Fraction-Free Computation of Matrix Rational Interpolants and Matrix GCDs

  • Authors:

  • Bernhard Beckermann;George Labahn

  • Affiliations:

  • -;-

  • Venue:
  • SIAM Journal on Matrix Analysis and Applications
  • Year:
  • 2000

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Abstract

We present a new set of algorithms for computation of matrix rational interpolants and one-sided matrix greatest common divisors. Examples of these interpolants include Padé approximants, Newton--Padé, Hermite--Padé, and simultaneous Padé approximants, and more generally M-Padé approximants along with their matrix generalizations. The algorithms are fast and compute all solutions to a given problem. Solutions for all (possibly singular) subproblems along offdiagonal paths in a solution table are also computed by stepping around singular blocks on a path corresponding to "closest" regular interpolation problems.The algorithms are suitable for computation in exact arithmetic domains where growth of coefficients in intermediate computations is a central concern. This coefficient growth is avoided by using fraction-free methods. At the same time, the methods are fast in the sense that they are at least an order of magnitude faster than existing fraction-free methods for the corresponding problems. The methods make use of linear systems having a special striped Krylov structure.