A Family of Block Preconditioners for Block Systems

Abstract

We study the solution of block system A mn x = b by the preconditioned conjugate gradient method where A mn is an m-by-m block matrix with n-by-n Toeplitz blocks. The preconditioner c (1) F (A mn) is a matrix that preserves the block structure of A mn. Speciically, it is deened to be the minimizer of kA mn ; C mn k F over all m-by-m block matrices C mn with n-by-n circulant blocks. We prove that if A mn is positive deenite, then c (1) F (A mn) is positive deenite too. We also show that c (1) F (A mn) is a good preconditioner for solving separable block systems with Toeplitz blocks and quadrantally symmetric block T oeplitz systems. We then discuss some of the spectral properties of the operator c (1) F. In particular, we s h o w that the operator norms jjc (1) F jj 2 = jjc (1) F jj F = 1. Abbreviated Title. Block Preconditioners