Band preconditioners for block-Toeplitz-Toeplitz-block systems Original Research Article

Preconditioned conjugate gradient methods are employed to solve symmetric positive definite m-by-m block Toeplitz with n-by-n Toeplitz block systems Am,nx = b where Am,n are generated by 2π-periodic nonnegative functions with zeros. Serra has proposed using band block Toeplitz with band Toeplitz block matrices Bm,n, with their external and internal bandwidths independent of m and n as preconditioners. Serra showed that if the Hessians of the generating function at the zeros are positive definite, then the condition number of B−1m,nAm,n is uniformly bounded by a constant independent of m and n, whereas the condition number of Am,n tends to infinity as m and n tend to infinity. In this paper, we provide a method for deriving band preconditioners for block-Toeplitz-Toeplitz-block matrices. Numerical examples are given to illustrate the performance of the method.