An Iterative Method Based on an A-Biorthogonalization Process for Nonsymmetric Linear Systems

摘要

The Conjugate Gradient (CG) method and the Conjugate Residual (CR) method are well-known Krylov subspace methods for solving symmetric (positive definite) linear systems. For solving nonsymmetric linear systems, Fletcher extended CG to nonsymmetric linear systems. However, the extended algorithm, known as Bi-CG, often shows irregular convergence behavior in the residual norm. The purpose of this paper is to extend CR to nonsymmetric linear systems based on an A-biorthogonalozation process. Since CR takes a minimum norm residual approach, the extended CR algorithm, named Bi-CR, can be expected to give smoother convergence behavior than Bi-CG. Numerical experiments show that Bi-CR is often more efficient than Bi-CG.