On Convergence Property of the Lanczos Method for Solving a Complex Shifted Hermitian Linear System

Abstract

We discuss the convergence property of the Lanczos method for solving a complex shifted Hermitian linear system $(α I + H)x = f$. By showing the colinear coefficient of two system's residuals, our convergence analysis reveals that under the condition $Re(α)+ λ _{min}(H)>0$, the method converges faster than that for the real shifted Hermitian linear system $(Re(α) I+H)x=f$. Numerical experiments verify such convergence property.