-08 562 182.4 181.6| 2.087e-08 1.866e-08 8.047e-08 0.467256877887659 6.597e-09 2.663e-08 563 180.9 180.2| 1.668e-08 1.490e-08 6.379e-08 0.467256883100795 5.213e-09 2.088e-08 564 179.4 178.7| 1.331e-08 1.188e-08 5.048e-08 0.467256887212996 4.112e-09 1.634e-08 565 177.9 177.2| 1.060e-08 9.455e-09 3.988e-08 0.467256890450928 3.238e-09 1.277e-08 566 176.4 175.7| 8.429e-09 7.510e-09 3.145e-08 0.467256892995885 2.545e-09 9.958e-09 567 175. 174.2| 6.689e-09 5.955e-09 2.476e-08 0.467256894992592 1.997e-09 7.754e-09 568 173.5 172.7| 5.300e-09 4.714e-09 1.946e-08 0.467256896556352 1.564e-09 6.026e-09 569 172.0 171.3| 4.191e-09 3.725e-09 1.527e-08 0.467256897778854 1.223e-09 4.676e-09 570 170.5 169.8| 3.309e-09 2.938e-09 1.196e-08 0.467256898732867 9.540e-10 3.622e-09 571 169.1 168.4| 2.608e-09 2.313e-09 9.353e-09 0.467256899476032 7.432e-10 2.801e-09 572 167.6 166.9| 2.052e-09 1.818e-09 7.301e-09 0.467256900053920 5.779e-10 2.162e-09 573 166.2 165.5| 1.611e-09 1.427e-09 5.690e-09 0.467256900502491 4.486e-10 1.666e-09 574 164.7 164.0| 1.263e-09 1.118e-09 4.427e-09 0.467256900850066 3.476e-10 1.281e-09 575 163.3 162.6| 9.885e-10 8.739e-10 3.438e-09 0.467256901118909 2.688e-10 9.841e-10 576 161.9 161.2| 7.723e-10 6.822e-10 2.666e-09 0.467256901326487 2.076e-10 7.544e-10 577 160.5 159.7| 6.023e-10 5.315e-10 2.064e-09 0.467256901486479 1.600e-10 5.774e-10 578 159.0 158.3| 4.689e-10 4.135e-10 1.595e-09 0.467256901609577 1.231e-10 4.411e-10 579 157.6 156.9| 3.644e-10 3.211e-10 1.231e-09 0.467256901704123 9.455e-11 3.365e-10 580 156.2 155.5| 2.828e-10 2.489e-10 9.478e-10 0.467256901776612 7.249e-11 2.562e-10 581 154.8 154.1| 2.190e-10 1.926e-10 7.288e-10 0.467256901832092 5.548e-11 1.947e-10 582 153.4 152.7| 1.693e-10 1.488e-10 5.594e-10 0.467256901874480 4.239e-11 1.478e-10 583 152.0 151.3| 1.307e-10 1.148e-10 4.287e-10 0.467256901906809 3.233e-11 1.119e-10 584 150.7 150.| 1.007e-10 8.835e-11 3.280e-10 0.467256901931424 2.461e-11 8.465e-11 585 149.3 148.6| 7.747e-11 6.790e-11 2.505e-10 0.467256901950132 1.871e-11 6.391e-11 586 147.9 147.2| 5.949e-11 5.210e-11 1.911e-10 0.467256901964327 1.419e-11 4.817e-11 587 146.6 145.9| 4.561e-11 3.991e-11 1.454e-10 0.467256901975078 1.075e-11 3.625e-11 588 145.2 144.5| 3.490e-11 3.051e-11 1.105e-10 0.467256901983207 8.129e-12 2.723e-11 589 143.9 143.2| 2.667e-11 2.329e-11 8.388e-11 0.467256901989343 6.136e-12 2.042e-11 590 142.5 141.8| 2.034e-11 1.775e-11 6.354e-11 0.467256901993966 4.623e-12 1.529e-11 591 141.2 140.5| 1.549e-11 1.350e-11 4.805e-11 0.467256901997444 3.478e-12 1.142e-11 592 139.9 139.2| 1.177e-11 1.026e-11 3.628e-11 0.467256902000055 2.611e-12 8.525e-12 593 138.5 137.9| 8.933e-12 7.777e-12 2.735e-11 0.467256902002012 1.958e-12 6.350e-12 594 137.2 136.6| 6.767e-12 5.886e-12 2.058e-11 0.467256902003477 1.465e-12 4.722e-12 595 135.9 135.3| 5.118e-12 4.448e-12 1.546e-11 0.467256902004572 1.094e-12 3.506e-12 596 134.6 134.| 3.864e-12 3.356e-12 1.160e-11 0.467256902005388 8.162e-13 2.599e-12 597 133.3 132.7| 2.913e-12 2.527e-12 8.686e-12 0.467256902005996 6.077e-13 1.923e-12 598 132.1 131.4| 2.192e-12 1.900e-12 6.494e-12 0.467256902006447 4.517e-13 1.420e-12 599 130.8 130.1| 1.647e-12 1.426e-12 4.848e-12 0.467256902006783 3.351e-13 1.048e-12 600 129.5 128.9| 1.235e-12 1.069e-12 3.613e-12 0.467256902007031 2.483e-13 7.713e-13 \--> tnc{sum} = 0.467256902007, tnc+pnorm(-del) = 0.467256902007, lower.t = 1 [1] 0.4672569 > pntR(30,30,ncp = 30:40) pnt(t= 30 , df= 30 , delta= 30 ) ==> x= 0.9677419 : p= 1.846942e-196 600 iter.: tnc{sum} = 0.467256902007, tnc+pnorm(-del) = 0.467256902007, lower.t = 1 pnt(t= 30 , df= 30 , delta= 31 ) ==> x= 0.9677419 : p= 1.048266e-209 633 iter.: tnc{sum} = 0.370447639204, tnc+pnorm(-del) = 0.370447639204, lower.t = 1 pnt(t= 30 , df= 30 , delta= 32 ) ==> x= 0.9677419 : p= 2.188746e-223 668 iter.: tnc{sum} = 0.282018230456, tnc+pnorm(-del) = 0.282018230456, lower.t = 1 pnt(t= 30 , df= 30 , delta= 33 ) ==> x= 0.9677419 : p= 1.68122e-237 703 iter.: tnc{sum} = 0.205862903802, tnc+pnorm(-del) = 0.205862903802, lower.t = 1 pnt(t= 30 , df= 30 , delta= 34 ) ==> x= 0.9677419 : p= 4.75072e-252 739 iter.: tnc{sum} = 0.143936755733, tnc+pnorm(-del) = 0.143936755733, lower.t = 1 pnt(t= 30 , df= 30 , delta= 35 ) ==> x= 0.9677419 : p= 4.938554e-267 776 iter.: tnc{sum} = 0.0963250973129, tnc+pnorm(-del) = 0.0963250973129, lower.t = 1 pnt(t= 30 , df= 30 , delta= 36 ) ==> x= 0.9677419 : p= 1.888625e-282 813 iter.: tnc{sum} = 0.0616699962855, tnc+pnorm(-del) = 0.0616699962855, lower.t = 1 pnt(t= 30 , df= 30 , delta= 37 ) ==> x= 0.9677419 : p= 2.657034e-298 851 iter.: tnc{sum} = 0.0377619966759, tnc+pnorm(-del) = 0.0377619966759, lower.t = 1 large 'df' or "large" 'ncp' ---> return()ing pnorm(*) = 0.01958007877837746 large 'df' or "large" 'ncp' ---> return()ing pnorm(*) = 0.010375072658058 large 'df' or "large" 'ncp' ---> return()ing pnorm(*) = 0.005196079382091164 [1] 0.467256902 0.370447639 0.282018230 0.205862904 0.143936756 0.096325097 [7] 0.061669996 0.037761997 0.019580079 0.010375073 0.005196079 > pntR(2, 10,ncp=1e5)#> C-code directly {underflow p=0, |ncp| = |delta| too large large 'df' or "large" 'ncp' ---> return()ing pnorm(*) = 0 [1] 0 > pntR(2,df=10,ncp=1e4)#> (ditto) large 'df' or "large" 'ncp' ---> return()ing pnorm(*) = 0 [1] 0 > > ## t --> 0 : is it problematic ? > df <- 1 > df <- 10 > ncp <- 1 > x <- 1e-12 > x <- 1e-6 > (pt1 <- pntR(x * sqrt((df+2)/df), df=df+2, ncp=ncp, errmax = 1e-14)) pnt(t= 1.095445e-06 , df= 12 , delta= 1 ) ==> x= 1e-13 : p= 0.3032653 1 iter.: tnc{sum} = 2.59606926552e-07, tnc+pnorm(-del) = 0.158655513538, lower.t = 1 [1] 0.1586555 > (pt2 <- pntR(x , df=df , ncp=ncp, errmax = 1e-14)) pnt(t= 1e-06 , df= 10 , delta= 1 ) ==> x= 1e-13 : p= 0.3032653 1 iter.: tnc{sum} = 2.36006285862e-07, tnc+pnorm(-del) = 0.158655489938, lower.t = 1 [1] 0.1586555 > c(pt1 - pt2, (pt1 - pt2)/x) [1] 2.360064e-08 2.360064e-02 > > if(FALSE) ## useful to see things + debug(pntR) > pntR(1e-8, df=10, ncp=1, errmax = 1e-14) pnt(t= 1e-08 , df= 10 , delta= 1 ) ==> x= 1e-17 : p= 0.3032653 1 iter.: tnc{sum} = 2.36006166036e-09, tnc+pnorm(-del) = 0.158655256292, lower.t = 1 [1] 0.1586553 > pntR(1e-15, df=10, ncp=1, errmax = 1e-14) pnt(t= 1e-15 , df= 10 , delta= 1 ) ==> x= 1e-31 : p= 0.3032653 1 iter.: tnc{sum} = 2.36006164827e-16, tnc+pnorm(-del) = 0.158655253931, lower.t = 1 [1] 0.1586553 > > plot(function(t)pntJW39(t,30,30)) > plot(function(t)pntJW39(t,100,30),-9,5, col="red", log="y") > plot(function(t)pt (t,100,30),-9,5, col="blue", log="y",add=TRUE) > plot(function(t)pntJW39(t,100,30),0,5, col="red", log="y")#~ lin > plot(function(t)pntJW39(t,100,30),0,15, col="red", log="y") > plot(function(t)pt(t,100,ncp=30),0,15, col="blue", log="y", add=TRUE) > > plot(function(t)pntJW39(t,100,30),0,25, col="red", log="y") > plot(function(t)pntJW39(t,100,30),0,25, col="red") > > plot(function(t)pntJW39(t,100,30),0,45, col="red") > > plot(function(t)pntJW39(t,100,30),20,50, col="red", log="y") > plot(function(t)pt (t,100,30),20,50, col="blue", log="y",add=TRUE) > > o <- par(las=1, mar=.1+c(4,4,4,4)) > plot(function(t)abs(pt(t,100,30)-pntJW39(t,100,30)),20,50, + col="red", log="y"); eaxis(4, at=10^-(2:8)) > par(o) > > ## xtended x-range > o <- par(las=1, mar=.1+c(4,4,4,4)) > plot(function(t)abs(pt(t,100,30)-pntJW39(t,100,30)), 17, 120, n=1001, + col="red", log="y"); eaxis(4) There were 50 or more warnings (use warnings() to see the first 50) > par(o) > > > ### --------------------- qt(), qtAppr(), qtNappr(), etc --------------------- > > ## df=1 is pretty bad... > plot(function(t)qt(t,df=1)) > plot(function(t)qtAppr(t,df=1,ncp=0),col="blue",add=TRUE) Warning message: In sqrt(den + ncp^2 * (1 - b2)) : NaNs produced > > ## df=2 quite a bit better.. > plot(function(t)qtAppr(t,df=2,ncp=0),col="blue") Warning message: In sqrt(den + ncp^2 * (1 - b2)) : NaNs produced > plot(function(t)qt(t,df=2),add=TRUE) > > ## df=4 a bit better.. still only for alpha ~ in (.1, .9) > plot(function(t)qtAppr(t,df=4,ncp=0),col="blue") Warning message: In sqrt(den + ncp^2 * (1 - b2)) : NaNs produced > plot(function(t)qt(t,df=4),add=TRUE) > > plot(function(t)abs(1-qt(t,df=4)/qtAppr(t,df=4,ncp=0)), main="rel.Error") Warning message: In sqrt(den + ncp^2 * (1 - b2)) : NaNs produced > plot(function(t)abs(1-qt(t,df=10)/qtAppr(t,df=10,ncp=0)), main="rel.Error") Warning message: In sqrt(den + ncp^2 * (1 - b2)) : NaNs produced > > ## max error: 10e-5, however... still NaN's sqrt() > plot(function(t)abs(1-qt(t,df=100)/qtAppr(t,df=100,ncp=0))) Warning message: In sqrt(den + ncp^2 * (1 - b2)) : NaNs produced > > plot(function(t)qtAppr(t,df=100,ncp=100)) Warning message: In sqrt(den + ncp^2 * (1 - b2)) : NaNs produced > ## catastrophe : !!!! > plot(function(t)qtAppr(t,df=1,ncp=100)) Warning message: In sqrt(den + ncp^2 * (1 - b2)) : NaNs produced > > > plot(function(t) t - pt (qtAppr(t,df=4,ncp=100),df=4,ncp=100)) Warning message: In sqrt(den + ncp^2 * (1 - b2)) : NaNs produced > > ## --> pntJW39() uses MUCH better asymptotic than Abramowitz&Stegun (in pnt). > plot(function(t) t - pntJW39(qtAppr(t,df=4,ncp=100),df=4,ncp=100), + col='red',add=TRUE) Warning message: In sqrt(den + ncp^2 * (1 - b2)) : NaNs produced > ## Absolute Error: very small -- just proves that the two ".appr" CORRESPOND! > plot(function(t) t - pntJW39(qtAppr(t,df=4,ncp=100),df=4,ncp=100)) Warning message: In sqrt(den + ncp^2 * (1 - b2)) : NaNs produced > > > plot(function(t)qtAppr(t,df=10,ncp=1e5)) Warning message: In sqrt(den + ncp^2 * (1 - b2)) : NaNs produced > > ## Shows that pt(,, ncp=1e5) uses asymptotic form alright: > plot(function(t) t - pt (qtAppr(t,df=10,ncp=1e5),df=10,ncp=1e5)) Warning message: In sqrt(den + ncp^2 * (1 - b2)) : NaNs produced > ## --> pntJW39() MUCH better ========== fitting to qtAppr !! > ## ------ not necessarily better asymptotic than Abramowitz&Stegun (in pnt). > plot(function(t) t - pntJW39(qtAppr(t,df=10,ncp=1e5),df=10,ncp=1e5), + col='red',add=TRUE) Warning message: In sqrt(den + ncp^2 * (1 - b2)) : NaNs produced > > > ###--- Diverse tests, some lifted from ../R/t-nonc-fn.R ------------------------ > > ## [MM: the next 2 'dntJKBch(.)' were 'dntRwrong(.)' originally ] > > print(f. <- .dntJKBch(1:6, df=3, ncp=5, check=TRUE), digits = 4) [1] 0.0005083 0.0260607 0.1191377 0.1761045 0.1657718 0.1305411 > ## now:[1] 0.0005083 0.0260607 0.1191377 0.1761045 0.1657718 0.1305411 > ## was:[1] 0.0032023 0.2728377 1.5174647 2.4481320 2.4106931 1.9481189 -- wrong "but sensible" > > x <- seq(-1,12, by=1/16) > if(FALSE) ## FIXME ?! -- internal logic error: + .dntJKBch(x, df=3, ncp=5, check=TRUE) > ## ## Error in (function (x, df, ncp, log = FALSE, M = 1000, check = FALSE, : > ## ## exp(lterms[ii]) and terms[ii] are not equal: > ## ## 'is.NA' value mismatch: 0 in current 340 in target > > ## -- This is now in help page examples >>> ../man/dnt.Rd <<< > ## fx <- dt(x, df=3, ncp=5) ~~~~~~~~~~~~~ > ## re1 <- 1 - .dntJKBch(x, df=3, ncp=5) / fx # with warnings > ## re2 <- 1 - dntJKBf (x, df=3, ncp=5) / fx > ## summary(warnings()) ## "In log(x * ncp * sqrt(2)/sqrt(df + x^2)) : NaNs produced" > ## all.equal(re1[!is.na(re1)], re2[!is.na(re1)], tol=0)## Mean relative ...: 2.068..e-5 > ## stopifnot(all.equal(re1[!is.na(re1)], re2[!is.na(re1)], tol=1e-6)) > ## matplot(x, log10(abs(cbind(re1, re2))), type = "o", cex = 1/4) > > > print(ft <- .dntJKBch(1:6, df=3, ncp=5), digits = 4) [1] 0.0005083 0.0260607 0.1191377 0.1761045 0.1657718 0.1305411 > ## [1] 0.0005083 0.0260607 0.1191377 0.1761045 0.1657718 0.1305411 > ## correct !! *with* the factorial ! > all.equal(ft, dt(1:6, df=3, ncp=5), tol = 0) # 1.4959e-12 [1] "Mean relative difference: 1.495917e-12" > stopifnot(all.equal(ft, dt(1:6, df=3, ncp=5), tol = 1e-11)) > > > ## From: Stephen Berman .. > ## To: > ## Subject: [R] Why does qt() return Inf with certain negative ncp values? > ## Date: Mon, 13 Jun 2022 23:47:29 +0200 > > ## Can anyone explain why Inf appears in the following results? > > ## MM: > options(nwarnings = 5e5) > > sapply(-1:-10, function(ncp) qt(1-1*(10^(-4+ncp)), 35, ncp)) [1] 3.6527153 3.0627759 2.4158355 1.7380812 1.0506904 0.3700821 [7] Inf -0.9279783 -1.5341759 -2.1085213 There were 1060 warnings (use warnings() to see them) > ## [1] 3.6527153 3.0627759 2.4158355 1.7380812 1.0506904 0.3700821 > ## [7] Inf -0.9279783 -1.5341759 -2.1085213 > ## There were 1060 warnings (use warnings() to see them) > summary(warnings()) 1060 identical warnings: In qt(1 - 1 * (10^(-4 + ncp)), 35, ncp) : full precision may not have been achieved in 'pnt{final}' > ## 1060 identical warnings: > ## In qt(1 - 1 * (10^(-4 + ncp)), 35, ncp) : > ## full precision may not have been achieved in 'pnt{final}' > > nc1 <- seq(-12, -1, by=1/32) > r1 <- vapply(nc1, function(ncp) qt(1 - 10^(-4+ncp), df=35, ncp=ncp), 1.23) There were 8264 warnings (use warnings() to see them) > ## There were 8264 warnings (use warnings() to see them) > summary(warnings()) 8264 identical warnings: In qt(1 - 10^(-4 + ncp), df = 35, ncp = ncp) : full precision may not have been achieved in 'pnt{final}' > ## 8264 identical warnings: > ## In qt(1 - 10^(-4 + ncp), df = 35, ncp = ncp) : > ## full precision may not have been achieved in 'pnt{final}' > > ## MM: better than "1 - r1. <- vapply(nc1, function(ncp) qt(10^(-4+ncp), df=35, ncp=ncp, lower.tail=FALSE), 1.23) There were 8267 warnings (use warnings() to see them) > summary(warnings()) 8267 identical warnings: In qt(10^(-4 + ncp), df = 35, ncp = ncp, lower.tail = FALSE) : full precision may not have been achieved in 'pnt{final}' > matplot(nc1, cbind(r1,r1.), type="l") > ## Zoom in "left tail": ==> {lower.tail=FALSE} did not help much > i <- nc1 <= -10 > matplot(nc1[i], cbind(r1,r1.)[i,], type="l") > > ## Zoom in around -7 > r2 <- sapply(seq(-6.9, -7.9, -0.1), function(ncp) qt(1-1*(10^(-4+ncp)), 35, ncp)) There were 3086 warnings (use warnings() to see them) > ## [1] -0.2268386 Inf Inf Inf -0.4857400 -0.5497784 > ## [7] -0.6135402 -0.6770143 -0.7401974 -0.8030853 -0.8656810 > summary(warnings()) 3086 identical warnings: In qt(1 - 1 * (10^(-4 + ncp)), 35, ncp) : full precision may not have been achieved in 'pnt{final}' > ## 3086 identical warnings: > ## In qt(1 - 1 * (10^(-4 + ncp)), 35, ncp) : > ## full precision may not have been achieved in 'pnt{final}' > > nc2 <- seq(-60, -55, by=1/32)/8 # {1/8, 1/32: exact decimals} > r2 <- vapply(nc2, function(ncp) qt(1 - 10^(-4+ncp), df=35, ncp=ncp), 1.23) There were 52444 warnings (use warnings() to see them) > ## There were 52444 warnings (use warnings() to see them) > summary(warnings()) 52444 identical warnings: In qt(1 - 10^(-4 + ncp), df = 35, ncp = ncp) : full precision may not have been achieved in 'pnt{final}' > ## 52444 identical warnings: .... > ## lower.tail=FALSE > r2. <- vapply(nc2, function(ncp) + qt(10^(-4+ncp), df=35, ncp=ncp, lower.tail=FALSE), 1.23) There were 52444 warnings (use warnings() to see them) > all.equal(r2., r2) ## "Mean relative difference: 6.740991e-06" [1] "Mean relative difference: 6.740991e-06" > matplot(nc2, cbind(r2,r2.), type="l") > ## not why such a locally linear function should not work... > > ## In case it matters: > > ## > sessionInfo() > ## R Under development (unstable) (2022-06-05 r82452) > ## Platform: x86_64-pc-linux-gnu (64-bit) > ## Running under: Linux From Scratch r11.0-165 > > ##--> MM: exploring > q2 <- seq(-3/4, -1/4, by=1/128) > pq2 <- pt(q2, 35, ncp=-7, lower.tail=FALSE) > # no warnings here! > ### ==> Providing qtU() a simple uniroot() - based inversion of pt() > qpqU <- qtU(pq2, 35, ncp=-7, lower.tail=FALSE, tol=1e-10) > stopifnot(all.equal(q2, qpqU, tol=1e-9)) # perfect! > > ## and qtAppr() is really poor in comparison > plot(pq2 ~ q2, type="l", col=2, lwd=2) > qpq2 <- qt (pq2, 35, ncp=-7, lower.tail=FALSE) # 6262 warnings There were 6262 warnings (use warnings() to see them) > qAp2a <- qtAppr(pq2, 35, ncp=-7, lower.tail=FALSE, method="a") > qAp2b <- qtAppr(pq2, 35, ncp=-7, lower.tail=FALSE, method="b") > qAp2c <- qtAppr(pq2, 35, ncp=-7, lower.tail=FALSE, method="c") > summary(warnings()) 6262 identical warnings: In qt(pq2, 35, ncp = -7, lower.tail = FALSE) : full precision may not have been achieved in 'pnt{final}' > ## 6262 identical warnings: ... > ## plot error -- interestingly *all* positive: .. then JUMP to Inf > plot (qpq2-q2 ~ q2, type="l", col=2, lwd=2); abline(h=0,lty=2) > > ## but the approximations are *very* bad, compared: > matplot(q2, cbind(qpq2, qAp2a,qAp2b,qAp2c)-q2, + type="l", col=2:5, lwd=2) > ## leave away the "b" which is particularly bad (here): > matplot(q2, cbind(qpq2, qAp2a,qAp2c)-q2, + type="l", col=2:5, lwd=2) > ## absolute values & log-scale: > matplot(q2, abs( cbind(qpq2, qAp2a,qAp2c) - q2), log="y", + type="l", col=2:5, lwd=2) > ## Idea: what about *starting* with qtAppr() and then do 1 or a few *Newton* steps? > > ## unfinished ... > mat2 <- cbind(q2, pq2, qpq2, Dq=qpq2-q2, relE=relErrV(q2, qpq2)) > tail(mat2, 10) q2 pq2 qpq2 Dq relE [56,] -0.3203125 1.218900e-11 -0.3203125 2.581537e-08 -8.059432e-08 [57,] -0.3125000 1.154225e-11 -0.3124999 1.019742e-07 -3.263174e-07 [58,] -0.3046875 1.092953e-11 -0.3046858 1.725959e-06 -5.664687e-06 [59,] -0.2968750 1.034907e-11 -0.2968736 1.358450e-06 -4.575833e-06 [60,] -0.2890625 9.799191e-12 Inf Inf Inf [61,] -0.2812500 9.278293e-12 Inf Inf Inf [62,] -0.2734375 8.784871e-12 Inf Inf Inf [63,] -0.2656250 8.317484e-12 Inf Inf Inf [64,] -0.2578125 7.874773e-12 Inf Inf Inf [65,] -0.2500000 7.455446e-12 Inf Inf Inf > > > ###=== Large df ; R currently uses "primitive" (<==> qtNappr(*, k=0)) for df > 1e20 > ## ======== > ##--- *CENTRAL* case (ncp == 0) ---------- > > if(!dev.interactive(orNone=TRUE)) { dev.off(); pdf("t-nonc_P-3.pdf") } > > ## originally from ../man/qtAppr.Rd : > qts <- function(p, df, lower.tail = TRUE, log.p = FALSE, tolUa = 1e-14) { + cbind(qt = qt (p, df=df, lower.tail=lower.tail, log.p=log.p) + , qtU = qtU (p, df=df, lower.tail=lower.tail, log.p=log.p) + , qtUa = qtU (p, df=df, lower.tail=lower.tail, log.p=log.p, tol = tolUa) + , qtN0 = qtNappr(p, df=df, lower.tail=lower.tail, log.p=log.p, k=0) + , qtN1 = qtNappr(p, df=df, lower.tail=lower.tail, log.p=log.p, k=1) + , qtN2 = qtNappr(p, df=df, lower.tail=lower.tail, log.p=log.p, k=2) + , qtN3 = qtNappr(p, df=df, lower.tail=lower.tail, log.p=log.p, k=3) + , qtN4 = qtNappr(p, df=df, lower.tail=lower.tail, log.p=log.p, k=4) + ) + } > > p.qtNappr <- function(qtsR, p, ip, errMax = 1e-17, do.diff=FALSE) { + stopifnot(is.matrix(qtsR), (n <- length(p)) == nrow(qtsR), 1 <= ip, ip <= n) + (iN <- startsWith(prefix="qtN", colnames(qtsR))) # also "drop" qtU()'s result + (dqN <- (qtsR[,iN] - qtsR[,1])[ip,]) + if(do.diff) ## difference + matplot(p[ip], dqN, type="l", col=2:6, + main = paste0("difference qtNappr(p,df) - qt(p,df), df=",df), xlab=quote(p)) + ## |difference| + matplot(p[ip], pmax(abs(dqN), errMax), log="y", type="l", col=2:6, + main = paste0("abs. difference |qtNappr(p,df) - qt(p,df)|, df=",df), xlab=quote(p)) + legend("bottomright", paste0("k=",0:4), col=2:6, lty=1:5, bty="n") + ## Rel.err. + matplot(p[ip], pmax(abs(dqN/qtsR[ip,"qt"]), errMax), log="y", type="l", col=2:6, + main = sprintf("rel.error qtNappr(p, df=%g, k=*)",df), xlab=quote(p)) + legend("left", paste0("k=",0:4), col=2:6, lty=1:5, bty="n") + invisible(dqN) + } > > p <- (0:100)/100 > ii <- 2:100 # drop p=0 & p=1 where q*(p, .) == +/- Inf > > df <- 100 # <<<<<<<<<<<<<<<<<<<<<<<<<<<<<<< > qsp1c <- qts(p, df = df) > matplot(p[ii], qsp1c[ii,], type="l") # "all on top" > p.qtNappr(qsp1c, p, ip=ii, do.diff=TRUE) > > df <- 2000 # <<<<<<<<<<<<<<<<<<<<<<<<<<<<<<< > qsp1c <- qts(p, df=df) > p.qtNappr(qsp1c, p, ip=ii) > ## qtAppr is really worse even than k=2 here (df=2000): > qtA <- qtAppr(p, df=df, ncp=0) Warning message: In sqrt(den + ncp^2 * (1 - b2)) : NaNs produced > lines(p, pmax(1e-17, abs(qtA/qsp1c[,1] - 1)), col=2, lwd=2, lty=2) > legend(0.6, 1e-9, "qtAppr()", bty="n",cex=.8, col=2, lwd=2, lty=2) > ## qtA is between k=1 and k=2 ... > print(digits=4, cbind(p, cbind(qsp1c[,-1], qtA) - qsp1c[,1])[ii,]) p qtU qtUa qtN0 qtN1 qtN2 qtN3 [1,] 0.01 5.029e-11 -4.441e-16 1.866e-03 1.431e-06 8.402e-10 3.522e-13 [2,] 0.02 1.625e-06 -4.441e-16 1.340e-03 8.531e-07 4.145e-10 1.399e-13 [3,] 0.03 2.080e-06 -2.220e-16 1.067e-03 5.986e-07 2.546e-10 7.327e-14 [4,] 0.04 -4.719e-08 -3.331e-15 8.900e-04 4.516e-07 1.721e-10 4.330e-14 [5,] 0.05 -7.722e-07 -4.441e-16 7.622e-04 3.552e-07 1.229e-10 2.709e-14 [6,] 0.06 -1.681e-07 -2.220e-16 6.644e-04 2.871e-07 9.086e-11 1.776e-14 [7,] 0.07 2.357e-07 -8.882e-16 5.865e-04 2.367e-07 6.878e-11 1.155e-14 [8,] 0.08 -2.475e-08 -4.441e-16 5.226e-04 1.979e-07 5.293e-11 7.994e-15 [9,] 0.09 -5.084e-07 4.441e-16 4.690e-04 1.674e-07 4.120e-11 5.551e-15 [10,] 0.10 2.959e-11 2.220e-16 4.234e-04 1.428e-07 3.232e-11 3.331e-15 [11,] 0.11 2.279e-10 -1.776e-15 3.841e-04 1.226e-07 2.547e-11 2.220e-15 [12,] 0.12 -2.166e-08 2.220e-16 3.498e-04 1.060e-07 2.011e-11 1.110e-15 [13,] 0.13 1.030e-07 -2.220e-16 3.195e-04 9.197e-08 1.587e-11 6.661e-16 [14,] 0.14 -2.278e-06 1.776e-15 2.927e-04 8.015e-08 1.248e-11 2.220e-16 [15,] 0.15 -2.132e-08 -8.882e-16 2.688e-04 7.007e-08 9.751e-12 -4.441e-16 [16,] 0.16 -6.324e-08 -9.992e-16 2.473e-04 6.142e-08 7.541e-12 -3.331e-16 [17,] 0.17 -3.580e-08 2.220e-16 2.279e-04 5.395e-08 