A FAMILY OF BLOCK PRECONDITIONERS FOR BLOCK SYSTEMS

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 parallel-to A(mn) - C(mn) parallel-to 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 parallel-to c(F)(1) parallel-to 2 = parallel-to c(F)(1) parallel-to F = 1.