On the optimal solution of large eigenpair problems (Q1080619)

For computing eigenvalues and eigenvectors of matrices, the so-called generalized minimal residual algorithm is sketched and compared to the Lanczos method. It is shown that this algorithm is almost optimal in reducing the residual \(\| Ax-\phi x\|\), using information from the Krylov subspaces \(A_ j=span\{b,Ab,...,A^{j-1}b\}\). In counting the numerical effort, only the matrix-vector multipliers are considered. Some generalizations are given for computing p eigenpairs of symmetric matrices simultaneously, and also some numerical examples.