R version 2.13.0 Patched (2011-04-19 r55507)
Copyright (C) 2011 The R Foundation for Statistical Computing
ISBN 3-900051-07-0
Platform: x86_64-unknown-linux-gnu (64-bit)
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> ## tests of R functions based on the lapack module
>
> options(digits=4)
>
> ## ------- examples from ?svd ---------
>
> hilbert <- function(n) { i <- 1:n; 1 / outer(i - 1, i, "+") }
> Eps <- 100 * .Machine$double.eps
>
> X <- hilbert(9)[,1:6]
> (s <- svd(X)); D <- diag(s$d)
$d
[1] 1.668e+00 2.774e-01 2.224e-02 1.085e-03 3.244e-05 5.235e-07
$u
[,1] [,2] [,3] [,4] [,5] [,6]
[1,] -0.7245 0.6266 0.27350 -0.08527 0.02074 -0.004025
[2,] -0.4282 -0.1299 -0.64294 0.55047 -0.27253 0.092816
[3,] -0.3122 -0.2804 -0.33633 -0.31418 0.61632 -0.440904
[4,] -0.2479 -0.3142 -0.06931 -0.44667 0.02945 0.530120
[5,] -0.2064 -0.3141 0.10786 -0.30242 -0.35567 0.237038
[6,] -0.1771 -0.3027 0.22106 -0.09042 -0.38879 -0.260449
[7,] -0.1553 -0.2877 0.29281 0.11551 -0.19286 -0.420945
[8,] -0.1384 -0.2722 0.33784 0.29313 0.11633 -0.160790
[9,] -0.1249 -0.2571 0.36543 0.43885 0.46497 0.434600
$v
[,1] [,2] [,3] [,4] [,5] [,6]
[1,] -0.7365 0.6225 0.2550 -0.06976 0.01328 -0.001588
[2,] -0.4433 -0.1819 -0.6867 0.50860 -0.19627 0.041117
[3,] -0.3275 -0.3509 -0.2611 -0.50474 0.61606 -0.259216
[4,] -0.2626 -0.3922 0.1044 -0.43748 -0.40834 0.638902
[5,] -0.2204 -0.3946 0.3510 0.01612 -0.46428 -0.675827
[6,] -0.1904 -0.3832 0.5111 0.53856 0.44664 0.257249
> stopifnot(abs(X - s$u %*% D %*% t(s$v)) < Eps)# X = U D V'
> stopifnot(abs(D - t(s$u) %*% X %*% s$v) < Eps)# D = U' X V
>
> X <- cbind(1, 1:7)
> (s <- svd(X)); D <- diag(s$d)
$d
[1] 12.07 1.16
$u
[,1] [,2]
[1,] 0.09762 0.674356
[2,] 0.17884 0.503717
[3,] 0.26006 0.333078
[4,] 0.34128 0.162438
[5,] 0.42250 -0.008201
[6,] 0.50372 -0.178840
[7,] 0.58494 -0.349479
$v
[,1] [,2]
[1,] 0.1979 0.9802
[2,] 0.9802 -0.1979
> stopifnot(abs(X - s$u %*% D %*% t(s$v)) < Eps)# X = U D V'
> stopifnot(abs(D - t(s$u) %*% X %*% s$v) < Eps)# D = U' X V
>
> # test nu and nv
> svd(X, nu = 0)
$d
[1] 12.07 1.16
$u
NULL
$v
[,1] [,2]
[1,] 0.1979 0.9802
[2,] 0.9802 -0.1979
> (s <- svd(X, nu = 7))
$d
[1] 12.07 1.16
$u
[,1] [,2] [,3] [,4] [,5] [,6] [,7]
[1,] 0.09762 0.674356 -0.38314 -0.35392 -0.32471 -0.29549 -0.26628
[2,] 0.17884 0.503717 -0.13857 0.05843 0.25543 0.45242 0.64942
