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.

. 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...

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.

The theory of the convergence of Krylov subspace iterations for linear systems of equations (conjugate gradients, biconjugate gradients, GMRES, QMR, Bi-CGSTAB, ...) is reviewed. For a computation of this kind, an estimated asymptotic convergence factor ae 1 can be derived by solving a problem of potential theory or conformal mapping. Six approximations are involved in relating the actual computation to this scalar estimate. These six approximations are discussed in a systematic way and illustrated by a sequence of examples computed with tools of numerical conformal mapping and semidefinite programming.

Eigenvalues and eigenfunctions of linear operators are important to many areas of ap-plied mathematics. The ability to approximate these quantities numerically is becoming increasingly important in a wide variety of applications. This increasing demand has fueled interest in the development of new methods and software for the numerical solution of large-scale algebraic eigenvalue problems. In turn, the existence of these new methods and software, along with the dramatically increased computational capabilities now available, has enabled the solution of problems that would not even have been posed five or ten years ago. Until very recently, software for large-scale nonsymmetric problems was virtually non-existent. Fortunately, the situation is improving rapidly. The purpose of this article is to provide an overview of the numerical solution of large-scale algebraic eigenvalue problems. The focus will be on a class of methods called Krylov subspace projection methods. The well-known Lanczos method is the premier member of this class. The Arnoldi method generalizes the Lanczos method to the nonsymmetric case. A recently developed variant of the Arnoldi/Lanczos scheme called the Implicitly Restarted Arnoldi Method is presented here in some depth. This method is highlighted because of its suitability as a basis for software development.

We describe accurate computations of three-dimensional periodic and relative periodic motions within plane Couette turbulence at Re = 400. To ensure that the computed solutions are true solutions of the Navier-Stokes equations, careful attention is paid to time discretization errors and to spatial resolution. All the computed solutions are linearly unstable. While direct numerical simulation helps us understand the statistics of turbulent fluid flows, elucidation of the geometry of turbulent flows in phase space requires the computation of steady states, traveling waves, periodic motions, and close recurrences. The computed solutions are used as a basis to discuss the manner in which the geometry of turbulent dynamics in phase space can be understood. The method used for computing these solutions is described in detail.

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this paper is to examine both the capabilities and limitations of such an approach, with particular focus on solving large problems on massively parallel computers using iterative methods. We accomplish our goal by solving a large variety of two and three dimensional problems of varying di#culty, comparing our results (whenever possible) to semi-analytical results. We also carefully explain how we successfully combined Cayley transformations with an Arnoldi based eigensolver and preconditioned Krylov methods for the necessary linear solves. For problems where the advective terms are not significant, we achieve excellent convergence of the computed eigenvalues as we refine the finite element mesh. We also successfully solve advectively dominated problems, but the convergence is slower. We believe that the main di#culties arise not from problems with the eigensolver, but from the accuracy of the finite element discretization. Therefore, we believe that our results are as reliable as using transient integration but are more e#ciently computed. The largest eigenvalue problem we solve has over 16 million unknowns on 2048 processors. Key Words: stability, Navier Stokes, eigenvalues, buoyancy driven flow, Arnoldi, bifurcation, finite element, massively parallel 1. INTRODUCTION Much of our understanding of fluid flow phenomena comes from linearized stability analyses of simple flows, such as the state of rest, Couette flow, or Poiseuille flow [3, 22, 10, 18]. Modern computational fluid dynamicists routinely analyze the stability of more complicated flows using a variety of methods (e.g. spectral methods, boundary integral methods). However, the majority of these calculations are 1 2 BURROUGHS, ROMERO, LEHOUCQ AND SALINGER done in such a way that the resulting linear systems...

In this paper we look at a number of approaches being investigated in the Center for Research on Parallel Computation (CRPC) to develop linear algebra software for high-performance computers. These approaches are exemplified by the LAPACK, templates, and ARPACK projects. LAPACK is a software library for performing dense and banded linear algebra computations, and was designed to run efficiently on high performance computers. We focus on the design of the distributed memory version of LAPACK, and on an object-oriented interface to LAPACK. The templates project aims at making the task of developing sparse linear algebra software simpler and easier. Reusable software templates are provided that the user can then customize to modify and optimize a particular algorithm, and hence build a more complex applications. ARPACK is a software package for solving large scale eigenvalue problems, and is based on an implicitly restarted variant of the Arnoldi scheme. The paper focuses on issues impact...

A crystal lattice with a small miscut from the plane of symmetry has a surface which consists of a series of atomic height steps separated by terraces. If the surface of this crystal is not in equilibrium with the surrounding medium, then its evolution is strongly mediated by the presence of these steps, which act as sites for attachment and detachment of diffusing adsorbed atoms (‘adatoms’). In the absence of material deposition and evaporation, steps move in response to two main physical effects: line tension, which is caused by curvature of the step edge, and step-step interactions