Verified error bounds for the singular values of structured matrices with applications to computer-assisted proofs for differential equations
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866912232093253632 |
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| author | Terao, Takeshi Watanabe, Yoshitaka Ozaki, Katsuhisa |
| author_facet | Terao, Takeshi Watanabe, Yoshitaka Ozaki, Katsuhisa |
| contents | This paper introduces two methods for verifying the singular values of the structured matrix denoted by $R^{-H}AR^{-1}$, where $R$ is a nonsingular matrix and $A$ is a general nonsingular square matrix. The first of the two methods uses the computed factors from a singular value decomposition (SVD) to verify all singular values; the second estimates a lower bound of the minimum singular value without performing the SVD. The proposed approach for verifying all singular values efficiently computes tight error bounds. The method for estimating a lower bound of the minimum singular value is particularly effective for sparse matrices. These methods have proven to be efficient in verifying solutions to differential equation problems, that were previously challenging due to the extensive computational time and memory requirements. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_09984 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Verified error bounds for the singular values of structured matrices with applications to computer-assisted proofs for differential equations Terao, Takeshi Watanabe, Yoshitaka Ozaki, Katsuhisa Numerical Analysis This paper introduces two methods for verifying the singular values of the structured matrix denoted by $R^{-H}AR^{-1}$, where $R$ is a nonsingular matrix and $A$ is a general nonsingular square matrix. The first of the two methods uses the computed factors from a singular value decomposition (SVD) to verify all singular values; the second estimates a lower bound of the minimum singular value without performing the SVD. The proposed approach for verifying all singular values efficiently computes tight error bounds. The method for estimating a lower bound of the minimum singular value is particularly effective for sparse matrices. These methods have proven to be efficient in verifying solutions to differential equation problems, that were previously challenging due to the extensive computational time and memory requirements. |
| title | Verified error bounds for the singular values of structured matrices with applications to computer-assisted proofs for differential equations |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2502.09984 |