Abstract
In this paper, we propose a palindromic quadratization approach, transforming a palindromic matrix polynomial of even degree to a palindromic quadratic pencil. Based on the \({(\mathcal{S}+ \mathcal{S}^{-1})}\) -transform and Patel’s algorithm, the structure-preserving algorithm can then be applied to solve the corresponding palindromic quadratic eigenvalue problem. Numerical experiments show that the relative residuals for eigenpairs of palindromic polynomial eigenvalue problems computed by palindromic quadratized eigenvalue problems are better than those via palindromic linearized eigenvalue problems or \({{\texttt {polyeig}}}\) in MATLAB.
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References
Betcke T.: Optimal scaling of generalized and polynomial eigenvalue problems. SIAM J. Matrix Anal. Appl. 30, 1320–1338 (2008)
Byers, R., Mackey, D.S., Mehrmann, V., Xu, H.: Symplectic, BVD, and palindromic approaches to discrete-time control problems. Technical report, Preprint 14-2008, Institute of Mathematics, Technische Universität Berlin (2008)
Chen T.-Y., Demmel J.W.: Balancing sparse matrices for computing eigenvalues. Linear Algebra Appl. 309, 261–287 (2000)
Chiang, C.-Y., Chu, E.K.-W., Li, T., Lin, W.-W.: The palindromic generalized eigenvalue problem A * x = λ A x: numerical solution and applications. Linear Algebra Appl. (2010, to appear)
Chu, E.K.-W., Huang, T.-M., Lin, W.-W.: Structured doubling algorithms for solving g-palindromic quadratic eigenvalue problems. Technical report, NCTS Preprints in Mathematics 2008-4-003, National Tsing Hua University, Hsinchu, Taiwan (2008)
Chu E.K.-W., Hwang T.-M., Lin W.-W., Wu C.-T.: Vibration of fast trains, palindromic eigenvalue problems and structure-preserving doubling algorithms. J. Comput. Appl. Math. 219, 237–252 (2008)
Fan H.-Y., Lin W.-W., Van Dooren P.: Normwise scaling of second order polynomial matrices. SIAM J. Matrix Anal. Appl. 26, 252–256 (2004)
Grammont, L., Higham, N.J., Tisseur, F.: A framework for analyzing nonlinear eigenproblems and parametrized linear systems. Technical report, The MIMS Secretary, School of Mathematics, the University of Manchester. MIMS EPrint: 2009.51 (2009)
Higham N.J., Tisseur F., Van Dooren P.M.: Detecting a definite hermitian pair and a hyperbolic or elliptic quadratic eigenvalue problem, and associated nearness problems. Linear Algebra Appl. 351–352, 455–474 (2002)
Highman N.J., Li R.-C., Tisseur F.: Backward error of polynomial eigenproblems solved by linearization. SIAM J. Matrix Anal. Appl. 29, 1218–1241 (2007)
Hilliges, A.: Numerische Lösung von quadratischen eigenwertproblemen mit Anwendungen in der Schiendynamik. Master’s thesis, Technical University Berlin, Germany, July 2004
Hilliges, A., Mehl, C., Mehrmann, V.: On the solution of palindramic eigenvalue problems. In: Proceedings 4th European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS). Jyväskylä, Finland (2004)
Huang T.-M., Lin W.-W., Qian J.: Structure-preserving algorithms for palindromic quadratic eigenvalue problems arising from vibration on fast trains. SIAM J. Matrix Anal. Appl. 30, 1566–1592 (2009)
Huang, T.-M., Lin, W.-W., Su, W.-S.: Palindromic quadratization and structure-preserving algorithm for palindromic matrix polynomials of even degree. Technical report, NCTS Preprints in Mathematics, National Tsing Hua University, Hsinchu, Taiwan, 2009-6-002 (2009)
Ipsen, C.F.: Accurate eigenvalues for fast trains. SIAM News, 37 (2004)
Lemonnier D., Van Dooren P.: Balancing regular matrix pencils. SIAM J. Matrix Anal. Appl. 28, 253–263 (2006)
Li, R.-L., Lin, W.-W., Wang, C.-S.: Structured backward error for palindromic polynomial eigenvalue problems. Technical report, NCTS Preprints in Mathematics, National Tsing Hua University, Hsinchu, Taiwan, 2008-7-002 (2008)
Lin W.-W.: A new method for computing the closed-loop eigenvalues of a discrete-time algebraic Riccatic equation. Linear Algebra Appl. 96, 157–180 (1987)
Mackey D.S., Mackey N., Mehl C., Mehrmann V.: Numerical methods for palindromic eigenvalue problems: computing the anti-triangular Schur form. Numer. Linear Algebra Appl. 16, 63–86 (2009)
Mackey D.S., Mackey N., Mehl C., Mehrmann V.: Structured polynomial eigenvalue problems: good vibrations from good linearizations. SIAM J. Matrix Anal. Appl. 28, 1029–1051 (2006)
Mackey D.S., Mackey N., Mehl C., Mehrmann V.: Vector spaces of linearizations for matrix polynomials. SIAM J. Matrix Anal. Appl. 28, 971–1004 (2006)
Patel R.V.: On computing the eigenvalues of a symplectic pencil. Linear Algebra Appl. 188, 591–611 (1993)
Schröder, C.: SKURV: a Matlab toolbox for the skew URV decomposition of a matrix triple. http://www.math.tu-berlin.de/~schroed/Software/skurv
Schröder, C.: A QR-like algorithm for the palindromic eigenvalue problem. Technical report, Preprint 388, TU Berlin, Matheon, Germany (2007)
Schröder, C.: URV decomposition based structured methods for palindromic and even eigenvalue problems. Technical report, Preprint 375, TU Berlin, MATHEON, Germany (2007)
Tisseur F., Meerbergen K.: A survey of the quadratic eigenvalue problem. SIAM Rev. 43, 234–286 (2001)
Xu H.: On equivalence of pencils from discrete-time and continuous-time control. Linear Algebra Appl. 414, 97–124 (2006)
Zaglmayr, S.: Eigenvalue problems in saw-filter simulations. Diplomarbeit, Institute of Computational Mathematics, Johannes Kepler University Linz, Linz, Austria (2002)
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Huang, TM., Lin, WW. & Su, WS. Palindromic quadratization and structure-preserving algorithm for palindromic matrix polynomials of even degree. Numer. Math. 118, 713–735 (2011). https://doi.org/10.1007/s00211-011-0370-7
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DOI: https://doi.org/10.1007/s00211-011-0370-7