[3,] 0.26006 0.333078 0.89399 -0.09282 -0.07963 -0.06644 -0.05326
[4,] 0.34128 0.162438 -0.10083 0.88314 -0.13289 -0.14891 -0.16494
[5,] 0.42250 -0.008201 -0.09566 -0.14090 0.81386 -0.23139 -0.27663
[6,] 0.50372 -0.178840 -0.09048 -0.16494 -0.23940 0.68614 -0.38831
[7,] 0.58494 -0.349479 -0.08531 -0.18898 -0.29265 -0.39633 0.50000
$v
[,1] [,2]
[1,] 0.1979 0.9802
[2,] 0.9802 -0.1979
> stopifnot(dim(s$u) == c(7,7))
> svd(X, nv = 0)
$d
[1] 12.07 1.16
$u
[,1] [,2]
[1,] 0.09762 0.674356
[2,] 0.17884 0.503717
[3,] 0.26006 0.333078
[4,] 0.34128 0.162438
[5,] 0.42250 -0.008201
[6,] 0.50372 -0.178840
[7,] 0.58494 -0.349479
$v
NULL
>
> # test of complex case
>
> X <- cbind(1, 1:7+(-3:3)*1i)
> s <- svd(X); D <- diag(s$d)
> stopifnot(abs(X - s$u %*% D %*% Conj(t(s$v))) < Eps)
> stopifnot(abs(D - Conj(t(s$u)) %*% X %*% s$v) < Eps)
>
>
>
> ## ------- tests of random real and complex matrices ------
> fixsign <- function(A) {
+ A[] <- apply(A, 2, function(x) x*sign(Re(x[1])))
+ A
+ }
> ## 100 may cause failures here.
> eigenok <- function(A, E, Eps=1000*.Machine$double.eps)
+ {
+ print(fixsign(E$vectors))
+ print(zapsmall(E$values))
+ V <- E$vectors; lam <- E$values
+ stopifnot(abs(A %*% V - V %*% diag(lam)) < Eps,
+ abs(lam[length(lam)]/lam[1]) < Eps || # this one not for singular A :
+ abs(A - V %*% diag(lam) %*% t(V)) < Eps)
+ }
>
> Ceigenok <- function(A, E, Eps=1000*.Machine$double.eps)
+ {
+ print(fixsign(E$vectors))
+ print(signif(E$values, 5))
+ V <- E$vectors; lam <- E$values
+ stopifnot(Mod(A %*% V - V %*% diag(lam)) < Eps,
+ Mod(A - V %*% diag(lam) %*% Conj(t(V))) < Eps)
+ }
>
> ## failed for some 64bit-Lapack-gcc combinations:
> sm <- cbind(1, 3:1, 1:3)
> eigenok(sm, eigen(sm))
[,1] [,2] [,3]
[1,] 0.5774 0.8452 0.9428
[2,] 0.5774 0.1690 -0.2357
[3,] 0.5774 -0.5071 -0.2357
[1] 5 1 0
> eigenok(sm, eigen(sm, sym=FALSE))
[,1] [,2] [,3]
[1,] 0.5774 0.8452 0.9428
[2,] 0.5774 0.1690 -0.2357
[3,] 0.5774 -0.5071 -0.2357
[1] 5 1 0
>
> set.seed(123)
> sm <- matrix(rnorm(25), 5, 5)
> sm <- 0.5 * (sm + t(sm))
> eigenok(sm, eigen(sm))
[,1] [,2] [,3] [,4] [,5]
[1,] 0.5899 0.1683 0.02315 0.471808 0.6329
[2,] 0.1936 0.2931 0.89217 -0.009784 -0.2838
[3,] 0.6627 -0.4812 -0.15825 0.082550 -0.5454
[4,] 0.1404 0.7985 -0.41848 0.094314 -0.3983
[5,] -0.3946 -0.1285 0.05768 0.872692 -0.2507
[1] 1.7814 1.5184 0.5833 -1.0148 -2.4908
