A design and analysis of circulant preconditioners

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-1T\\(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(NlogN) 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(NlogN) [2].