Abstract
Circulant preconditioning for symmetric Toeplitz systems has been well developed over the past few decades. For a large class of such systems, descriptive bounds on convergence for the conjugate gradient method can be obtained. For (real) nonsymmetric Toeplitz systems, much work had been focused on normalising the original systems until Pestana and Wathen (Siam J. Matrix Anal. Appl. 36(1):273–288 2015) recently showed that theoretic guarantees on convergence for the minimal residual method can be established via the simple use of reordering. The authors further proved that a suitable absolute value circulant preconditioner can be used to ensure rapid convergence. In this paper, we show that the related ideas can also be applied to the systems defined by analytic functions of (real) nonsymmetric Toeplitz matrices. For the systems defined by analytic functions of complex Toeplitz matrices, we also show that certain circulant preconditioners are effective. Numerical examples with the conjugate gradient method and the minimal residual method are given to support our theoretical results.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Axelsson, O., Lindskog, G.: On the rate of convergence of the preconditioned conjugate gradient method. Numer. Math. 48, 499–524 (1986)
Bini, D.A., Dendievel, S., Latouche, G., Meini, B.: Computing the exponential of large block-triangular block-Toeplitz matrices encountered in fluid queues. Linear Algebra Appl. 502(Supplement C), 387–419 (2016). Structured Matrices: Theory and Applications
Chan, R.H., Yeung, M.-C.: Circulant preconditioners for complex Toeplitz matrices. SIAM J. Numer. Anal. 30(4), 1193–1207 (1993)
Chan, T.F.: An optimal circulant preconditioner for Toeplitz systems. SIAM J. Sci. Stat. Comput. 9(4), 766–771 (1988)
Duffy, D.J.: Finite Difference Methods in Financial Engineering: a Partial Differential Equation Approach. The Wiley Finance Series. Wiley, New York (2013)
Higham, N.J.: Functions of matrices. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2008. Theory and computation
Higham, N.J., Kandolf, P.: Computing the action of trigonometric and hyperbolic matrix functions. SIAM J. Sci. Comput. 39(2), A613–A627 (2017)
Hon, S.: Optimal preconditioners for systems defined by functions of Toeplitz matrices. Under review
Kressner, D., Luce, R.: Fast computation of the matrix exponential for a Toeplitz matrix. Arxiv e-prints (2016)
Lee, S.T., Pang, H.-K., Sun, H.-W.: Shift-invert Arnoldi approximation to the Toeplitz matrix exponential. SIAM J. Sci. Comput. 32(2), 774–792 (2010)
McDonald, E., Hon, S., Pestana, J., Wathen, A.: Preconditioning for Nonsymmetry and Time-Dependence, pp. 81–91. Springer International Publishing Cham (2017)
Ng, M.K.: Iterative methods for Toeplitz systems. Numerical Mathematics and Scientific Computation. Oxford University Press, New York (2004)
Ng, M.K., Pan, J.: Approximate inverse circulant-plus-diagonal preconditioners for Toeplitz-plus-diagonal matrices. SIAM J. Sci. Comput. 32(3), 1442–1464 (2010)
Pestana, J., Wathen, A.J.: A preconditioned MINRES method for nonsymmetric Toeplitz matrices. SIAM J. Matrix Anal. Appl. 36(1), 273–288 (2015)
Sachs, E.W., Strauss, A.K.: Efficient solution of a partial integro-differential equation in finance. Appl. Numer. Math. 58(11), 1687–1703 (2008)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Hon, S., Wathen, A. Circulant preconditioners for analytic functions of Toeplitz matrices. Numer Algor 79, 1211–1230 (2018). https://doi.org/10.1007/s11075-018-0481-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11075-018-0481-7