A Design and Analysis of Circulant Preconditioners

Abstract

We propose a new type of preconditioners for symmetric Toeplitz system Tx = b . When applying iterative methods to solve linear system with matrix T, we often use some preconditioner C by the preconditioned conjugate gradient (PCG) method [3]. If T is a symmetric positive definite Toeplitz matrix, two kinds of preconditioners are investigated: the “optimal” one , which minimizes ||C – T || _F , and the “superoptimal” one , which minimize ||I – C ^{− − 1} T || _F [8]. In this paper, we present a general approach to the design of Toeplitz preconditioners based on the optimal investigating and also preconditioners C with preserving the characteristic of the given matrix T. Fast all resulting preconditioners can be inverted via fast transform algorithms with O(N log N) operations. For a wide class of problems, PCG method converges in a finite number of iterations independent of N so that the computational complexity for solving these Toeplitz systems is O(N log N ) [2].