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Backward Error Analysis for Eigenproblems Involving Conjugate Symplectic Matrices

Part of: Nonlinear algebraic or transcendental equations Numerical analysis: Ordinary differential equations Numerical linear algebra

Published online by Cambridge University Press:  10 November 2015

Wei-wei Xu ,
Wen Li  and
Xiao-qing Jin
Wei-wei Xu*
Affiliation:
School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, P.R. China
Wen Li
Affiliation:
School of Mathematical Sciences, South China Normal University, Guangzhou, 510631, P.R. China
Xiao-qing Jin
Affiliation:
Department of Mathematics, University of Macau, Macau
*
*Corresponding author. Email address:wwxl9840904@sina.com (W. Xu)
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Abstract

Conjugate symplectic eigenvalue problems arise in solving discrete linear-quadratic optimal control problems and discrete algebraic Riccati equations. In this article, backward errors of approximate pairs of conjugate symplectic matrices are obtained from their properties. Several numerical examples are given to illustrate the results.

Keywords

Backward errorapproximate eigenpairsconjugate symplectic matrix

MSC classification

Secondary: 65F10: Iterative methods for linear systems 65F15: Eigenvalues, eigenvectors 65H10: Systems of equations 65L15: Eigenvalue problems
Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 5 , Issue 4 , November 2015 , pp. 312 - 326
Copyright
Copyright © Global-Science Press 2015 

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References

[1]Tisseur, F., A chart of backward errors for singly and doubly structured eigenvalue problems, SIAM J. Matrix Anal. Appl. 24, 877–897 (2003).Google Scholar
[2]Bunse-Gerstner, A., Byers, R. and Mehrmann, V., A chart of numerical methods for structured eigenvalue problems, SIAM J. Matrix Anal. Appl. 13, 419–453 (1992).Google Scholar
[3]Sun, J.G., Backward Errors for the Unitary Eigenproblem, Tech. Report UMINF-97.25, Department of Computing Science, University of Umeå, Sweden (1997).Google Scholar
[4]Benner, P. and Fassbender, H., The symplectic eigenvalue problem, the butterfly form, the SR algorithm, and the Lanczos method, Linear Algebra Appl. 275/276, 19–47 (1998).Google Scholar
[5]Tisseur, F., Stability of structured Hamiltonian eigensolvers, SIAM J. Matrix Anal. Appl. 23, 103–125 (2001).Google Scholar
[6]Mehrmann, V. and Watkins, D., Structure-preserving methods for computing eigenpairs of large sparse skew-Hamiltonian/Hamiltonian pencils, SIAM J. Sci. Comp. 22, 1905–1925 (2001).CrossRefGoogle Scholar
[7]Benner, P. and Fassbender, H., An implicitly restarted symplectic Lanczos method for the Hamiltonian eigenvalue problem, Linear Algebra Appl. 263, 75–111 (1997).CrossRefGoogle Scholar
[8]Higham, D. J. and Higham, N. J., Structured backward error and condition of generalized eigenvalue problems, SIAM J. Matrix Anal. Appl. 20, 493–512 (1998).CrossRefGoogle Scholar
[9]Higham, D. J. and Higham, N. J., Backward error and condition of structured linear systems, SIAM J. Matrix Anal. Appl. 13, 162–175 (1992).CrossRefGoogle Scholar
[10]Sun, J. G., Backward perturbation analysis of certain characteristic subspaces, Numer. Math. 65, 357–382 (1993).CrossRefGoogle Scholar
[11]Xu, Y. H. and Jiang, E. X., An inverse eigenvalue problem for periodic Jacobi matrices, Inverse Problems. 23, 165–181 (2007).Google Scholar
[12]Xie, D. X. and Sheng, Y. P., Inverse eigenproblem of anti-symmetric and persymmetric matrices and its approximation, Inverse Problems. 19, 217–225 (2003).Google Scholar
[13]Gulliksson, M., Jin, X. Q., and Wei, Y. M., Perturbation bounds for constrained and weighted least squares problems, Linear Algebra Appl. 349, 221–232 (2002).Google Scholar
[14]Chu, M. T. and Golub, G. H., Structured inverse eigenvalue problems, Acta. Numer. 11, 1–71 (2002).Google Scholar
[15]Kahan, W., Parlett, B. N., and Jiang, E. X., Residual bounds on approximate eigensystems of non-normal matrices, SIAM J. Numer. Anal. 19, 470–484 (1982).CrossRefGoogle Scholar
[16]Wei, M. S., Theory and Computations for Generalized Least Squares Problems, Beijing: Science Press, in Chinese (2006).Google Scholar
[17]Tisseur, F. and Graillat, S., Structured condition numbers and backward errors in scalar product spaces, Electron. J. Linear Algebra., 15, 159–177 (2006).Google Scholar
[18]Byrnes, C. I. and Lindquist, A., Algebraic aspects of generalized eigenvalue problems for solving Riccati equations, in Computational and Combinatorial Methods in Systems Theory, pp. 213–227 (1986).Google Scholar
[19]Mehrmann, V., A symplectic orthogonal method for single input or single output discrete time optimal linear quadratic control problems, SIAM J. Matrix Anal. Appl. 9, 221–248 (1988).Google Scholar
[20]Pappas, T., Laub, A. J., and Sandell, N. R., On the numerical solution of the discrete-time algebraic Riccati equation, IEEE Trans. Auomat. Control. 25, 631–641 (1980).Google Scholar
[21]Payne, H. J. and Silverma, L. M., On the discrete time algebraic Riccati equation, IEEE Trans. Auomat. Control. 18, 226–234 (1973).CrossRefGoogle Scholar
[22]Xu, W., Li, W., Ching, W. and Chen, Y., Backward errors for two kinds of eigenvalue problems, J. Comput. Appl. Math. 235, 59–73 (2010).Google Scholar