R : Copyright 2005, The R Foundation for Statistical Computing
Version 2.1.1  (2005-06-20), ISBN 3-900051-07-0

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>  # rsvd.r         
>  #
>  # compute singular values and vectors 
>  #
>  # first matrix
>    A <- matrix(c(2,0,-1,2,1,-2.8,-2,4,2,.4,1,3),4,3)
>    A
     [,1] [,2] [,3]
[1,]    2  1.0  2.0
[2,]    0 -2.8  0.4
[3,]   -1 -2.0  1.0
[4,]    2  4.0  3.0
>    svdA <- svd(A)
>    svdA
$d
[1] 6.436702 3.000000 1.252545

$u
           [,1]      [,2]       [,3]
[1,] -0.3926071 0.4938534 -0.5166486
[2,]  0.3148308 0.6441566 -0.3183212
[3,]  0.2395126 0.5582691  0.7221964
[4,] -0.8302861 0.1717751  0.3319310

$v
           [,1]       [,2]       [,3]
[1,] -0.4171855  0.2576627 -0.8715309
[2,] -0.7883389 -0.5797410  0.2059663
[3,] -0.4521924  0.7729880  0.4449850

>    t(svdA$u) %*% A %*% svdA$v       # diagonal?
              [,1]          [,2]         [,3]
[1,]  6.436702e+00  4.709774e-16 4.624122e-16
[2,] -2.541370e-16  3.000000e+00 1.594861e-16
[3,]  1.317035e-16 -4.676435e-16 1.252545e+00
>  # second matrix
>    A <- matrix(1:12,4,3,byrow=T)
>    A
     [,1] [,2] [,3]
[1,]    1    2    3
[2,]    4    5    6
[3,]    7    8    9
[4,]   10   11   12
>    svdA <- svd(A)
>    svdA
$d
[1] 2.546241e+01 1.290662e+00 9.105382e-17

$u
           [,1]        [,2]       [,3]
[1,] -0.1408767 -0.82471435  0.4799123
[2,] -0.3439463 -0.42626394 -0.8366404
[3,] -0.5470159 -0.02781353  0.2335439
[4,] -0.7500855  0.37063688  0.1231842

$v
           [,1]        [,2]       [,3]
[1,] -0.5045331  0.76077568  0.4082483
[2,] -0.5745157  0.05714052 -0.8164966
[3,] -0.6444983 -0.64649464  0.4082483

>    t(svdA$u) %*% A %*% svdA$v       # diagonal?
             [,1]          [,2]          [,3]
[1,] 2.546241e+01 -1.097213e-15 -2.695327e-15
[2,] 1.566997e-15  1.290662e+00  6.475397e-16
[3,] 4.099391e-17  3.312136e-16  1.812987e-16
>  # third matrix
>    A <- matrix(c(1,1,2,1,1,-1,0,0, 3,6,9,12,0,0,0,0, -1,1,0,0,1,1,2,1,
+           0,0,0,0,3,6,9,12),8,4)
>    A
     [,1] [,2] [,3] [,4]
[1,]    1    3   -1    0
[2,]    1    6    1    0
[3,]    2    9    0    0
[4,]    1   12    0    0
[5,]    1    0    1    3
[6,]   -1    0    1    6
[7,]    0    0    2    9
[8,]    0    0    1   12
>    svdA <- svd(A)
>    svdA
$d
[1] 16.605301 16.605301  1.806652  1.806652

$u
             [,1]        [,2]        [,3]        [,4]
[1,]  0.186964610 0.004972061 -0.56613134  0.30743263
[2,]  0.365191519 0.036157226  0.51069590  0.06721240
[3,]  0.552156129 0.041129287 -0.05543544  0.37464504
[4,]  0.721645337 0.054839049 -0.07391392 -0.41322805
[5,] -0.004972061 0.186964610  0.30743263  0.56613134
[6,] -0.036157226 0.365191519  0.06721240 -0.51069590
[7,] -0.041129287 0.552156129  0.37464504  0.05543544
[8,] -0.054839049 0.721645337 -0.41322805  0.07391392

$v
              [,1]       [,2]        [,3]          [,4]
[1,]  1.450922e-01 0.00000000  0.00000000  9.894181e-01
[2,]  9.865038e-01 0.07588491 -0.01112806 -1.446648e-01
[3,]  4.212459e-18 0.14509216  0.98941815 -6.177314e-19
[4,] -7.588491e-02 0.98650380 -0.14466479  1.112806e-02

>    t(svdA$u) %*% A %*% svdA$v       # diagonal?
              [,1]          [,2]          [,3]          [,4]
[1,]  1.660530e+01  2.690990e-16  5.839784e-17  8.427046e-17
[2,] -4.208873e-16  1.660530e+01 -1.046255e-15 -6.261268e-17
[3,]  2.575158e-15  1.166602e-16  1.806652e+00 -2.521611e-16
[4,] -2.250620e-15 -7.973917e-16 -1.072649e-16  1.806652e+00
>  # done
>    rm(list=ls())
>    q()