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Computing the action of the matrix exponential, with an application to exponential integrators. (English) Zbl 1234.65028
The authors develop a new algorithm to evaluate $$e^{tA}B$$ for $$A$$ $$n$$-by-$$n$$ and $$B$$ $$n$$-by-$$n_0$$ with $$n_0 \ll n$$. Its cost consists mainly of that for multiplications of $$n$$-by-$$n$$ matrices times $$n$$-by-$$n_0$$ matrices. It uses the popular scaling method together with a truncated Taylor series approximation of the exponential function where the truncation degree is pre-determined from a few evaluations of $$\|A^k\|^{1/k}$$. Numerical experiments and an analysis of rounding errors and the conditioning of the problem are performed that show that this algorithm is often superior in speed and accuracy over existing ones. Algorithms of this kind find their main use as exponential integrators for ordinary differential equations.

##### MSC:
 65F60 Numerical computation of matrix exponential and similar matrix functions 15A16 Matrix exponential and similar functions of matrices 65G50 Roundoff error 65F35 Numerical computation of matrix norms, conditioning, scaling 65L05 Numerical methods for initial value problems 34A30 Linear ordinary differential equations and systems, general
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