SIAM Journal on Matrix Analysis and Applications vol:18 pages:1-20
For the computation of a few eigenvalues of Ax = mu Bx, the restarted Arnoldi method is often applied to transformations, e.g., the shift-invert transformation. Such transformations typically require the solution of linear systems. This paper presents an analysis of the application of the transformation (M(A) - alpha M(B))(-1)(A - lambda B) to Arnoldi's method where alpha and lambda are parameters and M(A) - alpha M(B) is some approximation to A - alpha B. In fact, (M(A) - alpha M(B))(-1) corresponds to an iterative linear system solver for the system (A - alpha B)x = b. The transformation is an alternative to the shift-invert transformation (A - alpha B)B--1 when direct system solvers are not available or not feasible. The restarted Amoldi method is analyzed in the case of detection of the rightmost eigenvalues of real nonsymmetric matrices. The method is compared to Davidson's method by use of numerical examples.