5.743e-12 -5.551e-16 [18,] 0.18 -1.497e-08 -5.551e-16 2.103e-04 4.748e-08 4.275e-12 -6.661e-16 [19,] 0.19 -3.459e-09 -4.441e-16 1.944e-04 4.184e-08 3.075e-12 -5.551e-16 [20,] 0.20 -2.452e-06 -6.661e-16 1.798e-04 3.691e-08 2.093e-12 -4.441e-16 [21,] 0.21 -1.712e-08 0.000e+00 1.664e-04 3.259e-08 1.290e-12 -6.661e-16 [22,] 0.22 -1.149e-08 -3.331e-16 1.541e-04 2.879e-08 6.349e-13 -6.661e-16 [23,] 0.23 4.518e-08 -1.110e-16 1.428e-04 2.544e-08 1.028e-13 -5.551e-16 [24,] 0.24 -9.008e-09 -2.220e-16 1.324e-04 2.249e-08 -3.267e-13 -6.661e-16 [25,] 0.25 -1.982e-07 -3.331e-16 1.227e-04 1.987e-08 -6.705e-13 -5.551e-16 [26,] 0.26 -1.761e-06 -2.220e-16 1.137e-04 1.756e-08 -9.421e-13 -5.551e-16 [27,] 0.27 2.477e-06 -6.661e-16 1.054e-04 1.550e-08 -1.153e-12 -6.661e-16 [28,] 0.28 5.232e-07 -2.220e-16 9.762e-05 1.368e-08 -1.313e-12 -4.441e-16 [29,] 0.29 -1.327e-06 -1.998e-15 9.037e-05 1.206e-08 -1.428e-12 -4.441e-16 [30,] 0.30 -4.474e-09 -5.551e-16 8.359e-05 1.062e-08 -1.507e-12 -4.441e-16 [31,] 0.31 7.048e-08 -6.106e-16 7.723e-05 9.342e-09 -1.554e-12 -4.441e-16 [32,] 0.32 1.319e-09 -1.110e-16 7.126e-05 8.206e-09 -1.575e-12 -3.331e-16 [33,] 0.33 -2.377e-06 -5.551e-17 6.564e-05 7.197e-09 -1.572e-12 -2.776e-16 [34,] 0.34 -2.211e-08 3.886e-16 6.034e-05 6.300e-09 -1.550e-12 -2.776e-16 [35,] 0.35 -2.017e-09 -5.551e-17 5.532e-05 5.503e-09 -1.511e-12 -2.776e-16 [36,] 0.36 -3.724e-08 -2.776e-16 5.057e-05 4.795e-09 -1.458e-12 -2.776e-16 [37,] 0.37 1.960e-06 -1.665e-16 4.605e-05 4.166e-09 -1.393e-12 -2.220e-16 [38,] 0.38 -9.275e-08 -1.110e-16 4.175e-05 3.608e-09 -1.317e-12 -2.220e-16 [39,] 0.39 -6.076e-07 -1.665e-16 3.764e-05 3.111e-09 -1.233e-12 -1.665e-16 [40,] 0.40 -6.688e-07 -5.551e-17 3.370e-05 2.669e-09 -1.141e-12 -1.665e-16 [41,] 0.41 -1.725e-07 -1.665e-16 2.992e-05 2.275e-09 -1.042e-12 -1.388e-16 [42,] 0.42 -1.448e-09 -1.388e-16 2.627e-05 1.924e-09 -9.383e-13 -1.665e-16 [43,] 0.43 -4.263e-08 -1.527e-15 2.273e-05 1.608e-09 -8.299e-13 -1.110e-16 [44,] 0.44 -1.148e-07 -2.776e-17 1.930e-05 1.323e-09 -7.178e-13 -1.110e-16 [45,] 0.45 -3.176e-09 -5.551e-17 1.596e-05 1.064e-09 -6.024e-13 -5.551e-17 [46,] 0.46 -9.304e-09 0.000e+00 1.268e-05 8.265e-10 -4.848e-13 -6.939e-17 [47,] 0.47 -2.827e-07 -1.110e-16 9.463e-06 6.055e-10 -3.652e-13 -4.163e-17 [48,] 0.48 -1.787e-09 -8.812e-16 6.285e-06 3.968e-10 -2.442e-13 -2.082e-17 [49,] 0.49 -5.216e-08 4.163e-17 3.136e-06 1.964e-10 -1.223e-13 -1.388e-17 [50,] 0.50 0.000e+00 0.000e+00 0.000e+00 0.000e+00 0.000e+00 0.000e+00 [51,] 0.51 5.216e-08 2.082e-17 -3.136e-06 -1.964e-10 1.223e-13 1.388e-17 [52,] 0.52 1.787e-09 8.743e-16 -6.285e-06 -3.968e-10 2.442e-13 2.082e-17 [53,] 0.53 2.827e-07 -2.776e-17 -9.463e-06 -6.055e-10 3.652e-13 4.163e-17 [54,] 0.54 9.304e-09 2.776e-17 -1.268e-05 -8.265e-10 4.848e-13 6.939e-17 [55,] 0.55 3.176e-09 -5.551e-17 -1.596e-05 -1.064e-09 6.024e-13 5.551e-17 [56,] 0.56 1.148e-07 1.388e-16 -1.930e-05 -1.323e-09 7.178e-13 1.110e-16 [57,] 0.57 4.263e-08 1.527e-15 -2.273e-05 -1.608e-09 8.299e-13 1.110e-16 [58,] 0.58 1.448e-09 5.551e-17 -2.627e-05 -1.924e-09 9.383e-13 1.665e-16 [59,] 0.59 1.725e-07 8.327e-17 -2.992e-05 -2.275e-09 1.042e-12 1.388e-16 [60,] 0.60 6.688e-07 -5.551e-17 -3.370e-05 -2.669e-09 1.141e-12 1.665e-16 [61,] 0.61 6.076e-07 3.331e-16 -3.764e-05 -3.111e-09 1.233e-12 1.665e-16 [62,] 0.62 9.275e-08 2.220e-16 -4.175e-05 -3.608e-09 1.317e-12 2.220e-16 [63,] 0.63 -1.960e-06 1.665e-16 -4.605e-05 -4.166e-09 1.393e-12 2.220e-16 [64,] 0.64 3.724e-08 1.110e-16 -5.057e-05 -4.795e-09 1.458e-12 2.776e-16 [65,] 0.65 2.017e-09 5.551e-17 -5.532e-05 -5.503e-09 1.511e-12 2.776e-16 [66,] 0.66 2.211e-08 -1.665e-16 -6.034e-05 -6.300e-09 1.550e-12 2.776e-16 [67,] 0.67 2.377e-06 2.220e-16 -6.564e-05 -7.197e-09 1.572e-12 2.776e-16 [68,] 0.68 -1.319e-09 1.665e-16 -7.126e-05 -8.206e-09 1.575e-12 3.331e-16 [69,] 0.69 -7.048e-08 6.106e-16 -7.723e-05 -9.342e-09 1.554e-12 4.996e-16 [70,] 0.70 4.474e-09 5.551e-16 -8.359e-05 -1.062e-08 1.507e-12 4.441e-16 [71,] 0.71 1.327e-06 2.554e-15 -9.037e-05 -1.206e-08 1.428e-12 4.441e-16 [72,] 0.72 -5.232e-07 2.220e-16 -9.762e-05 -1.368e-08 1.313e-12 4.441e-16 [73,] 0.73 -2.477e-06 5.551e-16 -1.054e-04 -1.550e-08 1.153e-12 6.661e-16 [74,] 0.74 1.761e-06 2.220e-16 -1.137e-04 -1.756e-08 9.421e-13 5.551e-16 [75,] 0.75 1.982e-07 0.000e+00 -1.227e-04 -1.987e-08 6.705e-13 5.551e-16 [76,] 0.76 9.008e-09 5.551e-16 -1.324e-04 -2.249e-08 3.267e-13 6.661e-16 [77,] 0.77 -4.518e-08 7.772e-16 -1.428e-04 -2.544e-08 -1.028e-13 5.551e-16 [78,] 0.78 1.149e-08 7.772e-16 -1.541e-04 -2.879e-08 -6.349e-13 6.661e-16 [79,] 0.79 1.712e-08 2.220e-16 -1.664e-04 -3.259e-08 -1.290e-12 6.661e-16 [80,] 0.80 2.452e-06 4.441e-16 -1.798e-04 -3.691e-08 -2.093e-12 4.441e-16 [81,] 0.81 3.459e-09 6.661e-16 -1.944e-04 -4.184e-08 -3.075e-12 6.661e-16 [82,] 0.82 1.497e-08 3.331e-16 -2.103e-04 -4.748e-08 -4.275e-12 6.661e-16 [83,] 0.83 3.580e-08 -5.551e-16 -2.279e-04 -5.395e-08 -5.743e-12 5.551e-16 [84,] 0.84 6.324e-08 8.882e-16 -2.473e-04 -6.142e-08 -7.541e-12 3.331e-16 [85,] 0.85 2.132e-08 8.882e-16 -2.688e-04 -7.007e-08 -9.751e-12 4.441e-16 [86,] 0.86 2.278e-06 -1.998e-15 -2.927e-04 -8.015e-08 -1.248e-11 -2.220e-16 [87,] 0.87 -1.030e-07 4.441e-16 -3.195e-04 -9.197e-08 -1.587e-11 -6.661e-16 [88,] 0.88 2.166e-08 -4.441e-16 -3.498e-04 -1.060e-07 -2.011e-11 -1.110e-15 [89,] 0.89 -2.279e-10 