Solving rational matrix equations

Massimiliano Fasi (The University of Manchester)

ATB Frank Adams 1,

We consider rational matrix equations of the form \(p(X)\;q(X)^{-1} = A\), where \(A\) is a complex square matrix and \(p\) and \(q\) are polynomials. It is easy to see that any solution \(X\) also satisfies \(p(X) = A\;q(X)\). Surprisingly enough, the other implication is also true, and we can conclude that the matrix inverse in the original equation is nonessential. We develop a novel Schur method for the computation of primary solutions to the latter equation, our approach generalises the algorithm of Smith [SIAM. J. Matrix Anal. & Appl., 24 (2003), pp. 971–989] for the computation of primary \(p\)th roots, and has a similar computational cost.
We show how this idea can be exploited to solve general matrix equations of the form \(f(X) = A\) by means of Padé approximation, and discuss techniques to improve the accuracy of the solution for matrices having widespread spectrum.
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