Preconditioning of Hermitian block-Toeplitz-Toeplitz-block matrices by level-1 preconditioners
Daniel Potts, Gabriele Steidl · Contemporary mathematics - American Mathematical Society · 2001
. In this paper, we are interested in the eigenvalue distribution of sequences of preconditioned Hermitian block{Toeplitz{Toeplitz{block (BTTB) matrices of size N1N2 N1N2 if N1 ; N2 !1. We focus on level{1 preconditioners constructed from circulant{like matrices. For some reasons we restrict our attention to BTTB matrices having the sum of nonnegative trigonometric polynomials p(x) and q(y) as generating function. We show that for some usual preconditioners, O(N 1 ) eigenvalues of the preconditioned matrices behave as the {th (0 < < 1) power of the reciprocal eigenvalues of the N1 N1 Toeplitz matrices generated by q. For example, if p and q have one zero of order 4, respectively, then the preconditioned BTTB matrices with level{1 preconditioners based on the sin-I transform have 2N1 eigenvalues which behave for N1 ; N2 !1 as N1=k (k = 1; : : : ; N1 ). 1. Introduction In this paper, we are interested in the solution of Hermitian block{Toeplitz{- Toeplitz{block (BTTB) systems AN...