A Lanczos Method for a Class of Nonsymmetric Systems of Linear Equations

Abstract

Let L be a real linear operator with a positive definite symmetric part M. In certain applications a number of problems of the form $Mv = g$ can be solved with less human or computational effort than the original equation $Lu = f$. An iterative Lanczos method, which requires no a priori information on the spectrum of the operators, is derived for such problems. The convergence of the method is established assuming only that $M^{ - 1} L$ is bounded. If $M^{ - 1} L$ differs from the identity mapping by a compact operator the convergence is shown to be superlinear. The method is particularly well suited for large sparse systems arising from elliptic problems. Results from a series of numerical experiments are presented. They indicate that the method is numerically stable and that the number of iterations can be accurately predicted by our error estimate.