Fails on singularity. about svd

[j2kun]

I can confirm that the output of this is nondeterministic.

When computing the third singular value for this matrix [[1,2,3], [4,5,6], [7,8,9]] I get (run a few times, to show it's nondeterministic):

5.00408346
1.87754565
5.07695215
11.89394783
16.00065575
1.68317765

This is due to numerical instability of the part where I subtract the outer product. For instance, when computing the SVD and printing the matrix in each 1d step, on the last one the matrix is reduced to:

[[ -4.44089210e-15  -5.78703752e-15  -6.43929354e-15]
 [ -1.36002321e-15  -1.59247615e-15  -1.85962357e-15]
 [  1.55431223e-15   1.78329573e-15   2.05391260e-15]]

So one fix can be to threshold each entry by epsilon, or threshold the entire matrix based on the max magnitude value. IIRC, if at any step the remaining matrix is zero, then all remaining singular values and vectors are zero.

However, I think this method is inherently not numerically stable. I'm not sure if thresholding is enough to fix the problem. A literature search on numerical stability for the power method might clarify this.