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Rev 81944 Rev 89335
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R Under development (unstable) (2022-03-19 r81942) -- "Unsuffered Consequences"
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R Under development (unstable) (2026-01-26 r89332) -- "Unsuffered Consequences"
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Copyright (C) 2022 The R Foundation for Statistical Computing
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Copyright (C) 2026 The R Foundation for Statistical Computing
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Platform: x86_64-pc-linux-gnu (64-bit)
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Platform: x86_64-pc-linux-gnu
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R is free software and comes with ABSOLUTELY NO WARRANTY.
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R is free software and comes with ABSOLUTELY NO WARRANTY.
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You are welcome to redistribute it under certain conditions.
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You are welcome to redistribute it under certain conditions.
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Type 'license()' or 'licence()' for distribution details.
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Type 'license()' or 'licence()' for distribution details.
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> ## Real Trig.:
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> ## Real Trig.:
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> cos(0) == 1
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> cos(0) == 1
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[1] TRUE
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[1] TRUE
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> sin(3*pi/2) == cos(pi)
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> sin(3*pi/2) == cos(pi)
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[1] TRUE
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[1] TRUE
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> x <- rnorm(99)
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> all( sin(-x) == - sin(x))
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[1] TRUE
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> all( cos(-x) == cos(x))
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[1] TRUE
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>
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>
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> x <- 1:99/100
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> x <- 1:99/100
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> ## IGNORE_RDIFF_BEGIN
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> all(abs(1 - x / asin(sin(x))) <= 2*Meps)# "== 2*" for HP-UX
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> table(Isin <- abs(1 - x / asin(sin(x)))/Meps)# mostly 0, often <= 1
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  0 0.5   1
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 88   2   9
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> table(Itan <- abs(1 - x / atan(tan(x)))/Meps)#  (ditto)
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  0 0.5   1
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 97   1   1
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> ## IGNORE_RDIFF_END
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> all(Isin <= 4) # typically <= 1, 3 needed for [mingw-w64 v12 on Win]
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[1] TRUE
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[1] TRUE
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> all(abs(1 - x / atan(tan(x))) <  2*Meps)
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> all(Itan <= 4)
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[1] TRUE
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[1] TRUE
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>
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>
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> ## Sun has asin(.) = acos(.) = 0 for these:
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> ## Sun has asin(.) = acos(.) = 0 for these:
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> ##	is.nan(acos(1.1)) && is.nan(asin(-2)) [!]
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> ##	is.nan(acos(1.1)) && is.nan(asin(-2)) [!]
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>
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>
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> stopifnot(all.equal( digamma(x), psigamma(x,0), tolerance = 2*Meps),
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> stopifnot(all.equal( digamma(x), psigamma(x,0), tolerance = 2*Meps),
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+           all.equal(trigamma(x), psigamma(x,1), tolerance = 2*Meps))# TRUE (+ NaN warnings)
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+           all.equal(trigamma(x), psigamma(x,1), tolerance = 2*Meps))# TRUE (+ NaN warnings)
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Warning messages:
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Warning messages:
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1: In digamma(x) : NaNs produced
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1: In digamma(x) : NaNs produced
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2: In psigamma(x, 0) : NaNs produced
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2: In psigamma(x, 0) : NaNs produced
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> ## very large x:
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> ## very large x,  compare with asymptotic approximation:
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> x <- 1e30 ^ (1:10)
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> x <- 1e30 ^ (1:10)
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> a.relE <- function(appr, true) abs(1 - appr/true)
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> arelEps <- function(appr, true) abs(1 - appr/true) / Meps
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> ## IGNORE_RDIFF_BEGIN
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> stopifnot(a.relE(digamma(x),   log(x)) < 1e-13,
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> sort( digamE <- arelEps( digamma(x), log(x))) # all 0 for glibc
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 [1] 0 0 0 0 0 0 0 0 0 0
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+           a.relE(trigamma(x),     1/x) < 1e-13)
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> sort(trigamE <- arelEps(trigamma(x),  1/x  )) # 11 22 ... 108 159 for glibc
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 [1]  11  22  32  43  54  65  75  86 108 159
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> x <- sqrt(x[2:6]); stopifnot(a.relE(psigamma(x,2), - 1/x^2) < 1e-13)
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> x <- sqrt(x[2:6]); sort(psiga2E <- arelEps(psigamma(x,2), - 1/x^2)) # 22 33 43 55 65
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[1] 22 33 43 55 65
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> x <- 10^(10*(2:6));stopifnot(a.relE(psigamma(x,5), +24/x^5) < 1e-13)
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> x <- 10^(10*(2:6));sort(psiga5E <- arelEps(psigamma(x,5), +24/x^5)) #  5 50.5 .. 106
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[1]   5.0  50.5  54.0  98.5 106.0
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> ## IGNORE_RDIFF_END
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> stopifnot(digamE  <   64, # seen '0'
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+           trigamE < 1300, # mfuzz: upto 671
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+           psiga2E <  430, # mfuzz: upto 212
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+           psiga5E < 1010) # mfuzz: upto 505.5
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>
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>
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> ## fft():
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> ## fft():
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> ok <- TRUE
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> ok <- TRUE
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> ##test EXTENSIVELY:	for(N in 1:100) {
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> ##test EXTENSIVELY:	for(N in 1:100) {
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>     cat(".")
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>     cat(".")
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[1] TRUE
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[1] TRUE
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>
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>
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> ## var():
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> ## var():
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> for(n in 2:10)
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> for(n in 2:10)
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+     print(all.equal(n*(n-1)*var(diag(n)),
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+     print(all.equal(n*(n-1)*var(diag(n)),
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+ 		    matrix(c(rep(c(n-1,rep(-1,n)),n-1), n-1), nr=n, nc=n),
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+ 		    matrix(c(rep(c(n-1,rep(-1,n)),n-1), n-1), nrow=n, ncol=n),
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+                                 ##----- use 0*Meps to see non-0 rel.error ==> all TRUE for glibc
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+ 		    tolerance = 20*Meps)) # use tolerance = 0 to see rel.error
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+ 		    tolerance = 20*Meps))
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[1] TRUE
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[1] TRUE
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[1] TRUE
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[1] TRUE
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[1] TRUE
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[1] TRUE
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[1] TRUE
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[1] TRUE
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[1] TRUE
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[1] TRUE
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> is.na(z)
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> is.na(z)
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[1] TRUE
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[1] TRUE
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>
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>
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> ## Last Line:
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> ## Last Line:
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> cat('Time elapsed: ', proc.time() - .proctime00,'\n')
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> cat('Time elapsed: ', proc.time() - .proctime00,'\n')
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Time elapsed:  0.677 0.028 0.714 0 0
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Time elapsed:  0.188 0.047 0.237 0 0
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>
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>