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R Under development (unstable) (2023-09-28 r85227) -- "Unsuffered Consequences"
Copyright (C) 2023 The R Foundation for Statistical Computing
Platform: x86_64-pc-linux-gnu
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.
> ### Tests of complex arithemetic.
>
> Meps <- .Machine$double.eps
> ## complex
> z <- 0i ^ (-3:3)
> stopifnot(Re(z) == 0 ^ (-3:3))
>
>
> ## powers, including complex ones
> a <- -4:12
> m <- outer(a +0i, b <- seq(-.5,2, by=.5), "^")
> dimnames(m) <- list(paste(a), "^" = sapply(b,format))
> round(m,3)
^
-0.5 0 0.5 1 1.5 2
-4 0.000-0.500i 1+0i 0.000+2.000i -4+0i 0.000-8.000i 16+0i
-3 0.000-0.577i 1+0i 0.000+1.732i -3+0i 0.000-5.196i 9+0i
-2 0.000-0.707i 1+0i 0.000+1.414i -2+0i 0.000-2.828i 4+0i
-1 0.000-1.000i 1+0i 0.000+1.000i -1+0i 0.000-1.000i 1+0i
0 Inf+0.000i 1+0i 0.000+0.000i 0+0i 0.000+0.000i 0+0i
1 1.000+0.000i 1+0i 1.000+0.000i 1+0i 1.000+0.000i 1+0i
2 0.707+0.000i 1+0i 1.414+0.000i 2+0i 2.828+0.000i 4+0i
3 0.577+0.000i 1+0i 1.732+0.000i 3+0i 5.196+0.000i 9+0i
4 0.500+0.000i 1+0i 2.000+0.000i 4+0i 8.000+0.000i 16+0i
5 0.447+0.000i 1+0i 2.236+0.000i 5+0i 11.180+0.000i 25+0i
6 0.408+0.000i 1+0i 2.449+0.000i 6+0i 14.697+0.000i 36+0i
7 0.378+0.000i 1+0i 2.646+0.000i 7+0i 18.520+0.000i 49+0i
8 0.354+0.000i 1+0i 2.828+0.000i 8+0i 22.627+0.000i 64+0i
9 0.333+0.000i 1+0i 3.000+0.000i 9+0i 27.000+0.000i 81+0i
10 0.316+0.000i 1+0i 3.162+0.000i 10+0i 31.623+0.000i 100+0i
11 0.302+0.000i 1+0i 3.317+0.000i 11+0i 36.483+0.000i 121+0i
12 0.289+0.000i 1+0i 3.464+0.000i 12+0i 41.569+0.000i 144+0i
> stopifnot(m[,as.character(0:2)] == cbind(1,a,a*a),
+ # latter were only approximate
+ all.equal(unname(m[,"0.5"]),
+ sqrt(abs(a))*ifelse(a < 0, 1i, 1),
+ tolerance = 20*Meps))
>
> ## 2.10.0-2.12.1 got z^n wrong in the !HAVE_C99_COMPLEX case
> z <- 0.2853725+0.3927816i
> z2 <- z^(1:20)
> z3 <- z^-(1:20)
> z0 <- cumprod(rep(z, 20))
> stopifnot(all.equal(z2, z0), all.equal(z3, 1/z0))
> ## was z^3 had value z^2 ....