1.554e-15 -3.841e-04 -1.226e-07 -2.547e-11 -2.220e-15 [90,] 0.90 -2.959e-11 -4.441e-16 -4.234e-04 -1.428e-07 -3.232e-11 -3.331e-15 [91,] 0.91 5.084e-07 -2.220e-16 -4.690e-04 -1.674e-07 -4.120e-11 -5.551e-15 [92,] 0.92 2.475e-08 2.220e-16 -5.226e-04 -1.979e-07 -5.293e-11 -8.216e-15 [93,] 0.93 -2.357e-07 0.000e+00 -5.865e-04 -2.367e-07 -6.878e-11 -1.177e-14 [94,] 0.94 1.681e-07 2.220e-16 -6.644e-04 -2.871e-07 -9.086e-11 -1.754e-14 [95,] 0.95 7.722e-07 1.776e-15 -7.622e-04 -3.552e-07 -1.229e-10 -2.687e-14 [96,] 0.96 4.719e-08 3.109e-15 -8.900e-04 -4.516e-07 -1.721e-10 -4.330e-14 [97,] 0.97 -2.080e-06 8.882e-16 -1.067e-03 -5.986e-07 -2.546e-10 -7.327e-14 [98,] 0.98 -1.625e-06 8.882e-16 -1.340e-03 -8.531e-07 -4.145e-10 -1.399e-13 [99,] 0.99 -5.029e-11 2.220e-15 -1.866e-03 -1.431e-06 -8.402e-10 -3.522e-13 qtN4 qtA [1,] 0.000e+00 -5.802e-07 [2,] 0.000e+00 -2.911e-07 [3,] -2.220e-16 -1.762e-07 [4,] -2.220e-16 -1.157e-07 [5,] 2.220e-16 -7.922e-08 [6,] 2.220e-16 -5.560e-08 [7,] -2.220e-16 -3.952e-08 [8,] 2.220e-16 -2.821e-08 [9,] 2.220e-16 -2.007e-08 [10,] 0.000e+00 -1.412e-08 [11,] 0.000e+00 -9.721e-09 [12,] 0.000e+00 -6.452e-09 [13,] 2.220e-16 -4.018e-09 [14,] 0.000e+00 -2.208e-09 [15,] -2.220e-16 -8.718e-10 [16,] -1.110e-16 1.037e-10 [17,] 0.000e+00 8.024e-10 [18,] -1.110e-16 1.288e-09 [19,] 2.220e-16 1.610e-09 [20,] 2.220e-16 1.806e-09 [21,] 1.110e-16 1.905e-09 [22,] 2.220e-16 1.932e-09 [23,] 1.110e-16 1.904e-09 [24,] 1.110e-16 1.836e-09 [25,] 2.220e-16 1.739e-09 [26,] 1.110e-16 1.623e-09 [27,] -1.110e-16 1.494e-09 [28,] 1.110e-16 1.359e-09 [29,] 0.000e+00 1.222e-09 [30,] 0.000e+00 1.087e-09 [31,] 0.000e+00 9.554e-10 [32,] 1.110e-16 8.304e-10 [33,] 1.110e-16 7.131e-10 [34,] 0.000e+00 6.047e-10 [35,] 5.551e-17 5.056e-10 [36,] 5.551e-17 4.164e-10 [37,] 5.551e-17 3.372e-10 [38,] 0.000e+00 2.678e-10 [39,] 0.000e+00 2.080e-10 [40,] 0.000e+00 1.573e-10 [41,] 0.000e+00 1.153e-10 [42,] -2.776e-17 8.126e-11 [43,] 0.000e+00 5.452e-11 [44,] -2.776e-17 3.429e-11 [45,] 2.776e-17 1.973e-11 [46,] 0.000e+00 9.958e-12 [47,] 0.000e+00 4.050e-12 [48,] 1.388e-17 1.066e-12 [49,] 3.469e-18 4.162e-14 [50,] 0.000e+00 0.000e+00 [51,] -3.469e-18 -4.162e-14 [52,] -1.388e-17 -1.066e-12 [53,] 0.000e+00 -4.050e-12 [54,] 0.000e+00 -9.958e-12 [55,] -2.776e-17 -1.973e-11 [56,] 2.776e-17 -3.429e-11 [57,] 0.000e+00 -5.452e-11 [58,] 2.776e-17 -8.126e-11 [59,] 0.000e+00 -1.153e-10 [60,] 0.000e+00 -1.573e-10 [61,] 0.000e+00 -2.080e-10 [62,] 0.000e+00 -2.678e-10 [63,] -5.551e-17 -3.372e-10 [64,] -5.551e-17 -4.164e-10 [65,] -5.551e-17 -5.056e-10 [66,] 0.000e+00 -6.047e-10 [67,] -1.110e-16 -7.131e-10 [68,] -1.110e-16 -8.304e-10 [69,] 5.551e-17 -9.554e-10 [70,] 0.000e+00 -1.087e-09 [71,] 0.000e+00 -1.222e-09 [72,] -1.110e-16 -1.359e-09 [73,] 1.110e-16 -1.494e-09 [74,] -1.110e-16 -1.623e-09 [75,] -2.220e-16 -1.739e-09 [76,] -1.110e-16 -1.836e-09 [77,] -1.110e-16 -1.904e-09 [78,] -2.220e-16 -1.932e-09 [79,] -1.110e-16 -1.905e-09 [80,] -2.220e-16 -1.806e-09 [81,] -1.110e-16 -1.610e-09 [82,] 1.110e-16 -1.288e-09 [83,] 0.000e+00 -8.024e-10 [84,] 1.110e-16 -1.037e-10 [85,] 2.220e-16 8.718e-10 [86,] 0.000e+00 2.208e-09 [87,] -2.220e-16 4.018e-09 [88,] 0.000e+00 6.452e-09 [89,] 0.000e+00 9.721e-09 [90,] 0.000e+00 1.412e-08 [91,] -2.220e-16 2.007e-08 [92,] -4.441e-16 2.821e-08 [93,] 0.000e+00 3.952e-08 [94,] 0.000e+00 5.560e-08 [95,] 0.000e+00 7.922e-08 [96,] 2.220e-16 1.157e-07 [97,] 2.220e-16 1.762e-07 [98,] 0.000e+00 2.911e-07 [99,] 0.000e+00 5.802e-07 > > df <- 1e8 # <<<<<<<<<<<<<<<<<<<<<<<<<<<<<<< large <<<<<<<<<<<< > ## more p-values close to {0, 1} > p. <- 2^-(2*(15:5)); p. <- sort(c(p., p, 1 - rev(p.))); plot(diff(p.)) > qspL <- qts(p., df=df) > ii <- which(is.finite(qspL[,"qt"])) > p.qtNappr(qspL, p=p., ip=ii) > lines(p., pmax(1e-17, abs(qspL[,"qtU"] /qspL[,"qt"] - 1)), col=2, lwd=2, lty=2) > legend(0.6, 1e-9, "qtU(tol=1e-5)", bty="n",cex=.8, col=2, lwd=2, lty=2) > lines(p., pmax(1e-17, abs(qspL[,"qtUa"]/qspL[,"qt"] - 1)), col=4, lwd=2, lty=3) > legend(0.6, 1e-14, "qtU(tol=1e-14)", bty="n",cex=.8, col=4, lwd=2, lty=3) > > print(qspL, digits=6) qt qtU qtUa qtN0 qtN1 qtN2 [1,] -Inf -61.1000000 -61.1000000 -Inf -Inf -Inf [2,] -6.0093541 -6.0093539 -6.0093541 -6.0093536 -6.0093541 -6.0093541 [3,] -5.7804398 -5.7804398 -5.7804398 -5.7804393 -5.7804398 -5.7804398 [4,] -5.5425945 -5.5425947 -5.5425945 -5.5425941 -5.5425945 -5.5425945 [5,] -5.2947045 -5.2947048 -5.2947045 -5.2947041 -5.2947045 -5.2947045 [6,] -5.0354063 -5.0354087 -5.0354063 -5.0354060 -5.0354063 -5.0354063 [7,] -4.7630013 -4.7630013 -4.7630013 -4.7630010 -4.7630013 -4.7630013 [8,] -4.4753287 -4.4753287 -4.4753287 -4.4753284 -4.4753287 -4.4753287 [9,] -4.1695695 -4.1695704 -4.1695695 -4.1695693 -4.1695695 -4.1695695 [10,] -3.8419308 -3.8419309 -3.8419308 -3.8419307 -3.8419308 -3.8419308 [11,] -3.4871042 -3.4871045 -3.4871042 -3.4871041 -3.4871042 -3.4871042 [12,] -3.0972692 -3.0972716 -3.0972692 -3.0972691 -3.0972692 -3.0972692 [13,] -2.3263479 -2.3263479 -2.3263479 -2.3263479 -2.3263479 -2.3263479 [14,] -2.0537489 -2.0537477 -2.0537489 -2.0537489 -2.0537489 -2.0537489 [15,] -1.8807936 -1.8807918 -1.8807936 -1.8807936 -1.8807936 -1.8807936 [16,] -1.7506861 -1.7506861 -1.7506861 -1.7506861 -1.7506861 -1.7506861 [17,] -1.6448536 -1.6448545 -1.6448536 -1.6448536 -1.6448536 -1.6448536 [18,] -1.5547736 -1.5547757 -1.5547736 -1.5547736 -1.5547736 -1.5547736 [19,] -1.4757910 -1.4757908 -1.4757910 -1.4757910 -1.4757910 -1.4757910 [20,] -1.4050716 -1.4050716 -1.4050716 -1.4050716 -1.4050716 -1.4050716 [21,] -1.3407550 -1.3407555 -1.3407550 -1.3407550 -1.3407550 -1.3407550 [22,] -1.2815516 -1.2815516 -1.2815516 -1.2815516 -1.2815516 -1.2815516 [23,] -1.2265281 -1.2265281 -1.2265281 -1.2265281 -1.2265281 -1.2265281 [24,] -1.1749868 -1.1749868 -1.1749868 -1.1749868 -1.1749868 -1.1749868 [25,] -1.1263911 -1.1263910 -1.1263911 -1.1263911 -1.1263911 -1.1263911 [26,] -1.0803193 -1.0803216 -1.0803193 -1.0803193 -1.0803193 -1.0803193 [27,] -1.0364334 -1.0364334 -1.0364334 -1.0364334 -1.0364334 -1.0364334 [28,] -0.9944579 -0.9944580 -0.9944579 -0.9944579 -0.9944579 -0.9944579 [29,] -0.9541653 -0.9541653 -0.9541653 -0.9541653 -0.9541653 -0.9541653 [30,] -0.9153651 -0.9153651 -0.9153651 -0.9153651 -0.9153651 -0.9153651 [31,] -0.8778963 -0.8778963 -0.8778963 -0.8778963 -0.8778963 -0.8778963 [32,] -0.8416212 -0.8416237 -0.8416212 -0.8416212 -0.8416212 -0.8416212 [33,] -0.8064213 -0.8064213 -0.8064213 -0.8064212 -0.8064213 -0.8064213 [34,] -0.7721932 -0.7721932 -0.7721932 -0.7721932 -0.7721932 -0.7721932 [35,] -0.7388469 -0.7388468 -0.7388469 -0.7388468 -0.7388469 -0.7388469 [36,] -0.7063026 -0.7063026 -0.7063026 -0.7063026 -0.7063026 -0.7063026 [37,] -0.6744898 -0.6744899 -0.6744898 -0.6744898 -0.6744898 -0.6744898 [38,] -0.6433454 -0.6433472 -0.6433454 -0.6433454 -0.6433454 -0.6433454 [39,] -0.6128130 -0.6128105 -0.6128130 -0.6128130 -0.6128130 -0.6128130 [40,] -0.5828415 -0.5828410 -0.5828415 -0.5828415 -0.5828415 -0.5828415 [41,] -0.5533847 -0.5533861 -0.5533847 -0.5533847 -0.5533847 -0.5533847 [42,] -0.5244005 -0.5244005 -0.5244005 -0.5244005 -0.5244005 -0.5244005 [43,] -0.4958503 -0.4958503 -0.4958503 -0.4958503 -0.4958503 -0.4958503 [44,] -0.4676988 -0.4676988 -0.4676988 -0.4676988 -0.4676988 -0.4676988 [45,] -0.4399132 -0.4399155 -0.4399132 -0.4399132 -0.4399132 -0.4399132 [46,] -0.4124631 -0.4124632 -0.4124631 -0.4124631 -0.4124631 -0.4124631 [47,] -0.3853205 -0.3853205 -0.3853205 -0.3853205 -0.3853205 -0.3853205 [48,] -0.3584588 -0.3584588 -0.3584588 -0.3584588 -0.3584588 -0.3584588 [49,] -0.3318533 -0.3318514 -0.3318533 -0.3318533 -0.3318533 -0.3318533 [50,] -0.3054808 -0.3054809 -0.3054808 -0.3054808 -0.3054808 -0.3054808 [51,] -0.2793190 -0.2793196 -0.2793190 -0.2793190 -0.2793190 -0.2793190 [52,] -0.2533471 -0.2533478 -0.2533471 -0.2533471 -0.2533471 -0.2533471 [53,] -0.2275450 -0.2275451 -0.2275450 -0.2275450 -0.2275450 -0.2275450 [54,] -0.2018935 -0.2018935 -0.2018935 -0.2018935 -0.2018935 -0.2018935 [55,] -0.1763742 -0.1763742 -0.1763742 -0.1763742 -0.1763742 -0.1763742 [56,] -0.1509692 -0.1509693 -0.1509692 -0.1509692 -0.1509692 -0.1509692 [57,] -0.1256613 -0.1256614 -0.1256613 -0.1256613 -0.1256613 -0.1256613 [58,] -0.1004337 -0.1004337 -0.1004337 -0.1004337 -0.1004337 -0.1004337 [59,] -0.0752699 -0.0752701 -0.0752699 -0.0752699 -0.0752699 -0.0752699 [60,] -0.0501536 -0.0501536 -0.0501536 -0.0501536 -0.0501536 -0.0501536 [61,] -0.0250689 -0.0250690 -0.0250689 -0.0250689 -0.0250689 -0.0250689 [62,] 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 [63,] 0.0250689 0.0250690 0.0250689 0.0250689 0.0250689 0.0250689 [64,] 0.0501536 0.0501536 0.0501536 0.0501536 0.0501536 0.0501536 [65,] 0.0752699 0.0752701 0.0752699 0.0752699 0.0752699 0.0752699 [66,] 0.1004337 0.1004337 0.1004337 0.1004337 0.1004337 0.1004337 [67,] 0.1256613 0.1256614 0.1256613 0.1256613 0.1256613 0.1256613 [68,] 0.1509692 0.1509693 0.1509692 0.1509692 0.1509692 0.1509692 [69,] 0.1763742 0.1763742 0.1763742 0.1763742 0.1763742 0.1763742 [70,] 0.2018935 0.2018935 0.2018935 0.2018935 0.2018935 0.2018935 [71,] 0.2275450 0.2275451 0.2275450 0.2275450 0.2275450 0.2275450 [72,] 0.2533471 0.2533478 0.2533471 0.2533471 0.2533471 0.2533471 [73,] 0.2793190 0.2793196 0.2793190 0.2793190 0.2793190 0.2793190 [74,] 0.3054808 0.3054809 0.3054808 0.3054808 0.3054808 0.3054808 [75,] 0.3318533 0.3318514 0.3318533 0.3318533 0.3318533 0.3318533 [76,] 0.3584588 0.3584588 0.3584588 0.3584588 0.3584588 0.3584588 [77,] 0.3853205 0.3853205 0.3853205 0.3853205 0.3853205 0.3853205 [78,] 0.4124631 0.4124632 0.4124631 0.4124631 0.4124631 0.4124631 [79,] 0.4399132 0.4399155 0.4399132 0.4399132 0.4399132 0.4399132 [80,] 0.4676988 0.4676988 0.4676988 0.4676988 0.4676988 0.4676988 [81,] 0.4958503 0.4958503 0.4958503 0.4958503 0.4958503 0.4958503 [82,] 0.5244005 0.5244005 0.5244005 0.5244005 0.5244005 0.5244005 [83,] 0.5533847 0.5533861 0.5533847 0.5533847 0.5533847 0.5533847 [84,] 0.5828415 0.5828410 0.5828415 0.5828415 0.5828415 0.5828415 [85,] 0.6128130 0.6128105 0.6128130 0.6128130 0.6128130 0.6128130 [86,] 0.6433454 0.6433472 0.6433454 0.6433454 0.6433454 0.6433454 [87,] 0.6744898 0.6744899 0.6744898 0.6744898 0.6744898 0.6744898 [88,] 0.7063026 0.7063026 0.7063026 0.7063026 0.7063026 0.7063026 [89,] 0.7388469 0.7388468 0.7388469 0.7388468 0.7388469 0.7388469 [90,] 0.7721932 0.7721932 0.7721932 0.7721932 0.7721932 0.7721932 [91,] 0.8064213 0.8064213 0.8064213 0.8064212 0.8064213 0.8064213 [92,] 0.8416212 0.8416237 0.8416212 0.8416212 0.8416212 0.8416212 [93,] 0.8778963 0.8778963 0.8778963 0.8778963 0.8778963 0.8778963 [94,] 0.9153651 0.9153651 0.9153651 0.9153651 