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The evaluation of a matrix exponential function is a classic problem ofcomputational linear algebra. Many different methods have been employed for itsnumerical evaluation [Moler C and van Loan C 1978 SIAM Review 20 4], none ofwhich produce a definitive algorithm which is broadly applicable andsufficiently accurate, as well as being reasonably fast. Herein, we employ amethod which evaulates a matrix exponential as the solution to a first-orderinitial value problem in a fictitious time variable. The new aspect of thepresent implementation of this method is to use finite elements in thefictitious time variable. [Weatherford C A, Red E, and Wynn A 2002 Journal ofMolecular Structure 592 47] Then using an expansion in a properly chosen timebasis, we are able to make accurate calculations of the exponential of anygiven matrix as the solution to a set of simultaneous equations.

Evaluation of the matrix exponential function using finite elements in time