عنوان

Algorithms for the matrix exponential and its Fréchet derivative

TLets518467

Algorithms for the matrix exponential and its Fréchet derivative

[Thesis]

Al-Mohy, Awad

Higham, Nicholas ; Thatcher, Ronald

University of Manchester

2011

Ph.D.

University of Manchester

2011

New algorithms for the matrix exponential and its Fréchet derivative are presented. First, we derive a new scaling and squaring algorithm (denoted expm[new]) for computing eA, where A is any square matrix, that mitigates the overscaling problem. The algorithm is built on the algorithm of Higham [SIAM J.Matrix Anal. Appl., 26(4): 1179-1193, 2005] but improves on it by two key features. The first, specific to triangular matrices, is to compute the diagonal elements in the squaring phase as exponentials instead of powering them. The second is to base the backward error analysis that underlies the algorithm on members of the sequence {||Ak||1/k} instead of ||A||. The terms ||Ak||1/k are estimated without computing powers of A by using a matrix 1-norm estimator. Second, a new algorithm is developed for computing the action of the matrix exponential on a matrix, etAB, where A is an n x n matrix and B is n x n₀ with n₀ << n. The algorithm works for any A, its computational cost is dominated by the formation of products of A with n x n₀ matrices, and the only input parameter is a backward error tolerance. The algorithm can return a single matrix etAB or a sequence etkAB on an equally spaced grid of points tk. It uses the scaling part of the scaling and squaring method together with a truncated Taylor series approximation to the exponential. It determines the amount of scaling and the Taylor degree using the strategy of expm[new].Preprocessing steps are used to reduce the cost of the algorithm. An important application of the algorithm is to exponential integrators for ordinary differential equations. It is shown that the sums of the form \sum_{k=0}^p\varphi_k(A)u_k that arise in exponential integrators, where the \varphi_k are related to the exponential function, can be expressed in terms of a single exponential of a matrix of dimension

matrix exponential

matrix function

Pad\'e approximation

scaling and squaring method

Al-Mohy, Awad

Higham, Nicholas

Thatcher, Ronald

University of Manchester

p

[Thesis]

276903

a

Y