# Trigonometric Preconditioners for Block Toeplitz Systems

Abstract

This paper is concerned with the solution of a system of linear equations T
M, N
X=b, where T
M, N
denotes a positive definite doubly symmetric block-Toeplitz matrix with Toeplitz blocks arising from a generating function f of the Wiener class. We derive optimal and Strang-type trigonometric preconditioners P
M, N
of T
M, N
from the Fejér and Fourier sum of f, respectively. Using relations between trigonometric transforms and Toeplitz matrices, we prove that for all ε > 0 and sufficiently large M, N, at most O(M) + O(N) eigenvalues of lie outside the interval (1 — ε, l + ε) such that the preconditioned conjugate gradient method converges in at most O(M) + O(N) steps.

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