Computing matrix functions arising in engineering models with orthogonal matrix polynomials

Trigonometric matrix functions play a fundamental role in the solution of second order differential equations. Hermite series truncation together with Paterson–Stockmeyer method and the double angle formula technique allow efficient computation of the matrix cosine. A careful error bound analysis of the Hermite approximation is given and a theoretical estimate for the optimal value of its parameters is obtained. Based on the ideas above, an efficient and highly-accurate Hermite algorithm is presented. A MATLAB implementation of this algorithm has also been developed and made available online. This implementation has been compared to other efficient state-of-the-art implementations on a large class of matrices for different dimensions, obtaining higher accuracy and lower computational costs in the majority of cases.