Rational Krylov: A Practical Algorithm for Large Sparse Nonsymmetric Matrix Pencils

Abstract

The rational Krylov algorithm computes eigenvalues and eigenvectors of a regular not necessarily symmetric matrix pencil. It is a generalization of the shifted and inverted Arnoldi algorithm, where several factorizations with different shifts are used in one run. It computes an orthogonal basis and a small Hessenberg pencil. The eigensolution of the Hessenberg pencil approximates the solution of the original pencil. Different types of Ritz values and harmonic Ritz values are described and compared. Periodical purging of uninteresting directions reduces the size of the basis and makes it possible to compute many linearly independent eigenvectors and principal vectors of pencils with multiple eigenvalues. Relations to iterative methods are established. Results are reported for two large test examples. One is a symmetric pencil coming from a finite element approximation of a membrane; the other is a nonsymmetric matrix modeling an idealized aircraft stability problem.