Precision of SVD?

This is a "what's going on under the hood?" kind of question. I have this symmetric positive-definite matrix C, and I'm interested in its eigenvalues. Since it's SPD, I'm using svd to calculate the eigenvalues.

My question is this: I'm getting a smallest eigenvalue of 10^-35, and second-smallest of order 10^-17, both of which are below machine precision. How is MATLAB calculating the smallest elements here? Additionally, how suspicious of these numbers should I be?

For the curious, I've attached a copy of the matrix I'm working with.

Cheers, Zach

[Matt J]

For the curious, this is an imprecise copy of the matrix I'm working with. It won't give exactly the same results, but will do qualitatively the same thing.

Note that you can attach files here and could attach a .mat file for us with the exact matrix C.

[Zachary]

Excellent -- I've gone ahead and done just that.

Answer 1

[the cyclist]

This is a non-trivial question, and is related to the condition number of the eigenvector matrix. (Not to be confused with the condition number of the matrix itself.)

I suggest you look at this book chapter from Cleve Moler's book, particular Section 10.6 which is about eigenvalue sensitivity and accuracy. This chapter points out some useful functions for estimating accuracy, such as condest.

[Matt J]

I'm getting a smallest eigenvalue of 10^-35, and second-smallest of order 10^-17, both of which are below machine precision.

I agree with cyclist, but one thing I'd comment is that 10^-35 and 10^-17 are not necessarily out of the reach of machine precision, as premised. For example, here is an example where they are the exact eigenvalues, and are exactly recovered.

    >> A=diag([1e-17,1e-35]); 

    >> error = nnz(svd(A)-[1e-17;1e-35])

    error =

         0