Currently there does not exist reliable mathematical software to compute the exponential of a matrix. Several methods have been proposed, each with its weaknesses, and this has held back the use of the matrix exponential in application. In this study we give careful attention to some of the better methods and point out improvements to frequently-employed techniques, and analyze a few surprising results in detail.;We give the definitions of matrix functions and the expression used to measure their sensitivity at particular matrix arguments. We show the role of the departure from normality in connection with the exponential.;The Taylor series has a bad reputation as a numerical method. We show that when used with proper shifting and scaling it is an effective technique. Ward's method uses Pade approximation on the unit circle. It is one of the most effective candidates for matrices with modest norms. Walz's method uses Euler's definition of the exponential combined with Richardson extrapolation. We provide an error analysis and show sensitivity of the output with respect to depth in the extrapolation table.;The Schur form of the given matrix permits use of more elaborate approaches. Newton interpolation from the Schur form appears to be the most reliable among the methods. Its success is due to accurate computation of divided differences.;Parlett's recurrence is fast and may be executed in double precision in less time than some of its rivals. We show how to extend the class of matrices on which this recurrence is reliable.;We conclude with some comparisons of these methods on matrices of various sizes. A hybrid of techniques by Li and Ng is most effective in computing the exponential.
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