Calculate eigenfunctions to known eigenvalues

Hello,

I need to find eigenfunctions and eigenvalues of an operator . It would require to much storage to calculate the matrix depiction of , so I am using the eigs command and calculate the eigenvalues and eigenfunctions of by writing the operator as a function (the input of this function is a test function f and the output is ). So far so good. Everything works.

No I have run the code and I forgot to save the eigenvectors. I could run the code again to get the eigenvectors, but I wondering: Is there a more efficient way of getting the eigenvectors if the eigenvalues are known?

Thanks for help!

Answer 1

[Christine Tobler]

Unfortunately, there isn't a painless way to do this. That is, it's likely that writing new code to get eigenvectors given the eigenvalues will take you longer to get right, then just re-running eigs.

If you had the eigenvectors and needed the eigenvalues, that's quite painlessly possible using x'*A(x) for each eigenvector x.

The other way around, the usual way to do this would be to solve (A - lambda_mod*I) * x = rand, where lambda_mod is slightly different from lambda to avoid a singular linear system. You might need to do this in multiple iterations (inverse iteration as Torsten mentioned). But with your matrix being given as a function handle, I imagine solving a linear system with it is non-trivial.

Answer 2

[Torsten]

https://en.wikipedia.org/wiki/Inverse_iteration

You could also try

null(A-lambda*eye(size(A)))

where lambda is a given eigenvalue.

Are you talking about eigenfunctions or eigenvectors ? You mix up the two terms in your question.

[Malte]

Thanks for the answer. I want to calculate an eigenfunction. I dont think what you have suggested works for me. I can not write the operator as a matrix. It would require too much storage. It is of the size (). So what I am actually looking for is a solution that you have provided me in your answer, where A is a function.

[Torsten]

I think in this case you will have to numerically solve for the function f in the equation

A^ * f = lambda * f

Maybe the resulting equation is a PDE - I don't know what kind of differential operator A^ is.