An extended collection of matrix derivative results for forward and reverse mode automatic differentiation

This paper collects together a number of matrix derivative results which are very useful in forward and reverse mode algorithmic differentiation (AD). It highlights in particular the remarkable contribution of a 1948 paper by Dwyer and Macphail which derives the linear and adjoint sensitivities of a matrix product, inverse and determinant, and a number of related results motivated by applications in multivariate analysis in statistics.\ud \ud This is an extended version of a paper which will appear in the proceedings of AD2008, the 5th International Conference on Automatic Differentiation

A note on performance profiles for benchmarking software

In recent years, performance profiles have become a popular and widely used tool for benchmarking and evaluating the performance of several solvers when run on a large test set. Here we use data from a real application as well as a simple artificial example to illustrate that caution should be exercised when trying to interpret performance profiles to assess the relative performance of the solvers

Accurate Approximation of the Matrix Hyperbolic Cosine Using Bernoulli Polynomials

[EN] This paper presents three different alternatives to evaluate the matrix hyperbolic cosine using Bernoulli matrix polynomials, comparing them from the point of view of accuracy and computational complexity. The first two alternatives are derived from two different Bernoulli series expansions of the matrix hyperbolic cosine, while the third one is based on the approximation of the matrix exponential by means of Bernoulli matrix polynomials. We carry out an analysis of the absolute and relative forward errors incurred in the approximations, deriving corresponding suitable values for the matrix polynomial degree and the scaling factor to be used. Finally, we use a comprehensive matrix testbed to perform a thorough comparison of the alternative approximations, also taking into account other current state-of-the-art approaches. The most accurate and efficient options are identified as results.This research was supported by the Vicerrectorado de Investigacion de la Universitat Politecnica de Valencia (PAID-11-21).Alonso Abalos, JM.; Ibáñez González, JJ.; Defez Candel, E.; Alvarruiz Bermejo, F. (2023). Accurate Approximation of the Matrix Hyperbolic Cosine Using Bernoulli Polynomials. Mathematics. 11(3):1-22. https://doi.org/10.3390/math1103052012211

Exploiting Kronecker structure in exponential integrators: Fast approximation of the action of phi-functions of matrices via quadrature

In this article, we propose an algorithm for approximating the action of  $\varphi$-functions of matrices against vectors, which is a key operation in exponential time integrators. In particular, we consider matrices with Kronecker sum structure, which arise from problems admitting a tensor product representation. The method is based on quadrature approximations of the integral form of the  $\varphi$-functions combined with a scaling and modified squaring method. Owing to the Kronecker sum representation, only actions of 1D matrix exponentials are needed at each quadrature node and assembly of the full matrix can be avoided. Additionally, we derive a priori bounds for the quadrature error, which show that, as expected by classical theory, the rate of convergence of our method is supergeometric. Guided by our analysis, we construct a fast and robust method for estimating the optimal scaling factor and number of quadrature nodes that minimizes the total cost for a prescribed error tolerance. We investigate the performance of our algorithm by solving several linear and semilinear time-dependent problems in 2D and 3D. The results show that our method is accurate and orders of magnitude faster than the current state-of-the-art

Exploiting Kronecker structure in exponential integrators: fast approximation of the action of φ\varphi-functions of matrices via quadrature

In this article, we propose an algorithm for approximating the action of $\varphi-$functions of matrices against vectors, which is a key operation in exponential time integrators. In particular, we consider matrices with Kronecker sum structure, which arise from problems admitting a tensor product representation. The method is based on quadrature approximations of the integral form of the $\varphi-$functions combined with a scaling and modified squaring method. Owing to the Kronecker sum representation, only actions of 1D matrix exponentials are needed at each quadrature node and assembly of the full matrix can be avoided. Additionally, we derive \emph{a priori} bounds for the quadrature error, which show that, as expected by classical theory, the rate of convergence of our method is supergeometric. Guided by our analysis, we construct a fast and robust method for estimating the optimal scaling factor and number of quadrature nodes that minimizes the total cost for a prescribed error tolerance. We investigate the performance of our algorithm by solving several linear and semilinear time-dependent problems in 2D and 3D. The results show that our method is accurate and orders of magnitude faster than the current state-of-the-art.Comment: 20 pages, 3 figures, 7 table