Any reason why Octave, R, Numpy and LAPACK yield different SVD results on the same matrix?

Question

I'm using Octave and R to compute SVD using a simple matrix and getting two different answers! The code is listed below:

R

> a<-matrix(c(1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,1,1,0,0,0,0,0,0,0,0,0,1,1,1), 9, 4)
> a
 [,1] [,2] [,3] [,4]
 [1,]    1    1    0    0
 [2,]    1    1    0    0
 [3,]    1    1    0    0
 [4,]    1    0    1    0
 [5,]    1    0    1    0
 [6,]    1    0    1    0
 [7,]    1    0    0    1
 [8,]    1    0    0    1
 [9,]    1    0    0    1
> a.svd <- svd(a)
> a.svd$d
[1] 3.464102e+00 1.732051e+00 1.732051e+00 1.922963e-16
> a.svd$u
       [,1]       [,2]          [,3]          [,4]
 [1,] -0.3333333  0.4714045 -1.741269e-16  7.760882e-01
 [2,] -0.3333333  0.4714045 -3.692621e-16 -1.683504e-01
 [3,] -0.3333333  0.4714045 -5.301858e-17 -6.077378e-01
 [4,] -0.3333333 -0.2357023 -4.082483e-01  6.774193e-17
 [5,] -0.3333333 -0.2357023 -4.082483e-01  6.774193e-17
 [6,] -0.3333333 -0.2357023 -4.082483e-01  6.774193e-17
 [7,] -0.3333333 -0.2357023  4.082483e-01  5.194768e-17
 [8,] -0.3333333 -0.2357023  4.082483e-01  5.194768e-17
 [9,] -0.3333333 -0.2357023  4.082483e-01  5.194768e-17
> a.svd$v
      [,1]       [,2]          [,3] [,4]
[1,] -0.8660254  0.0000000 -4.378026e-17  0.5
[2,] -0.2886751  0.8164966 -2.509507e-16 -0.5
[3,] -0.2886751 -0.4082483 -7.071068e-01 -0.5
[4,] -0.2886751 -0.4082483  7.071068e-01 -0.5

Octave

octave:32> a = [ 1, 1, 1, 1, 1, 1, 1, 1, 1; 1, 1, 1, 0, 0, 0, 0, 0, 0; 0, 0, 0, 1, 1, 1, 0, 0, 0; 0, 0, 0, 0, 0, 0, 1, 1, 1 ]
a =

  1   1   1   1   1   1   1   1   1
  1   1   1   0   0   0   0   0   0
  0   0   0   1   1   1   0   0   0
  0   0   0   0   0   0   1   1   1

octave:33> [u, s, v] = svd (a)
u =

 -8.6603e-01  -1.0628e-16   2.0817e-17  -5.0000e-01
 -2.8868e-01   5.7735e-01  -5.7735e-01   5.0000e-01
 -2.8868e-01  -7.8868e-01  -2.1132e-01   5.0000e-01
 -2.8868e-01   2.1132e-01   7.8868e-01   5.0000e-01

s =

Diagonal Matrix

 3.4641e+00            0            0            0            0
          0   1.7321e+00            0            0            0
          0            0   1.7321e+00            0            0
          0            0            0   3.7057e-16            0
          0            0            0            0            0

v =

 -3.3333e-01   3.3333e-01  -3.3333e-01   7.1375e-01
 -3.3333e-01   3.3333e-01  -3.3333e-01  -1.3472e-02
 -3.3333e-01   3.3333e-01  -3.3333e-01  -7.0027e-01
 -3.3333e-01  -4.5534e-01  -1.2201e-01  -9.0583e-17
 -3.3333e-01  -4.5534e-01  -1.2201e-01   2.0440e-17
 -3.3333e-01  -4.5534e-01  -1.2201e-01   2.0440e-17
 -3.3333e-01   1.2201e-01   4.5534e-01   8.3848e-17
 -3.3333e-01   1.2201e-01   4.5534e-01   8.3848e-17
 -3.3333e-01   1.2201e-01   4.5534e-01   8.3848e-17

I'm new to both Octave and R so my first question is am I doing this right? If so, which one is "correct"? Are they both right? I've also tried this out in numpy and calling the LAPACK functions dgesdd and dgesvd directly. Numpy give me an answer similar to Octave and calling the LAPACK functions gives me an answer similar to R.

Thanks!

Answer 1

In SVD decomposition $A=UDV^T$ only $D$ is unique (up to reordering). It is more or less easy to see that $cU$ and $\frac{1}{c}V$ will give the same decomposition. So it is not surprising that different algorithms can give different results. What matters is that $D$ must be the same for all algorithms.