A fast, preconditioned conjugate gradient toeplitz solver

Abstract

We give a simple factorization of an arbitrary hermitian, positive definite matrix in which the factors are well-conditioned, hermitian, and positive definite. In fact, given knowledge of the extreme eigenvalues of the original matrix A, we can achieve an optimal improvement, making the condition numbers of each of the two factors equal to the square root of the condition number of A. We apply this technique to the solution of hermitian, positive definite Toeplitz systems. Large linear systems with hermitian, positive definite Toeplitz matrices arise in some signal processing applications. We give a stable fast algorithm for solving these systems that is based on the preconditioned conjugate gradient method. The algorithm exploits Toeplitz structure to reduce the cost of an iteration to O( nlog n) by applying the fast Fourier Transform to compute matrix-vector products. We use our matrix factorization as a preconditioner.