Computing the matrix exponential with the double exponential formula
This article considers the computation of the matrix exponential eA{{\rm{e}}}^{A} with numerical quadrature. Although several quadrature-based algorithms have been proposed, they focus on (near) Hermitian matrices. In order to deal with non-Hermitian matrices, we use another integral representation...
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| Format: | Article |
| Language: | English |
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De Gruyter
2024-10-01
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| Series: | Special Matrices |
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| Online Access: | https://doi.org/10.1515/spma-2024-0013 |
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| _version_ | 1850180819776176128 |
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| author | Tatsuoka Fuminori Sogabe Tomohiro Kemmochi Tomoya Zhang Shao-Liang |
| author_facet | Tatsuoka Fuminori Sogabe Tomohiro Kemmochi Tomoya Zhang Shao-Liang |
| author_sort | Tatsuoka Fuminori |
| collection | DOAJ |
| description | This article considers the computation of the matrix exponential eA{{\rm{e}}}^{A} with numerical quadrature. Although several quadrature-based algorithms have been proposed, they focus on (near) Hermitian matrices. In order to deal with non-Hermitian matrices, we use another integral representation including an oscillatory term and consider applying the double exponential (DE) formula specialized to Fourier integrals. The DE formula transforms the given integral into another integral whose interval is infinite, and therefore, it is necessary to truncate the infinite interval. In this article, to utilize the DE formula, we analyze the truncation error and propose two algorithms. The first one approximates eA{{\rm{e}}}^{A} with the fixed mesh size, which is a parameter in the DE formula affecting the accuracy. The second one computes eA{{\rm{e}}}^{A} based on the first one with automatic selection of the mesh size depending on the given error tolerance. |
| format | Article |
| id | doaj-art-7c76cdb981cc4f7fa61c7c9f9fead7b9 |
| institution | OA Journals |
| issn | 2300-7451 |
| language | English |
| publishDate | 2024-10-01 |
| publisher | De Gruyter |
| record_format | Article |
| series | Special Matrices |
| spelling | doaj-art-7c76cdb981cc4f7fa61c7c9f9fead7b92025-08-20T02:18:03ZengDe GruyterSpecial Matrices2300-74512024-10-0112197098910.1515/spma-2024-0013Computing the matrix exponential with the double exponential formulaTatsuoka Fuminori0Sogabe Tomohiro1Kemmochi Tomoya2Zhang Shao-Liang3Department of Applied Physics, Graduate School of Engineering, Nagoya University, Furo-cho, Chikusa-ku, Nagoya 464-8603, JapanDepartment of Applied Physics, Graduate School of Engineering, Nagoya University, Furo-cho, Chikusa-ku, Nagoya 464-8603, JapanDepartment of Applied Physics, Graduate School of Engineering, Nagoya University, Furo-cho, Chikusa-ku, Nagoya 464-8603, JapanDepartment of Applied Physics, Graduate School of Engineering, Nagoya University, Furo-cho, Chikusa-ku, Nagoya 464-8603, JapanThis article considers the computation of the matrix exponential eA{{\rm{e}}}^{A} with numerical quadrature. Although several quadrature-based algorithms have been proposed, they focus on (near) Hermitian matrices. In order to deal with non-Hermitian matrices, we use another integral representation including an oscillatory term and consider applying the double exponential (DE) formula specialized to Fourier integrals. The DE formula transforms the given integral into another integral whose interval is infinite, and therefore, it is necessary to truncate the infinite interval. In this article, to utilize the DE formula, we analyze the truncation error and propose two algorithms. The first one approximates eA{{\rm{e}}}^{A} with the fixed mesh size, which is a parameter in the DE formula affecting the accuracy. The second one computes eA{{\rm{e}}}^{A} based on the first one with automatic selection of the mesh size depending on the given error tolerance.https://doi.org/10.1515/spma-2024-0013matrix functionmatrix exponentialnumerical quadraturedouble exponential formula65f6065d30 |
| spellingShingle | Tatsuoka Fuminori Sogabe Tomohiro Kemmochi Tomoya Zhang Shao-Liang Computing the matrix exponential with the double exponential formula Special Matrices matrix function matrix exponential numerical quadrature double exponential formula 65f60 65d30 |
| title | Computing the matrix exponential with the double exponential formula |
| title_full | Computing the matrix exponential with the double exponential formula |
| title_fullStr | Computing the matrix exponential with the double exponential formula |
| title_full_unstemmed | Computing the matrix exponential with the double exponential formula |
| title_short | Computing the matrix exponential with the double exponential formula |
| title_sort | computing the matrix exponential with the double exponential formula |
| topic | matrix function matrix exponential numerical quadrature double exponential formula 65f60 65d30 |
| url | https://doi.org/10.1515/spma-2024-0013 |
| work_keys_str_mv | AT tatsuokafuminori computingthematrixexponentialwiththedoubleexponentialformula AT sogabetomohiro computingthematrixexponentialwiththedoubleexponentialformula AT kemmochitomoya computingthematrixexponentialwiththedoubleexponentialformula AT zhangshaoliang computingthematrixexponentialwiththedoubleexponentialformula |