Preconditioned Conjugate Gradient is M-Error-Reducing

Abstract

The Preconditioned Conjugate Gradient (PCG) method has proven to be extremely powerful for solving symmetric positive definite linear systems. This method can also be applied to nonsymmetric linear systems when combined with the NR/NE techniques. It has been shown in [1] that the CGNR algorithm, which is a nonsymmetric variant of the Conjugate Gradient (CG) method, is error-reducing with respect to the Euclidean norm. However, in practice the simple CGNR algorithm is seldom used because of the squared condition number of the iteration matrix. Preconditioning is frequently needed to overcome this difficulty. In the present paper we give a much richer result concerning the error-reducing property of the CG procedure. Assume that the preconditioner M is also symmetric positive definite. It is shown that the PCG method is error-reducing with respect to the M-norm.