A Family of Block Preconditioners for Block Systems
Raymond H. Chan, Xiaoqing Jin · SIAM Journal on Scientific and Statistical Computing · 1992
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 is studied. The preconditioner $c_F^{(1)} (A_{mn} )$ is a matrix that preserves the block structure of $A_{mn} $. Specifically, it is defined as the minimizer of $||A_{mn} - C_{mn} ||_F $ over all m-by-m block matrices $C_{mn} $ with n-by-n circulant blocks. We prove that if $A_{mn} $ is positive definite, then $c_F^{(1)} (A_{mn} )$ is positive definite too. We also show that $c_F^{(1)} (A_{mn} )$ is a good preconditioner for solving separable block systems with Toeplitz blocks and quadrantally symmetric block Toeplitz systems. We then discuss some of the spectral properties of the operator $c_F^{(1)} $. In particular, we show that the operator norms $||c_F^{(1)} ||_2 = ||c_F^{(1)} ||_F = 1$.