Polynomial approximations for the matrix logarithm with computation graphs, E. Jarlebring, J. Sastre, J. Ibáñez, Linear Algebra Applications, in Press (open access), 2024. https://doi.org/10.1016/j.laa.2024.10.024, https://arxiv.org/abs/2401.10089, code.
In this article the matrix logarithm is computed by using matrix polynomial approximations evaluated by using matrix polynomial multiplications and additions. The most popular method for computing the matrix logarithm is a combination of the inverse scaling and squaring method in conjunction with a Padé approximation, sometimes accompanied by the Schur decomposition. The main computational effort lies in matrix-matrix multiplications and left matrix division. In this work we illustrate that the number of such operations can be substantially reduced, by using a graph based representation of an efficient polynomial evaluation scheme. A technique to analyze the rounding error is proposed, and backward error analysis is adapted. We provide substantial simulations illustrating competitiveness both in terms of computation time and rounding errors.