Eigs Function on Function Handle Not Converged

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Sam Sun
Sam Sun 2018-10-12
评论： Sam Sun 2018-10-23
my little fun problem looks like:
F is some neat linear function handle. I need to find its largest real eigenvalues. (it basically looks like A*x where the construction of A involves unpleasant large matrix inversion and is huge (size of tens of thousands squared)
So I called eigs and it did not converge when specified to calculate 10 largestreal eigenvalues but converged when specified to calculate 10 smallestreal. I need both so any reason this might be it?
Or what is the threshold that determines convergence or not behind the curtains?
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Sam Sun
Sam Sun 2018-10-12
Thanks for taking the time! I'd specify more about my linear operator but the structure isn't quite so simple (just an excuse for unable to wield more powerful mathematics like many wizards in the field do)... but nevertheless, thank you!
Bruno Luong
Bruno Luong 2018-10-12
It doesn't matter what things you do behind the scene as long as it represents a linear operator.
Sometime you have to do some extra specific work to understand your operator, and might write your own eigs() to have a more suitable method.
In my youth I solved some big system of second order linearized Navier Stokes system with more 200k unknown and feed it through a Lanczos method to pull out the eigen mode of the system in order study the sensitivity of environmental global climate and it works just fine, and that was 30 year ago, and no one has cared this problem at the time.

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Christine Tobler
Christine Tobler 2018-10-15
Hi Sam,
The most likely reason for the convergence problems is that the eigenvalues are close together (or even multiples). The convergence speed of the internal method (for the 'largestabs' case) depends on the ratio between the smallest chosen eigenvalue and the largest eigenvalue that was not chosen.
Because of this, increasing the number of eigenvalues you are asking for (or just the 'SubspaceDimension') can help with convergence.
Another factor is that the 'largestreal' and 'smallestreal' options have some issues compared to 'largestabs' and 'smallestabs'. The inner iteration tends to go for the largest eigenvalues by absolute value, and has to be called back on every outer iteration to go for largest or smallest real part instead. So if the spectrum of your matrix is larger in the imaginary axes than the real axes, that could also be a reason for the convergence problems. Unfortunately, there's not much I can think of to improve that case.
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Bruno Luong
Bruno Luong 2018-10-18
编辑：Bruno Luong 2018-10-18
If you know 0 is an eigen value, so why not remove the kernel, meaning compute the eigen-values of the orthogonal projection on the kernel? Using function NULL to get the basis of the kernel.
It's is impossible that all the eigen values are 0 unless your matrix is 0.
Sam Sun
Sam Sun 2018-10-23
Yes good idea but I was in fact trying to prove this case that 0 is indeed an eigenvalue. I will lose the point once I suppress that output.

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