Substitution algorithms for rational matrix equations
Massimiliano Fasi, Bruno Iannazzo · ETNA - Electronic Transactions on Numerical Analysis · 2020
We study equations of the form r(X)=A, where r is a rational function and A and X are square matrices of the same size. We develop two techniques for solving these equations by inverting (through a substitution strategy) two schemes for the evaluation of rational functions of matrices. For triangular matrices, the new methods yield the same computational cost as the evaluation schemes from which they are obtained. A general equation can be reduced to upper triangular form by exploiting the Schur decomposition of the given matrix. For real data, the algorithms rely on the real Schur decomposition in order to compute real solutions using only real arithmetic. Numerical experiments show that our implementations are faster than existing alternatives without sacrificing accuracy.