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Nineteen Dubious Ways to Compute the Exponential of a Matrix
 SIAM Review
, 1978
"... 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 efficienc ..."
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Cited by 228 (0 self)
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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.
Exponential Integrators For Large Systems Of Differential Equations
 SIAM J. Sci. Comput
, 1997
"... . We study the numerical integration of large stiff systems of differential equations by methods that use matrixvector 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 ..."
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Cited by 88 (1 self)
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. We study the numerical integration of large stiff systems of differential equations by methods that use matrixvector 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 constantcoefficient equations. The implementation of the methods is discussed. Numerical experiments with reactiondiffusion problems and a timedependent Schrodinger equation are included. Key words. Numerical integrator, highdimensional differential equations, matrix exponential, Krylov subspace methods. AMS(MOS) subject classifications. 65L05, 65M15, 65F10. 1. Introduction. The idea to use the exp...
From Potential Theory To Matrix Iterations In Six Steps
 SIAM REVIEW
"... The theory of the convergence of Krylov subspace iterations for linear systems of equations (conjugate gradients, biconjugate gradients, GMRES, QMR, BiCGSTAB, ...) is reviewed. For a computation of this kind, an estimated asymptotic convergence factor ae 1 can be derived by solving a problem of pot ..."
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Cited by 35 (4 self)
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The theory of the convergence of Krylov subspace iterations for linear systems of equations (conjugate gradients, biconjugate gradients, GMRES, QMR, BiCGSTAB, ...) 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.
Calculation Of Pseudospectra By The Arnoldi Iteration
, 1996
"... 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 nonnormal matrices or operators, such as hydrodynamic stability, pseudospectra may be phys ..."
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Cited by 34 (5 self)
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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 nonnormal matrices or operators, such as hydrodynamic stability, pseudospectra may be physically more significant than spectra.
Recurrent motions within plane Couette turbulence
 Journal of Fluid Mechanics
"... We describe accurate computations of threedimensional periodic and relative periodic motions within plane Couette turbulence at Re = 400. To ensure that the computed solutions are true solutions of the NavierStokes equations, careful attention is paid to time discretization errors and to spatial r ..."
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Cited by 23 (5 self)
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We describe accurate computations of threedimensional periodic and relative periodic motions within plane Couette turbulence at Re = 400. To ensure that the computed solutions are true solutions of the NavierStokes 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.
Solving multidimensional evolution problems with localized structures using second generation wavelets
 Int. J. Comput. Fluid Dynamics
, 2003
"... A dynamically adaptive numerical method for solving multidimensional evolution problems with localized structures is developed. The method is based on the general class of multidimensional secondgeneration wavelets and is an extension of the secondgeneration wavelet collocation method of Vasilye ..."
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Cited by 22 (9 self)
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A dynamically adaptive numerical method for solving multidimensional evolution problems with localized structures is developed. The method is based on the general class of multidimensional secondgeneration wavelets and is an extension of the secondgeneration wavelet collocation method of Vasilyev and Bowman to two and higher dimensions and irregular sampling intervals. Wavelet decomposition is used for grid adaptation and interpolation, while O(N) hierarchical finite difference scheme, which takes advantage of wavelet multilevel decomposition, is used for derivative calculations. The prowess and computational efficiency of the method are demonstrated for the solution of a number of twodimensional test problems.
Implicitly restarted Arnoldi/Lanczos Methods for Large Scale Eigenvalue Calculations
, 1996
"... Eigenvalues and eigenfunctions of linear operators are important to many areas of applied 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 m ..."
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Cited by 20 (3 self)
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Eigenvalues and eigenfunctions of linear operators are important to many areas of applied 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 largescale 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 largescale nonsymmetric problems was virtually nonexistent. Fortunately, the situation is improving rapidly. The purpose of this article is to provide an overview of the numerical solution of largescale algebraic eigenvalue problems. The focus will be on a class of methods called Krylov subspace projection methods. The wellknown 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.
Oscillatory modes in an enclosed swirling flow
 J. Fluid Mech
, 2001
"... The flow in a completely filled cylinder driven by a rotating endwall has multiple timedependent stable states when the endwall rotation exceeds a critical value. These states have been observed experimentally and computed numerically elsewhere. In this article, the linear stability of the basic st ..."
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Cited by 15 (8 self)
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The flow in a completely filled cylinder driven by a rotating endwall has multiple timedependent stable states when the endwall rotation exceeds a critical value. These states have been observed experimentally and computed numerically elsewhere. In this article, the linear stability of the basic state, which is a nontrivial axisymmetric flow, is analysed at parameter values where the unsteady solutions exist. We show that the basic state undergoes a succession of Hopf bifurcations and the corresponding eigenvalues and eigenvectors of these excited modes describe most of the characteristics of the observed timedependent states. 1.
Large Scale Eigenvalue Calculations for Computing the Stability of Buoyancy Driven Flows
, 2001
"... 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, co ..."
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Cited by 7 (2 self)
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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 semianalytical 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...