A Note on the Complexity of the PCG Algorithm for Solving Toeplitz ...

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25030Number of iterations: y=f(x)20015010050d=0.37T. Chan PreconditionerNo PreconditionerNumber of iterations: y=f(x)252015105d=0.1T. Chan PreconditionerNo Preconditioner00 5 10 15The number of matrix size (n=2 x )00 5 10 15The number of matrix size (n=2 x )Figure 1: Two examples of iteration numbers.All the experiments were performed in MATLAB. We used the MATLAB-provided M-file“pcg” to solve the system. In our tests, the zero vector is the initial guess and the stoppingcriterion is ‖r j ‖ 2 /‖r 0 ‖ 2 < 10 −10 where r j is the residual after the j-th iteration. We considerthe following two cases in Problem 1:Case 1: Choose d = 0.37 and σ 2 s = 0.27.Case 2: Choose d = 0.1 and σ 2 s = 0.27.In Table 1, we give the number of iterations for convergence. In the table, n is the matrixsize. We compare the preconditioned system by using T. Chan’s preconditoner with thesystem with no preconditioner. We note that when n increases, the number of iterations ofthe preconditioned system is much less than that of the system with no preconditioner.n 2 5 2 6 2 7 2 8 2 9 2 10 2 11 2 12 2 13 2 14 2 15Case1No preconditionerCase1T. Chan’s preconditionerCase2No preconditionerCase2T. Chan’s preconditioner18 25 32 40 52 65 81 103 130 168 2147 7 8 8 8 9 9 10 10 10 1211 13 14 16 17 19 20 22 23 25 286 6 7 7 7 7 7 8 8 8 9Table 1: Number of iterations for PCG algorithm.
Acknowledgments. The authors would like to thank Prof. Clifford M. Hurvich for hisuseful suggestions on this topic.References[1] P. Brockwell and R. Davis, Time Series: Theory and Methods, Springer-Verlag, NewYork, 1991.[2] R. Chan and M. Ng, Conjugate Gradient Methods for Toeplitz Systems, SIAM Review,Vol. 38 (1996), pp. 427–482.[3] T. Chan, An Optimal Circulant Preconditioner for Toeplitz Systems, SIAM J. Sci.Statist. Comput., Vol. 9 (1988), pp. 766–771.[4] M. Fisher and R. Hartwig, Toeplitz Determinants–Some Applications, Theorems, andConjectures, Adv. Chem. Phys., Vol. 32 (1968), pp. 190–225.[5] G. Golub and C. Loan, Matrix Computations, 3rd edition, Johns Hopkins UniversityPress, Baltimore, 1996.[6] X. Jin, Developments and Applications of Block Toeplitz Iterative Solvers, Kluwer AcademicPublishers, Dordrecht; and Science Press, Beijing, 2002.[7] X. Jin and Y. Wei, Numerical Linear Algebra and Its Applications, Science Press, Beijing,2004.[8] R. King, M. Ahmadi, R. Gorgui-Naguib, A. Kwabwe and M. Azimi-Sadjadi, DigitalFiltering in One and Two Dimensions: Design and Applications, Plenum, New York,1989.[9] Y. Lu and C. Hurvich, On the complexity of the Preconditioned Conjugate GradientAlgorithm for Solving Toeplitz Systems with a Fisher–Hartwig Singularity, SIAM J.Matrix Anal. Appl., Vol. 27 (2005), pp. 638–653.[10] M. Ng, Iterative Methods for Toeplitz Systems, Oxford University Press, Oxford, 2004.[11] G. Strang, A Proposal for Toeplitz Matrix Calculations, Stud. Appl. Math., Vol. 74(1986), pp. 171–176.
Page 3 and 4: ⎧(n − k)t ⎪⎨k + kt k−n, i
Page 5: By using Lemmas 1 and 2, we notice

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