>
> ## fft():
> for(n in 1:30) cat("\nn=",n,":", round(fft(1:n), 8),"\n")
n= 1 : 1+0i
n= 2 : 3+0i -1+0i
n= 3 : 6+0i -1.5+0.8660254i -1.5-0.8660254i
n= 4 : 10+0i -2+2i -2+0i -2-2i
n= 5 : 15+0i -2.5+3.440955i -2.5+0.8122992i -2.5-0.8122992i -2.5-3.440955i
n= 6 : 21+0i -3+5.196152i -3+1.732051i -3+0i -3-1.732051i -3-5.196152i
n= 7 : 28+0i -3.5+7.267825i -3.5+2.791157i -3.5+0.7988522i -3.5-0.7988522i -3.5-2.791157i -3.5-7.267825i
n= 8 : 36+0i -4+9.656854i -4+4i -4+1.656854i -4+0i -4-1.656854i -4-4i -4-9.656854i
n= 9 : 45+0i -4.5+12.36365i -4.5+5.362891i -4.5+2.598076i -4.5+0.7934714i -4.5-0.7934714i -4.5-2.598076i -4.5-5.362891i -4.5-12.36365i
n= 10 : 55+0i -5+15.38842i -5+6.88191i -5+3.632713i -5+1.624598i -5+0i -5-1.624598i -5-3.632713i -5-6.88191i -5-15.38842i
n= 11 : 66+0i -5.5+18.73128i -5.5+8.558167i -5.5+4.765777i -5.5+2.511766i -5.5+0.7907806i -5.5-0.7907806i -5.5-2.511766i -5.5-4.765777i -5.5-8.558167i -5.5-18.73128i
n= 12 : 78+0i -6+22.3923i -6+10.3923i -6+6i -6+3.464102i -6+1.607695i -6+0i -6-1.607695i -6-3.464102i -6-6i -6-10.3923i -6-22.3923i
n= 13 : 91+0i -6.5+26.37154i -6.5+12.38472i -6.5+7.336983i -6.5+4.486626i -6.5+2.465125i -6.5+0.7892429i -6.5-0.7892429i -6.5-2.465125i -6.5-4.486626i -6.5-7.336983i -6.5-12.38472i -6.5-26.37154i
n= 14 : 105+0i -7+30.669i -7+14.53565i -7+8.777722i -7+5.582314i -7+3.371022i -7+1.597704i -7+0i -7-1.597704i -7-3.371022i -7-5.582314i -7-8.777722i -7-14.53565i -7-30.669i
n= 15 : 120+0i -7.5+35.28473i -7.5+16.84528i -7.5+10.32286i -7.5+6.75303i -7.5+4.330127i -7.5+2.436898i -7.5+0.7882818i -7.5-0.7882818i -7.5-2.436898i -7.5-4.330127i -7.5-6.75303i -7.5-10.32286i -7.5-16.84528i -7.5-35.28473i
n= 16 : 136+0i -8+40.21872i -8+19.31371i -8+11.97285i -8+8i -8+5.345429i -8+3.313709i -8+1.591299i -8+0i -8-1.591299i -8-3.313709i -8-5.345429i -8-8i -8-11.97285i -8-19.31371i -8-40.21872i
n= 17 : 153+0i -8.5+45.47098i -8.5+21.94103i -8.5+13.72797i -8.5+9.324056i -8.5+6.418902i -8.5+4.232497i -8.5+2.418459i -8.5+0.787641i -8.5-0.787641i -8.5-2.418459i -8.5-4.232497i -8.5-6.418902i -8.5-9.324056i -8.5-13.72797i -8.5-21.94103i -8.5-45.47098i
n= 18 : 171+0i -9+51.04154i -9+24.7273i -9+15.58846i -9+10.72578i -9+7.551897i -9+5.196152i -9+3.275732i -9+1.586943i -9+0i -9-1.586943i -9-3.275732i -9-5.196152i -9-7.551897i -9-10.72578i -9-15.58846i -9-24.7273i -9-51.04154i
n= 19 : 190+0i -9.5+56.93038i -9.5+27.67255i -9.5+17.55446i -9.5+12.2056i -9.5+8.745366i -9.5+6.20666i -9.5+4.167086i -9.5+2.405727i -9.5+0.7871924i -9.5-0.7871924i -9.5-2.405727i -9.5-4.167086i -9.5-6.20666i -9.5-8.745366i -9.5-12.2056i -9.5-17.55446i -9.5-27.67255i -9.5-56.93038i
n= 20 : 210+0i -10+63.13752i -10+30.77684i -10+19.62611i -10+13.76382i -10+10i -10+7.265425i -10+5.095254i -10+3.249197i -10+1.583844i -10+0i -10-1.583844i -10-3.249197i -10-5.095254i -10-7.265425i -10-10i -10-13.76382i -10-19.62611i -10-30.77684i -10-63.13752i
n= 21 : 231+0i -10.5+69.66295i -10.5+34.04016i -10.5+21.80347i -10.5+15.40067i -10.5+11.31631i -10.5+8.373471i -10.5+6.062178i -10.5+4.120946i -10.5+2.396556i -10.5+0.7868662i -10.5-0.7868662i -10.5-2.396556i -10.5-4.120946i -10.5-6.062178i -10.5-8.373471i -10.5-11.31631i -10.5-15.40067i -10.5-21.80347i -10.5-34.04016i -10.5-69.66295i
