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possible_helping_document - SIAM REVIEW Vol 45 No 1 pp 3000...

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SIAM R EVIEW c 2003 Society for Industrial and Applied Mathematics Vol. 45, No. 1, pp. 3–000 Nineteen Dubious Ways to Compute the Exponential of a Matrix, Twenty-Five Years Later * Cleve Moler Charles Van Loan Abstract. In principle, the exponential of a matrix could be computed in many ways. Methods involv- ing 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 bibliog- raphy, describes a few recent developments. Key words. matrix, exponential, roundoff error, truncation error, condition AMS subject classifications. 15A15, 65F15, 65F30, 65L99 PII. S0036144502418010 1. Introduction. Mathematical models of many physical, biological, and eco- nomic processes involve systems of linear, constant coefficient ordinary differential equations ˙ x ( t ) = Ax ( t ) . Here A is a given, fixed, real or complex n -by- n matrix. A solution vector x ( t ) is sought which satisfies an initial condition x (0) = x 0 . In control theory, A is known as the state companion matrix and x ( t ) is the system response. In principle, the solution is given by x ( t ) = e tA x 0 where e tA can be formally defined by the convergent power series e tA = I + tA + t 2 A 2 2! + · · · . * Published electronically February 3, 2003. A portion of this paper originally appeared in SIAM Review , Volume 20, Number 4, 1978, pages 801–836. http://www.siam.org/journals/sirev/45-1/41801.html The MathWorks, Inc., 3 Apple Hill Drive, Natick, MA 01760-2098 ([email protected]). Department of Computer Science, Cornell University, 4130 Upson Hall, Ithaca, NY 14853-7501 ([email protected]). 1
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2 CLEVE MOLER AND CHARLES VAN LOAN The effective computation of this matrix function is the main topic of this survey. We will primarily be concerned with matrices whose order n is less than a few hundred, so that all the elements can be stored in the main memory of a contemporary computer. Our discussion will be less germane to the type of large, sparse matrices which occur in the method of lines for partial differential equations. Dozens of methods for computing e tA can be obtained from more or less classical results in analysis, approximation theory, and matrix theory. Some of the methods have been proposed as specific algorithms, while others are based on less constructive characterizations. Our bibliography concentrates on recent papers with strong algo- rithmic content, although we have included a fair number of references which possess historical or theoretical interest. In this survey we try to describe all the methods that appear to be practical, clas- sify them into five broad categories, and assess their relative effectiveness. Actually, each of the “methods” when completely implemented might lead to many different computer programs which differ in various details. Moreover, these details might have more influence on the actual performance than our gross assessment indicates. Thus,
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