Design and analysis of Toeplitz preconditioners

The solution of symmetric positive definite Toeplitz systems Ax = b by the preconditioned conjugate gradient (PCG) method was recently proposed by Strang and analyzed by R. Chan and Strang. The convergence rate of the PCG method depends heavily on the choice of preconditioners for the given Toeplitz matrices. In this paper, we present a general approach to the design of Toeplitz preconditioners based on the idea to approximate a partially characterized linear deconvolution with circular deconvolutions. All resulting preconditioners can therefore be inverted via various fast transform algorithms with O(N log N ) operations. For a wide class of problems, the 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 ) .