> eigenok(sm, eigen(sm, sym=FALSE))
[,1] [,2] [,3] [,4] [,5]
[1,] 0.6329 0.5899 0.1683 0.471808 0.02315
[2,] -0.2838 0.1936 0.2931 -0.009784 0.89217
[3,] -0.5454 0.6627 -0.4812 0.082550 -0.15825
[4,] -0.3983 0.1404 0.7985 0.094314 -0.41848
[5,] -0.2507 -0.3946 -0.1285 0.872692 0.05768
[1] -2.4908 1.7814 1.5184 -1.0148 0.5833
>
> sm[] <- as.complex(sm)
> Ceigenok(sm, eigen(sm))
[,1] [,2] [,3] [,4] [,5]
[1,] 0.5899+0i 0.1683+0i 0.02315+0i 0.471808+0i 0.6329+0i
[2,] 0.1936+0i 0.2931+0i 0.89217+0i -0.009784+0i -0.2838+0i
[3,] 0.6627+0i -0.4812+0i -0.15825+0i 0.082550+0i -0.5454+0i
[4,] 0.1404+0i 0.7985+0i -0.41848+0i 0.094314+0i -0.3983+0i
[5,] -0.3946+0i -0.1285+0i 0.05768+0i 0.872692+0i -0.2507+0i
[1] 1.7814 1.5184 0.5833 -1.0148 -2.4908
> Ceigenok(sm, eigen(sm, sym=FALSE))
[,1] [,2] [,3] [,4] [,5]
[1,] 0.6329+0i 0.5899+0i 0.1683+0i 0.471808+0i 0.02315+0i
[2,] -0.2838+0i 0.1936+0i 0.2931+0i -0.009784+0i 0.89217+0i
[3,] -0.5454+0i 0.6627+0i -0.4812+0i 0.082550+0i -0.15825+0i
[4,] -0.3983+0i 0.1404+0i 0.7985+0i 0.094314+0i -0.41848+0i
[5,] -0.2507+0i -0.3946+0i -0.1285+0i 0.872692+0i 0.05768+0i
[1] -2.4908+0i 1.7814+0i 1.5184+0i -1.0148+0i 0.5833+0i
>
> sm[] <- sm + rnorm(25) * 1i
> sm <- 0.5 * (sm + Conj(t(sm)))
> Ceigenok(sm, eigen(sm))
[,1] [,2] [,3] [,4]
[1,] 0.5373+0.0000i 0.3338+0.0000i 0.02834+0.0000i 0.4378+0.0000i
[2,] 0.3051+0.0410i -0.0264-0.1175i -0.43963+0.7256i -0.0474+0.2975i
[3,] 0.3201-0.3756i 0.3379+0.4760i -0.09325-0.3281i 0.0536+0.2447i
[4,] 0.3394+0.2330i -0.1044-0.6839i 0.09966-0.3629i 0.1894+0.1979i
[5,] -0.2869+0.3483i -0.0766+0.2210i -0.14602+0.0132i 0.7449-0.1576i
[,5]
[1,] 0.6383+0.0000i
[2,] -0.1909-0.2093i
[3,] -0.4788-0.0861i
[4,] -0.3654+0.0418i
[5,] -0.2229-0.3012i
[1] 2.4043 1.3934 0.7854 -1.4050 -2.8006
> Ceigenok(sm, eigen(sm, sym=FALSE))
[,1] [,2] [,3] [,4]
[1,] 0.6383+0.0000i 0.5373+0.0000i 0.4283+0.0907i 0.0504-0.3300i
[2,] -0.1909-0.2093i 0.3051+0.0410i -0.1080+0.2813i -0.1201+0.0084i
[3,] -0.4788-0.0861i 0.3201-0.3756i 0.0018+0.2505i 0.5216-0.2622i
[4,] -0.3654+0.0418i 0.3394+0.2330i 0.1443+0.2329i -0.6918+0.0000i
[5,] -0.2229-0.3012i -0.2869+0.3483i 0.7614+0.0000i 0.2069+0.1091i
[,5]
[1,] 0.01468+0.02424i
[2,] -0.84836+0.00000i
[3,] 0.23232-0.24980i
[4,] 0.36200-0.10282i
[5,] -0.08698-0.11804i
[1] -2.8006+0i 2.4043+0i -1.4050+0i 1.3934+0i 0.7854+0i
>