I am dealing with a generalized eignevalue problem which I need to solve every time-step.
Although I used the Matlab function eigs, the computation is still expensive since I need to generate a large sparse matrix A and apply boundary conditions.
Now I am trying to implement the Arnoldi iteration, this way I do not have to create the matrix A, since we are only interested in the action of A on the vector v.
My questions are:
Will I be treating the Hessenberg matrix as my new A, i.e. applying boundary conditions to it and using it to solve for the desired eigenvalues and eigenvectors? As far as I know, H is a projection of A and its eigenvalues are related to the eigenvalues of A.
Furthermore, if I have to solve for H, can I use gsvd to target a specific eigenvalue and its corresponding eigenvector like I do with eigs? (H is not a square matrix so I am unable now to use eig or eigs)
Lastly, how do I go about the fact that the eigenvectors of H have a different size (less) than my initial matrix A or the eigenvectors obtained from it?
Any insight regarding any of my questions is highly appreciated.