n= 22 : 253+0i -11+76.50668i -11+37.46256i -11+24.08664i -11+17.11633i -11+12.69468i -11+9.531554i -11+7.069271i -11+5.023532i -11+3.229891i -11+1.581561i -11+0i -11-1.581561i -11-3.229891i -11-5.023532i -11-7.069271i -11-9.531554i -11-12.69468i -11-17.11633i -11-24.08664i -11-37.46256i -11-76.50668i
n= 23 : 276+0i -11.5+83.66871i -11.5+41.04404i -11.5+26.47566i -11.5+18.91094i -11.5+14.1354i -11.5+10.74025i -11.5+8.117586i -11.5+5.95882i -11.5+4.087101i -11.5+2.389727i -11.5+0.7866216i -11.5-0.7866216i -11.5-2.389727i -11.5-4.087101i -11.5-5.95882i -11.5-8.117586i -11.5-10.74025i -11.5-14.1354i -11.5-18.91094i -11.5-26.47566i -11.5-41.04404i -11.5-83.66871i
n= 24 : 300+0i -12+91.14905i -12+44.78461i -12+28.97056i -12+20.78461i -12+15.6387i -12+12i -12+9.207924i -12+6.928203i -12+4.970563i -12+3.21539i -12+1.57983i -12+0i -12-1.57983i -12-3.21539i -12-4.970563i -12-6.928203i -12-9.207924i -12-12i -12-15.6387i -12-20.78461i -12-28.97056i -12-44.78461i -12-91.14905i
n= 25 : 325+0i -12.5+98.94769i -12.5+48.68429i -12.5+31.5714i -12.5+22.73742i -12.5+17.20477i -12.5+13.31115i -12.5+10.3409i -12.5+7.932741i -12.5+5.882054i -12.5+4.061496i -12.5+2.384503i -12.5+0.7864333i -12.5-0.7864333i -12.5-2.384503i -12.5-4.061496i -12.5-5.882054i -12.5-7.932741i -12.5-10.3409i -12.5-13.31115i -12.5-17.20477i -12.5-22.73742i -12.5-31.5714i -12.5-48.68429i -12.5-98.94769i
n= 26 : 351+0i -13+107.0646i -13+52.74307i -13+34.27818i -13+24.76943i -13+18.83375i -13+14.67397i -13+11.517i -13+8.973252i -13+6.822926i -13+4.93025i -13+3.204212i -13+1.578486i -13+0i -13-1.578486i -13-3.204212i -13-4.93025i -13-6.822926i -13-8.973252i -13-11.517i -13-14.67397i -13-18.83375i -13-24.76943i -13-34.27818i -13-52.74307i -13-107.0646i
n= 27 : 378+0i -13.5+115.4999i -13.5+56.96098i -13.5+37.09095i -13.5+26.88071i -13.5+20.52575i -13.5+16.08867i -13.5+12.73659i -13.5+10.05038i -13.5+7.794229i -13.5+5.823332i -13.5+4.041635i -13.5+2.380414i -13.5+0.7862855i -13.5-0.7862855i -13.5-2.380414i -13.5-4.041635i -13.5-5.823332i -13.5-7.794229i -13.5-10.05038i -13.5-12.73659i -13.5-16.08867i -13.5-20.52575i -13.5-26.88071i -13.5-37.09095i -13.5-56.96098i -13.5-115.4999i
n= 28 : 406+0i -14+124.2534i -14+61.33801i -14+40.0097i -14+29.0713i -14+22.28087i -14+17.55544i -14+14i -14+11.16463i -14+8.796783i -14+6.742045i -14+4.898812i -14+3.195409i -14+1.577421i -14+0i -14-1.577421i -14-3.195409i -14-4.898812i -14-6.742045i -14-8.796783i -14-11.16463i -14-14i -14-17.55544i -14-22.28087i -14-29.0713i -14-40.0097i -14-61.33801i -14-124.2534i
n= 29 : 435+0i -14.5+133.3253i -14.5+65.87416i -14.5+43.03447i -14.5+31.34124i -14.5+24.09919i -14.5+19.07442i -14.5+15.30746i -14.5+12.31641i -14.5+9.831244i -14.5+7.687413i -14.5+5.777328i -14.5+4.025905i -14.5+2.377154i -14.5+0.7861672i -14.5-0.7861672i -14.5-2.377154i -14.5-4.025905i -14.5-5.777328i -14.5-7.687413i -14.5-9.831244i -14.5-12.31641i -14.5-15.30746i -14.5-19.07442i -14.5-24.09919i -14.5-31.34124i -14.5-43.03447i -14.5-65.87416i -14.5-133.3253i
n= 30 : 465+0i -15+142.7155i -15+70.56945i -15+46.16525i -15+33.69055i -15+25.98076i -15+20.64573i -15+16.65919i -15+13.50606i -15+10.89814i -15+8.660254i -15+6.67843i -15+4.873795i -15+3.188348i -15+1.576564i -15+0i -15-1.576564i -15-3.188348i -15-4.873795i -15-6.67843i -15-8.660254i -15-10.89814i -15-13.50606i -15-16.65919i -15-20.64573i -15-25.98076i -15-33.69055i -15-46.16525i -15-70.56945i -15-142.7155i
>
>
> ## polyroot():
> stopifnot(abs(1 + polyroot(choose(8, 0:8))) < 1e-10)# maybe smaller..
