Abstract
In this work we consider the problem to compute the vector \(y={\it \Phi}_{m,n}(A)x\) where \({\it \Phi}_{m,{\it n}}\)(z) is a rational function, x is a vector and A is a matrix of order N, usually nonsymmetric. The problem arises when we need to compute the matrix function f(A), being f(z) a complex analytic function and \({\it \Phi}_{m,{\it n}}\)(z) a rational approximation of f. Hence \({\it \Phi}_{m,{\it n}}\)(A) is a approximation for f(A) cheaper to compute. We consider the problem to compute first the Schur decomposition of A then the matrix rational function exploting the partial fractions expansion. In this case it is necessary to solve a sequence of linear systems with the shifted coefficient matrix (A − z j I)y = b.
Chapter PDF
Similar content being viewed by others
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
References
Åstrom, K.J., Wittenmark, B.: Computer-Controlled Systems: Theory and Design. Prentice-Hall, Englewoods Ciffs (1997)
Baldwin, C., Freund, R.W., Gallopoulos, E.: A Parallel Iterative Method for Exponential Propagation. In: Bailey, D.H., et al. (eds.) Proceedings of the Seventh SIAM Conference on Parallel Processing for Scientific Computing, pp. 534–539. SIAM, Philadelphia (1995)
Calvetti, D., Gallopoulos, E., Reichel, L.: Incomplete Partial Fractions for Parallel Evaluation of Rational Matrix Functions. J. Comp. Appl. Math. 59, 349–380 (1995)
Davies, P.J., Higham, N.J.: A Schur-Parlett Algorithm for Computing Matrix Functions. SIAM J. Matr. Anal. Appl. 25(2), 464–485 (2003)
Del Buono, N., Lopez, L., Politi, T.: Computation of functions of Hamiltonian and skew-symmetric matrices. Preprint (2006)
Golub, G.H., Van Loan, C.F.: Matrix Computation. The John Hopkins Univ. Press, Baltimore (1996)
Higham, N.J.: Functions of Matrices. MIMS EPrint 2005.21 The University of Manchester
Iserles, A., Munthe-Kaas, H., Nørsett, S., Zanna, A.: Lie-Group Methods. Acta Numerica 9, 215–365 (2000)
Lopez, L., Simoncini, V.: Analysis of projection methods for rational function approximation to the matrix exponential. SIAM J. Numer. Anal. (to appear)
Parlett, B.N.: A Recurrence among the Elements of Functions of Triangular Matrices. Lin. Alg. Appl. 14, 117–121 (1976)
Saad, Y.: Analysis of some Krylov subspace approximation to the matrix exponential operator. SIAM J. Numer. Anal. 29(1), 209–228 (1992)
Schmelzer, T.: Rational approximations in scientific computing. Computing Laboratory. Oxford University, U.K. (2005)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2006 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Politi, T., Popolizio, M. (2006). Schur Decomposition Methods for the Computation of Rational Matrix Functions. In: Alexandrov, V.N., van Albada, G.D., Sloot, P.M.A., Dongarra, J. (eds) Computational Science – ICCS 2006. ICCS 2006. Lecture Notes in Computer Science, vol 3994. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11758549_96
Download citation
DOI: https://doi.org/10.1007/11758549_96
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-34385-1
Online ISBN: 978-3-540-34386-8
eBook Packages: Computer ScienceComputer Science (R0)