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Efficient and accurate computation for the \(\varphi\)-functions arising from exponential integrators

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Abstract

In this paper, we develop efficient and accurate algorithms for evaluating both \(\varphi _l(A)\) and \(\varphi _l(A)b,\) where \(\varphi _l(\cdot )\) is the function related to the exponential defined by \(\varphi _l(z)\equiv \sum \nolimits ^{\infty }_{k=0}\frac{z^k}{(l+k)!}\), A is an \(N\times N\) matrix and b is an N dimensional vector. Such matrix functions play a key role in a class of numerical methods well-known as exponential integrators. The algorithms use the modified scaling and squaring procedure combined with truncated Taylor series. A quasi-backward error analysis is presented to find the optimal value of the scaling and the degree of the Taylor approximation. Some useful techniques are employed for reducing the computational cost. Numerical comparisons with state-of-the-art algorithms show that the algorithms perform well in both accuracy and efficiency.

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Acknowledgements

The authors would like to express their gratitude to the referees and the editor for their helpful suggestions and comments. This work was supported in part by the Jilin Scientific and Technological Development Program (Grant No. 20200201276JC) and the Natural Science Foundation of Jilin Province (Grant No. 20200822KJ), and the Scientific Startup Foundation for Doctors of Changchun Normal University (Grant No. 002006059).

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Authors and Affiliations

  1. Department of Mathematics, Changchun Normal University, Changchun, 130032, People’s Republic of China

    Dongping Li, Siyu Yang & Jiamei Lan

  2. Department of Mathematics, Jilin University, Changchun, 130012, People’s Republic of China

    Dongping Li

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  1. Dongping Li
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  2. Siyu Yang
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Correspondence to Dongping Li.

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Li, D., Yang, S. & Lan, J. Efficient and accurate computation for the \(\varphi\)-functions arising from exponential integrators. Calcolo 59, 11 (2022). https://doi.org/10.1007/s10092-021-00453-2

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