# linear system solution, iterative vs direct

Dear all, I have systems like

$(A - \lambda B) X = F$

where lambda is being updated inside a loop. I also have a limited number of eigenvectors of the matrix pair (A, B), say 40 eigenpair from a previous analysis step.

I could get the results with direct solution however due to repeated factorization of the operator matrix, I run into the O(n^3) wall. I was wondering if that could be possible to solve this system by some iterative methods. I, myself, tried some iterative methods, cg, minres, and others firstly in MATLAB, however cg like methods do not even converge due to the nature of the operator matrix since that can become indefinite in the loop, minres seems to converge hovewer that also does not converge in reasonable iteration counts. Also in these tries I had to use a PD preconditioner and do a factorization, which also induces other costs.

Basically are there some kind of a mathematical tricks for this kind of problem, where some limited number of eigenpairs are available from a previous analysis step?

As a poor engineer I tried all I know, so any further ideas?

TIA, Umut

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It is a vague suggestion, rather than an answer, but perhaps the eigenpairs you know for A,B will help in constructing a pre-conditioner for a conjugate gradient iteration. I don't recall anything quite like what you describe. – hardmath Jan 1 '11 at 5:24
You may be able to use the Gauss-Seidel with Successive Over-Relaxation (SOR)? – Steve Tranby May 7 '12 at 5:10