>
> ## precision of complex numbers
> signif(1.678932e80+0i, 5)
[1] 1.6789e+80+0i
> signif(1.678932e-300+0i, 5)
[1] 1.6789e-300+0i
> signif(1.678932e-302+0i, 5)
[1] 1.6789e-302+0i
> signif(1.678932e-303+0i, 5)
[1] 1.6789e-303+0i
> signif(1.678932e-304+0i, 5)
[1] 1.6789e-304+0i
> signif(1.678932e-305+0i, 5)
[1] 1.6789e-305+0i
> signif(1.678932e-306+0i, 5)
[1] 1.6789e-306+0i
> signif(1.678932e-307+0i, 5)
[1] 1.6789e-307+0i
> signif(1.678932e-308+0i, 5)
[1] 1.6789e-308+0i
> signif(1.678932-1.238276i, 5)
[1] 1.6789-1.2383i
> signif(1.678932-1.238276e-1i, 5)
[1] 1.6789-0.1238i
> signif(1.678932-1.238276e-2i, 5)
[1] 1.6789-0.0124i
> signif(1.678932-1.238276e-3i, 5)
[1] 1.6789-0.0012i
> signif(1.678932-1.238276e-4i, 5)
[1] 1.6789-1e-04i
> signif(1.678932-1.238276e-5i, 5)
[1] 1.6789+0i
> signif(8.678932-9.238276i, 5)
[1] 8.6789-9.2383i
> ## prior to 2.2.0 rounded real and imaginary parts separately.
>
>
> ## Complex Trig.:
> abs(Im(cos(acos(1i))) - 1) < 2*Meps
[1] TRUE
> abs(Im(sin(asin(1i))) - 1) < 2*Meps
[1] TRUE
> ##P (1 - Im(sin(asin(Ii))))/Meps
> ##P (1 - Im(cos(acos(Ii))))/Meps
> abs(Im(asin(sin(1i))) - 1) < 2*Meps
[1] TRUE
> all.equal(cos(1i), cos(-1i)) # i.e. Im(acos(*)) gives + or - 1i:
[1] TRUE
> abs(abs(Im(acos(cos(1i)))) - 1) < 4*Meps
[1] TRUE
>
>
> set.seed(123) # want reproducible output
> Isi <- Im(sin(asin(1i + rnorm(100))))
> all(abs(Isi-1) < 100* Meps)
[1] TRUE
> ##P table(2*abs(Isi-1) / Meps)
> Isi <- Im(cos(acos(1i + rnorm(100))))
> all(abs(Isi-1) < 100* Meps)
[1] TRUE
> ##P table(2*abs(Isi-1) / Meps)
> Isi <- Im(atan(tan(1i + rnorm(100)))) #-- tan(atan(..)) does NOT work (Math!)
> all(abs(Isi-1) < 100* Meps)
[1] TRUE
> ##P table(2*abs(Isi-1) / Meps)
>
> set.seed(123)
> z <- complex(real = rnorm(100), imag = rnorm(100))
> stopifnot(Mod ( 1 - sin(z) / ( (exp(1i*z)-exp(-1i*z))/(2*1i) )) < 20 * Meps)
> ## end of moved from complex.Rd
>
>
> ## PR#7781
> ## This is not as given by e.g. glibc on AMD64
> (z <- tan(1+1000i)) # 0+1i from R's own code.
