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Band preconditioners for block-Toeplitz-Toeplitz-block systems

1997; Elsevier BV; Volume: 259; Linguagem: Inglês

10.1016/s0024-3795(96)00295-9

1873-1856

Michael K. Ng,

Advanced Antenna and Metasurface Technologies

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.