Accurate matrix exponential computation to solve coupled differential models in engineering

Abstract

a b s t r a c tThe matrix exponential plays a fundamental role in linear systems arising in engineering, mechanics and control theory. This work presents a new scaling-squaring algorithm for matrix exponential computation. It uses forward and backward error analysis with improved bounds for normal and nonnormal matrices. Applied to the Taylor method, it has presented a lower or similar cost compared to the state-of-the-art Padé algorithms with better accuracy results in the majority of test matrices, avoiding Padé's denominator condition problems.

Accurate and efficient matrix exponential computation

“…However, in [3] the authors presented a scaling and squaring Taylor algorithm based on an improved mixed backward and forward error analysis, which was more accurate than existing stateof-the-art Padé algorithms [5,6] in the majority of test matrices with a lower or similar cost. Moreover, modifications to Padé algorithm had to be carried out in [6, p. 983] to improve the denominator conditioning, whereas Taylor algorithms have no denominator.…”

“…Later, [6] presents an algorithm that reduces the overscaling problem choosing parameter s, based on the norms of low powers of matrix A instead of in ||A||, and computes the diagonal elements in the squaring phase as exponentials instead of from powers of the diagonal Padé approximation for the case of triangular matrices. In [3], estimations of norms of higher powers of matrix A (greater than or equal to m+1) are used to obtain the scaling parameter s, and similar ideas to those in [6] can be used in the case of Taylor approximations to compute the diagonal elements for triangular matrices.…”

“…Analogously to Sastre et al [3], in the proposed scaling algorithm it will be necessary that the same powers of A are used for the two last orders m M −1 and m M , i.e. A i , i = 2, 3, .…”

“…This work improves the scaling and squaring algorithm presented in [3] providing a competitive scaling and squaring algorithm for matrix exponential computation. The new algorithm employs an improved version of Theorem 1 from [3] to bound the norm of matrix power series. A new formula for the forward relative error of Taylor approximation in exact arithmetic and new sharp bounds for forward and backward relative errors are given.…”

“…Section 3 deals with numerical tests and gives some conclusions. The following theorem will be used to bound the norm of matrix power series, see [3,Th. 1].…”