Schur Decomposition Methods for the Computation of Rational Matrix Functions

  • T. Politi
  • M. Popolizio
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3994)


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.


Rational Approximation Matrix Function Partial Fraction Krylov Subspace Chebyshev Approximation 
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  1. 1.
    Åstrom, K.J., Wittenmark, B.: Computer-Controlled Systems: Theory and Design. Prentice-Hall, Englewoods Ciffs (1997)Google Scholar
  2. 2.
    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)Google Scholar
  3. 3.
    Calvetti, D., Gallopoulos, E., Reichel, L.: Incomplete Partial Fractions for Parallel Evaluation of Rational Matrix Functions. J. Comp. Appl. Math. 59, 349–380 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Davies, P.J., Higham, N.J.: A Schur-Parlett Algorithm for Computing Matrix Functions. SIAM J. Matr. Anal. Appl. 25(2), 464–485 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Del Buono, N., Lopez, L., Politi, T.: Computation of functions of Hamiltonian and skew-symmetric matrices. Preprint (2006)Google Scholar
  6. 6.
    Golub, G.H., Van Loan, C.F.: Matrix Computation. The John Hopkins Univ. Press, Baltimore (1996)Google Scholar
  7. 7.
    Higham, N.J.: Functions of Matrices. MIMS EPrint 2005.21 The University of ManchesterGoogle Scholar
  8. 8.
    Iserles, A., Munthe-Kaas, H., Nørsett, S., Zanna, A.: Lie-Group Methods. Acta Numerica 9, 215–365 (2000)CrossRefGoogle Scholar
  9. 9.
    Lopez, L., Simoncini, V.: Analysis of projection methods for rational function approximation to the matrix exponential. SIAM J. Numer. Anal. (to appear)Google Scholar
  10. 10.
    Parlett, B.N.: A Recurrence among the Elements of Functions of Triangular Matrices. Lin. Alg. Appl. 14, 117–121 (1976)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Saad, Y.: Analysis of some Krylov subspace approximation to the matrix exponential operator. SIAM J. Numer. Anal. 29(1), 209–228 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Schmelzer, T.: Rational approximations in scientific computing. Computing Laboratory. Oxford University, U.K. (2005)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • T. Politi
    • 1
  • M. Popolizio
    • 2
  1. 1.Dipartimento di MatematicaPolitecnico di BariBariItaly
  2. 2.Dipartimento di MatematicaUniversità degli Studi di BariBariItaly

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