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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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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