0.9153651 0.9153651 [95,] 0.9541653 0.9541653 0.9541653 0.9541653 0.9541653 0.9541653 [96,] 0.9944579 0.9944580 0.9944579 0.9944579 0.9944579 0.9944579 [97,] 1.0364334 1.0364334 1.0364334 1.0364334 1.0364334 1.0364334 [98,] 1.0803193 1.0803216 1.0803193 1.0803193 1.0803193 1.0803193 [99,] 1.1263911 1.1263910 1.1263911 1.1263911 1.1263911 1.1263911 [100,] 1.1749868 1.1749868 1.1749868 1.1749868 1.1749868 1.1749868 [101,] 1.2265281 1.2265281 1.2265281 1.2265281 1.2265281 1.2265281 [102,] 1.2815516 1.2815516 1.2815516 1.2815516 1.2815516 1.2815516 [103,] 1.3407550 1.3407555 1.3407550 1.3407550 1.3407550 1.3407550 [104,] 1.4050716 1.4050716 1.4050716 1.4050716 1.4050716 1.4050716 [105,] 1.4757910 1.4757908 1.4757910 1.4757910 1.4757910 1.4757910 [106,] 1.5547736 1.5547757 1.5547736 1.5547736 1.5547736 1.5547736 [107,] 1.6448536 1.6448545 1.6448536 1.6448536 1.6448536 1.6448536 [108,] 1.7506861 1.7506861 1.7506861 1.7506861 1.7506861 1.7506861 [109,] 1.8807936 1.8807918 1.8807936 1.8807936 1.8807936 1.8807936 [110,] 2.0537489 2.0537477 2.0537489 2.0537489 2.0537489 2.0537489 [111,] 2.3263479 2.3263479 2.3263479 2.3263479 2.3263479 2.3263479 [112,] 3.0972692 3.0972716 3.0972692 3.0972691 3.0972692 3.0972692 [113,] 3.4871042 3.4871045 3.4871042 3.4871041 3.4871042 3.4871042 [114,] 3.8419308 3.8419309 3.8419308 3.8419307 3.8419308 3.8419308 [115,] 4.1695695 4.1695704 4.1695695 4.1695693 4.1695695 4.1695695 [116,] 4.4753287 4.4753287 4.4753287 4.4753284 4.4753287 4.4753287 [117,] 4.7630013 4.7630013 4.7630013 4.7630010 4.7630013 4.7630013 [118,] 5.0354063 5.0354087 5.0354063 5.0354060 5.0354063 5.0354063 [119,] 5.2947045 5.2947048 5.2947045 5.2947041 5.2947045 5.2947045 [120,] 5.5425945 5.5425947 5.5425945 5.5425941 5.5425945 5.5425945 [121,] 5.7804398 5.7804398 5.7804398 5.7804393 5.7804398 5.7804398 [122,] 6.0093541 6.0093539 6.0093541 6.0093536 6.0093541 6.0093541 [123,] Inf 10.0000000 10.0000000 Inf Inf Inf qtN3 qtN4 [1,] -Inf -Inf [2,] -6.0093541 -6.0093541 [3,] -5.7804398 -5.7804398 [4,] -5.5425945 -5.5425945 [5,] -5.2947045 -5.2947045 [6,] -5.0354063 -5.0354063 [7,] -4.7630013 -4.7630013 [8,] -4.4753287 -4.4753287 [9,] -4.1695695 -4.1695695 [10,] -3.8419308 -3.8419308 [11,] -3.4871042 -3.4871042 [12,] -3.0972692 -3.0972692 [13,] -2.3263479 -2.3263479 [14,] -2.0537489 -2.0537489 [15,] -1.8807936 -1.8807936 [16,] -1.7506861 -1.7506861 [17,] -1.6448536 -1.6448536 [18,] -1.5547736 -1.5547736 [19,] -1.4757910 -1.4757910 [20,] -1.4050716 -1.4050716 [21,] -1.3407550 -1.3407550 [22,] -1.2815516 -1.2815516 [23,] -1.2265281 -1.2265281 [24,] -1.1749868 -1.1749868 [25,] -1.1263911 -1.1263911 [26,] -1.0803193 -1.0803193 [27,] -1.0364334 -1.0364334 [28,] -0.9944579 -0.9944579 [29,] -0.9541653 -0.9541653 [30,] -0.9153651 -0.9153651 [31,] -0.8778963 -0.8778963 [32,] -0.8416212 -0.8416212 [33,] -0.8064213 -0.8064213 [34,] -0.7721932 -0.7721932 [35,] -0.7388469 -0.7388469 [36,] -0.7063026 -0.7063026 [37,] -0.6744898 -0.6744898 [38,] -0.6433454 -0.6433454 [39,] -0.6128130 -0.6128130 [40,] -0.5828415 -0.5828415 [41,] -0.5533847 -0.5533847 [42,] -0.5244005 -0.5244005 [43,] -0.4958503 -0.4958503 [44,] -0.4676988 -0.4676988 [45,] -0.4399132 -0.4399132 [46,] -0.4124631 -0.4124631 [47,] -0.3853205 -0.3853205 [48,] -0.3584588 -0.3584588 [49,] -0.3318533 -0.3318533 [50,] -0.3054808 -0.3054808 [51,] -0.2793190 -0.2793190 [52,] -0.2533471 -0.2533471 [53,] -0.2275450 -0.2275450 [54,] -0.2018935 -0.2018935 [55,] -0.1763742 -0.1763742 [56,] -0.1509692 -0.1509692 [57,] -0.1256613 -0.1256613 [58,] -0.1004337 -0.1004337 [59,] -0.0752699 -0.0752699 [60,] -0.0501536 -0.0501536 [61,] -0.0250689 -0.0250689 [62,] 0.0000000 0.0000000 [63,] 0.0250689 0.0250689 [64,] 0.0501536 0.0501536 [65,] 0.0752699 0.0752699 [66,] 0.1004337 0.1004337 [67,] 0.1256613 0.1256613 [68,] 0.1509692 0.1509692 [69,] 0.1763742 0.1763742 [70,] 0.2018935 0.2018935 [71,] 0.2275450 0.2275450 [72,] 0.2533471 0.2533471 [73,] 0.2793190 0.2793190 [74,] 0.3054808 0.3054808 [75,] 0.3318533 0.3318533 [76,] 0.3584588 0.3584588 [77,] 0.3853205 0.3853205 [78,] 0.4124631 0.4124631 [79,] 0.4399132 0.4399132 [80,] 0.4676988 0.4676988 [81,] 0.4958503 0.4958503 [82,] 0.5244005 0.5244005 [83,] 0.5533847 0.5533847 [84,] 0.5828415 0.5828415 [85,] 0.6128130 0.6128130 [86,] 0.6433454 0.6433454 [87,] 0.6744898 0.6744898 [88,] 0.7063026 0.7063026 [89,] 0.7388469 0.7388469 [90,] 0.7721932 0.7721932 [91,] 0.8064213 0.8064213 [92,] 0.8416212 0.8416212 [93,] 0.8778963 0.8778963 [94,] 0.9153651 0.9153651 [95,] 0.9541653 0.9541653 [96,] 0.9944579 0.9944579 [97,] 1.0364334 1.0364334 [98,] 1.0803193 1.0803193 [99,] 1.1263911 1.1263911 [100,] 1.1749868 1.1749868 [101,] 1.2265281 1.2265281 [102,] 1.2815516 1.2815516 [103,] 1.3407550 1.3407550 [104,] 1.4050716 1.4050716 [105,] 1.4757910 1.4757910 [106,] 1.5547736 1.5547736 [107,] 1.6448536 1.6448536 [108,] 1.7506861 1.7506861 [109,] 1.8807936 1.8807936 [110,] 2.0537489 2.0537489 [111,] 2.3263479 2.3263479 [112,] 3.0972692 3.0972692 [113,] 3.4871042 3.4871042 [114,] 3.8419308 3.8419308 [115,] 4.1695695 4.1695695 [116,] 4.4753287 4.4753287 [117,] 4.7630013 4.7630013 [118,] 5.0354063 5.0354063 [119,] 5.2947045 5.2947045 [120,] 5.5425945 5.5425945 [121,] 5.7804398 5.7804398 [122,] 6.0093541 6.0093541 [123,] Inf Inf > > proc.time() user system elapsed 17.54 0.25 17.75