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
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.