Mixed Precision Orthogonalization-Free Projection Methods for Eigenvalue and Singular Value Problems
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arXiv
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| Main Authors: | , , , , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866909598600921088 |
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| author | Xu, Tianshi Zhang, Zechen Chen, Jie Saad, Yousef Xi, Yuanzhe |
| author_facet | Xu, Tianshi Zhang, Zechen Chen, Jie Saad, Yousef Xi, Yuanzhe |
| contents | Mixed-precision arithmetic offers significant computational advantages for large-scale matrix computation tasks, yet preserving accuracy and stability in eigenvalue problems and the singular value decomposition (SVD) remains challenging. This paper introduces an approach that eliminates orthogonalization requirements in traditional Rayleigh-Ritz projection methods. The proposed method employs non-orthogonal bases computed at reduced precision, resulting in bases computed without inner-products. A primary focus is on maintaining the linear independence of the basis vectors. Through extensive evaluation with both synthetic test cases and real-world applications, we demonstrate that the proposed approach achieves the desired accuracy while fully taking advantage of mixed-precision arithmetic. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_00281 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Mixed Precision Orthogonalization-Free Projection Methods for Eigenvalue and Singular Value Problems Xu, Tianshi Zhang, Zechen Chen, Jie Saad, Yousef Xi, Yuanzhe Numerical Analysis 15A23, 65F25, 65Y05, 68W10 Mixed-precision arithmetic offers significant computational advantages for large-scale matrix computation tasks, yet preserving accuracy and stability in eigenvalue problems and the singular value decomposition (SVD) remains challenging. This paper introduces an approach that eliminates orthogonalization requirements in traditional Rayleigh-Ritz projection methods. The proposed method employs non-orthogonal bases computed at reduced precision, resulting in bases computed without inner-products. A primary focus is on maintaining the linear independence of the basis vectors. Through extensive evaluation with both synthetic test cases and real-world applications, we demonstrate that the proposed approach achieves the desired accuracy while fully taking advantage of mixed-precision arithmetic. |
| title | Mixed Precision Orthogonalization-Free Projection Methods for Eigenvalue and Singular Value Problems |
| topic | Numerical Analysis 15A23, 65F25, 65Y05, 68W10 |
| url | https://arxiv.org/abs/2505.00281 |