An iterated shift-and-invert Arnoldi algorithm for quadratic matrix eigenvalue problems

For solving the large scale quadratic eigenvalue problem L(λ)x: = (Aλ² + Bλ + C)x = 0, a direct projection method based on the Krylov subspaces generated by a single matrix A^-1B using the standard Arnoldi algorithm is considered. It is shown that, when iteratively combined with the shift-and-invert technique, it results in a fast converging algorithm. The important situations of inexact shift-and-invert are also discussed and numerical examples are presented to illustrate the new method.