[1] 0+1i
> stopifnot(is.finite(z))
> ##
>
>
> ## Branch cuts in complex inverse trig functions
> atan(2)
[1] 1.107149
> atan(2+0i)
[1] 1.107149+0i
> tan(atan(2+0i))
[1] 2+0i
> ## should not expect exactly 0i in result
> round(atan(1.0001+0i), 7)
[1] 0.7854482+0i
> round(atan(0.9999+0i), 7)
[1] 0.7853482+0i
> ## previously not as in Abramowitz & Stegun.
>
>
> ## typo in z_atan2.
> (z <- atan2(0+1i, 0+0i))
[1] 1.570796+0i
> stopifnot(all.equal(z, pi/2+0i))
> ## was NA in 2.1.1
>
>
> ## Hyperbolic
> x <- seq(-3, 3, len=200)
> Meps <- .Machine$double.eps
> stopifnot(
+ Mod(cosh(x) - cos(1i*x)) < 20*Meps,
+ Mod(sinh(x) - sin(1i*x)/1i) < 20*Meps
+ )
> ## end of moved from Hyperbolic.Rd
>
> ## values near and on branch cuts
> options(digits=5)
> z <- c(2+0i, 2-0.0001i, -2+0i, -2+0.0001i)
> asin(z)
[1] 1.5708-1.317i 1.5707-1.317i -1.5708+1.317i -1.5707+1.317i
> acos(z)
[1] 0.0000e+00+1.317i 5.7735e-05+1.317i 3.1416e+00-1.317i 3.1415e+00-1.317i
> atanh(z)
[1] 0.54931-1.5708i 0.54931-1.5708i -0.54931+1.5708i -0.54931+1.5708i
> z <- c(0+2i, 0.0001+2i, 0-2i, -0.0001i-2i)
> asinh(z)
[1] 1.317+1.5708i 1.317+1.5707i -1.317-1.5708i -1.317-1.5708i
> acosh(z)
[1] 1.4436+1.5708i 1.4436+1.5708i -1.4436+1.5708i -1.4437+1.5708i
> atan(z)
[1] 1.5708+0.54931i 1.5708+0.54931i -1.5708-0.54931i -1.5708-0.54927i
> ## According to C99, should have continuity from the side given if there
> ## are not signed zeros.
> ## Both glibc 2.12 and macOS 10.6 used continuity from above in the first set
> ## but they seem to assume signed zeros.
> ## Windows gave incorrect (NaN) values on the cuts.
>
> stopifnot(identical(tanh(356+0i), 1+0i))
> ## Used to be NaN+0i on Windows
>
> ## Not a regression test, but rather one of the good cases:
> (cNaN <- as.complex("NaN"))
[1] NaN+0i
> stopifnot(identical(cNaN, complex(re = NaN)), is.nan(Re(cNaN)), Im(cNaN) == 0)
> dput(cNaN) ## (real = NaN, imaginary = 0)
complex(real=NaN, imaginary=0)
> ## Partly new behavior:
> (c0NaN <- complex(real=0, im=NaN))
[1] 0+NaNi
> (cNaNaN <- complex(re=NaN, im=NaN))
[1] NaN+NaNi
> stopifnot(identical(cNaN, as.complex(NaN)),
+ identical(vapply(c(cNaN, c0NaN, cNaNaN), format, ""),
+ c("NaN+0i", "0+NaNi", "NaN+NaNi")),
+ identical(cNaN, NaN + 0i),
+ identical(cNaN, Conj(cNaN)),
+ identical(cNaN, cNaN+cNaN),
+
+ identical(cNaNaN, 1i * NaN),
+ identical(cNaNaN, complex(modulus= NaN)),
+ identical(cNaNaN, complex(argument= NaN)),
+ identical(cNaNaN, complex(arg=NaN, mod=NaN)),
+
+ identical(c0NaN, c0NaN+c0NaN), # !
+ ## Platform dependent, not TRUE e.g. on F21 gcc 4.9.2:
+ ## identical(NA_complex_, NaN + NA_complex_ ) ,
+ ## Probably TRUE, but by a standard ??
+ ## identical(cNaNaN, 2 * c0NaN), # C-library arithmetic
+ ## identical(cNaNaN, 2 * cNaN), # C-library arithmetic
+ ## identical(cNaNaN, NA_complex_ * Inf),
+ TRUE)
>
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