Research Seminar in Mathematics - Substitution algorithms for rational matrix equations

Speaker

Massimiliano Fasi, University of Manchester, UK.

Abstract

Functions of matrices defined as solutions to matrix equations play an important role in many applications. As rational approximation is a customary tool in algorithms for evaluating this class of functions, one may wonder whether it be possible to solve numerically equations of the form $r(X)=A$, where $A$ and $X$ are square matrices and $r$ is a rational function.

As it turns out, accurate and efficient techniques for this problem can be derived by inverting, through a substitution strategy, computational schemes for evaluating rational matrix functions. The resulting methods all exploit the Schur decomposition of the input matrix to reduce the problem to upper triangular form, and for triangular matrices they yield the same computational cost as the evaluation schemes from which they are obtained. This suggests that solving rational matrix equations is not more difficult than evaluating rational functions at a matrix argument.

These methods can be used in a natural way as building blocks in algorithms for computing functions of matrices defined via matrix equation of the type $f(X) = A$, where $f$ is a primary matrix function.