Abstract. In principle, the exponential of a matrix could be computed in many ways. Methods involving approximation theory, differential equations, the matrix eigenvalues, and the matrix characteristic polynomial have been proposed. In practice, consideration of computational stability and efficiency indicates that some of the methods are preferable to others but that none are completely satisfactory. Most of this paper was originally published in 1978. An update, with a separate bibliography, describes a few recent developments.

Many differential equations of practical interest evolve on Lie groups or on manifolds acted upon by Lie groups. The retention of Lie-group structure under discretization is often vital in the recovery of qualitatively correct geometry and dynamics and in the minimization of numerical error. Having introduced requisite elements of differential geometry, this paper surveys the novel theory of numerical integrators that respect Lie-group structure, highlighting theory, algorithmic issues and a number of applications.

. We study the numerical integration of large stiff systems of differential equations by methods that use matrix-vector products with the exponential or a related function of the Jacobian. For large problems, these can be approximated by Krylov subspace methods, which typically converge faster than those for the solution of the linear systems arising in standard stiff integrators. The exponential methods also offer favorable properties in the integration of differential equations whose Jacobian has large imaginary eigenvalues. We derive methods up to order 4 which are exact for linear constant-coefficient equations. The implementation of the methods is discussed. Numerical experiments with reaction-diffusion problems and a time-dependent Schrodinger equation are included. Key words. Numerical integrator, high-dimensional differential equations, matrix exponential, Krylov subspace methods. AMS(MOS) subject classifications. 65L05, 65M15, 65F10. 1. Introduction. The idea to use the exp...

Expokit provides a set of routines aimed at computing matrix exponentials. More precisely, it computes either a small matrix exponential in full, the action of a large sparse matrix exponential on an operand vector, or the solution of a system of linear ODEs with constant inhomogeneity. The backbone of the sparse routines consists of Krylov subspace projection methods (Arnoldi and Lanczos processes) and that is why the toolkit is capable of coping with sparse matrices of large dimension. The software handles real and complex matrices and provides specific routines for symmetric and Hermitian matrices. The computation of matrix exponentials is a numerical issue of critical importance in the area of Markov chains and furthermore, the computed solution is subject to probabilistic constraints. In addition to addressing general matrix exponentials, a distinct attention is assigned to the computation of transient states of Markov chains.

We show that there exist classes of explicit numerical integration methods that can handle very stiff problems if the eigenvalues are separated into two clusters, one containing the "stiff", or fast components, and one containing the slow components. These methods have large average step sizes relative to the fast components. Conventional implicit methods involve the solution of non-linear equations at each step, which for large problems requires significant communication between processors on a multiprocessor machine. For such problems the methods proposed here have significant potential for speed improvement.

Abstract. Consider a differential equation y ′ = A(t, y)y, y(0) = y0 with y0 ∈ GandA: R + × G → g, wheregis a Lie algebra of the matricial Lie group G. Every B ∈ g canbemappedtoGbythematrixexponentialmap exp (tB) witht∈R. Most numerical methods for solving ordinary differential equations (ODEs) on Lie groups are based on the idea of representing the approximation yn of the exact solution y(tn), tn ∈ R +, by means of exact exponentials of suitable elements of the Lie algebra, applied to the initial value y0. This ensures that yn ∈ G. When the exponential is difficult to compute exactly, as is the case when the dimension is large, an approximation of exp (tB) plays an important role in the numerical solution of ODEs on Lie groups. In some cases rational or polynomial approximants are unsuitable and we consider alternative techniques, whereby exp (tB) is approximated by a product of simpler exponentials. In this paper we present some ideas based on the use of the Strang splitting for the approximation of matrix exponentials. Several cases of g and G are considered, in tandem with general theory. Order conditions are discussed, and a number of numerical experiments conclude the paper. 1.

The Arnoldi iteration, usually viewed as a method for calculating eigenvalues, can also be used to estimate pseudospectra. This possibility may be of practical importance, for in applications involving highly non-normal matrices or operators, such as hydrodynamic stability, pseudospectra may be physically more significant than spectra.

Many advances in the development of Krylov subspace methods for the iterative solution of linear systems during the last decade and a half are reviewed. These new developments include different versions of restarted, augmented, deflated, flexible, nested, and inexact methods. Also reviewed are methods specifically tailored to systems with special properties such as special forms of symmetry and those depending on one or more parameters.

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