Design and analysis of Toeplitz preconditioners

A general approach is presented for constructing preconditioners P for a symmetric positive-definite Toeplitz matrix A so that the Toeplitz system Ax=b can be solved efficiently by the preconditioned conjugate-gradient (PCG) method. The work is a generalization of recent work by G. Strang (1986) and R. Chan (1988). It is shown that the new preconditioners can be inverted directly by means of various fast transform methods with O(N log N) operations and that the eigenvalues of the preconditioned matrices P^-1 A are clustered around one so that the PCG method converges very quickly