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On-the-fly backward error estimate for matrix exponential approximation by Taylor algorithm. (English) Zbl 1452.65082
Summary: In this paper we show that it is possible to estimate the backward error for the approximation of the matrix exponential on-the-fly, without the need to precompute in high precision quantities related to specific accuracies. In this way, the scaling parameter \(s\) and the degree \(m\) of the truncated Taylor series (the underlying method) are adapted to the input matrix and not to a class of matrices sharing some spectral properties, such as with the classical backward error analysis. The result is a very flexible method which can approximate the matrix exponential at any desired accuracy. Moreover, the risk of overscaling, that is the possibility to select a value \(s\) larger than necessary, is mitigated by a new choice as the sum of two powers of two. Finally, several numerical experiments in MATLAB with data in double and variable precision and with different tolerances confirm that the method is accurate and often faster than available good alternatives.

MSC:
65F60 Numerical computation of matrix exponential and similar matrix functions
15A16 Matrix exponential and similar functions of matrices
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