Fraction-Free Computation of Matrix Rational Interpolants and Matrix GCDs

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.