The Restarted Arnoldi Method Applied to Iterative Linear System Solvers for the Computation of Rightmost Eigenvalues

Abstract

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 $(A-\alpha B)^{-1}B$ when direct system solvers are not available or not feasible. The restarted Arnoldi 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.