Convergence analysis of superoptimal PCG algorithm for Toeplitz systems with a Fisher–Hartwig singularity

Abstract

Recently, Lu and Hurvich [Y. Lu, C. Hurvich, On the complexity of the preconditioned conjugate gradient algorithm for solving toeplitz systems with a Fisher–Hartwig singularity, SIAM J. Matrix Anal. Appl. 27 (2005) 638–653] used the preconditioned conjugate gradient method with the optimal circulant preconditioner proposed in Chan [T. Chan, An optimal circulant preconditioner for Toeplitz systems, SIAM J. Sci. Statist. Comput. 9 (1988) 766–771] for solving the Toeplitz system T n ( f ) x = b where the generating function f is given by f ( ω ) = | 1 - e - i ω | - 2 d h ( ω ) with d ∈ - 1 2 , 1 2 ⧹ { 0 } . The function h ( ω ) is positive continuous on [ - π , π ] and differentiable on [ - π , π ] ⧹ { 0 } . In this paper, we will use the superoptimal circulant preconditioner proposed by Tyrtyshnikov [E. Tyrtyshnikov, Optimal and superoptimal circulant preconditioners, SIAM J. Matrix Anal. Appl. 13 (1992) 459–473] to solve the same problem when 0 < d < 1 / 2 . Our convergence analysis shows that the number of iterations is bounded by O ( log 3 n ) and therefore the complexity of our algorithm is O ( n log 4 n ) . We notice that the numerical performance of superoptimal preconditioner is almost the same as that of the optimal preconditioner.