Scaled and squared subdiagonal Padé approximation for the matrix exponential
The scaling and squaring method is the most widely used algorithm for computing the exponential of a square matrix A. We introduce an efficient variant that uses a much smaller squaring factor when ||A|| » 1 and a subdiagonal Padé approximant of low degree, thereby significantly re...
| Asıl Yazarlar: | , |
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| Materyal Türü: | Journal article |
| Dil: | English |
| Baskı/Yayın Bilgisi: |
Society for Industrial and Applied Mathematics
2016
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| _version_ | 1826304355716825088 |
|---|---|
| author | Guettel, S Nakatsukasa, Y |
| author_facet | Guettel, S Nakatsukasa, Y |
| author_sort | Guettel, S |
| collection | OXFORD |
| description | The scaling and squaring method is the most widely used algorithm for computing the exponential of a square matrix A. We introduce an efficient variant that uses a much smaller squaring factor when ||A|| » 1 and a subdiagonal Padé approximant of low degree, thereby significantly reducing the overall cost and avoiding the potential instability caused by overscaling, while giving forward error of the same magnitude as that of the standard algorithm. The new algorithm performs well if a rough estimate of the rightmost eigenvalue of A is available and the rightmost eigenvalues do not have widely varying imaginary parts, and it achieves significant speedup over the conventional algorithm especially when A is of large norm. Our algorithm uses the partial fraction form to evaluate the Padé approximant, which makes it suitable for parallelization and directly applicable to computing the action of the matrix exponential exp(A)b, where b is a vector or a tall skinny matrix. For this problem the significantly smaller squaring factor has an even more pronounced benefit for efficiency when evaluating the action of the Padé approximant. |
| first_indexed | 2024-03-07T06:16:33Z |
| format | Journal article |
| id | oxford-uuid:f1454c2d-d9a4-4be9-b749-3cff395788c4 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-07T06:16:33Z |
| publishDate | 2016 |
| publisher | Society for Industrial and Applied Mathematics |
| record_format | dspace |
| spelling | oxford-uuid:f1454c2d-d9a4-4be9-b749-3cff395788c42022-03-27T11:54:46ZScaled and squared subdiagonal Padé approximation for the matrix exponentialJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:f1454c2d-d9a4-4be9-b749-3cff395788c4EnglishSymplectic ElementsSociety for Industrial and Applied Mathematics2016Guettel, SNakatsukasa, YThe scaling and squaring method is the most widely used algorithm for computing the exponential of a square matrix A. We introduce an efficient variant that uses a much smaller squaring factor when ||A|| » 1 and a subdiagonal Padé approximant of low degree, thereby significantly reducing the overall cost and avoiding the potential instability caused by overscaling, while giving forward error of the same magnitude as that of the standard algorithm. The new algorithm performs well if a rough estimate of the rightmost eigenvalue of A is available and the rightmost eigenvalues do not have widely varying imaginary parts, and it achieves significant speedup over the conventional algorithm especially when A is of large norm. Our algorithm uses the partial fraction form to evaluate the Padé approximant, which makes it suitable for parallelization and directly applicable to computing the action of the matrix exponential exp(A)b, where b is a vector or a tall skinny matrix. For this problem the significantly smaller squaring factor has an even more pronounced benefit for efficiency when evaluating the action of the Padé approximant. |
| spellingShingle | Guettel, S Nakatsukasa, Y Scaled and squared subdiagonal Padé approximation for the matrix exponential |
| title | Scaled and squared subdiagonal Padé approximation for the matrix exponential |
| title_full | Scaled and squared subdiagonal Padé approximation for the matrix exponential |
| title_fullStr | Scaled and squared subdiagonal Padé approximation for the matrix exponential |
| title_full_unstemmed | Scaled and squared subdiagonal Padé approximation for the matrix exponential |
| title_short | Scaled and squared subdiagonal Padé approximation for the matrix exponential |
| title_sort | scaled and squared subdiagonal pade approximation for the matrix exponential |
| work_keys_str_mv | AT guettels scaledandsquaredsubdiagonalpadeapproximationforthematrixexponential AT nakatsukasay scaledandsquaredsubdiagonalpadeapproximationforthematrixexponential |