Session A5 - Geometric Integration and Computational Mechanics
July 12, 18:00 ~ 18:30 - Room 111
An improved algorithm to compute the exponential of a matrix
Universidad Politécnica de Valencia, Spain - firstname.lastname@example.org
Exponential integrators have shown to be highly efficient geometric integrators, in particular as Lie group methods. However, for matrices of moderate size, their performance depends on an efficient computation of the matrix exponential. This can be achieved by the scaling-squaring technique with an appropriate approximation to the Taylor expansion of the exponential. Padé approximants allows to approximate polynomial matrix functions with relatively few products of matrices (the cost to compute the inverse of a matrix is 4/3 the cost of one product). Recently, the Paterson-Stockmeyer scheme has shown to be superior to Padé in many cases, since only requires $2(k-1)$ products to compute a polynomial of degree $k^2$.
Taking into account that with $k$ products one can build matrix polynomials of degree $2^k$, we consider the inverse problem. Given a matrix polynomial, we analyse how to factorize it with the minimum number of products. For example, a polynomial of degree 8 can be obtained with only 3 products. We consider the decomposition of higher order polynomials and its application to approximate the exponential of a matrix and its performance with respect to the algorithm "expm" used in Matlab.
Joint work with Fernando Casas and Philipp Bader.