A skew-symmetric Lanczos bidiagonalization method for computing several largest eigenpairs of a large skew-symmetric matrix
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2022
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866929462089613312 |
|---|---|
| author | Huang, Jinzhi Jia, Zhongxiao |
| author_facet | Huang, Jinzhi Jia, Zhongxiao |
| contents | The spectral decomposition of a real skew-symmetric matrix $A$ can be mathematically transformed into a specific structured singular value decomposition (SVD) of $A$. Based on such equivalence, a skew-symmetric Lanczos bidiagonalization (SSLBD) method is proposed for the specific SVD problem that computes extreme singular values and the corresponding singular vectors of $A$, from which the eigenpairs of $A$ corresponding to the extreme conjugate eigenvalues in magnitude are recovered pairwise in real arithmetic. A number of convergence results on the method are established, and accuracy estimates for approximate singular triplets are given. In finite precision arithmetic, it is proven that the semi-orthogonality of each set of basis vectors and the semi-biorthogonality of two sets of basis vectors suffice to compute the singular values accurately. A commonly used efficient partial reorthogonalization strategy is adapted to maintaining the needed semi-orthogonality and semi-biorthogonality. For a practical purpose, an implicitly restarted SSLBD algorithm is developed with partial reorthogonalization. Numerical experiments illustrate the effectiveness and overall efficiency of the algorithm. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2212_07574 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | A skew-symmetric Lanczos bidiagonalization method for computing several largest eigenpairs of a large skew-symmetric matrix Huang, Jinzhi Jia, Zhongxiao Numerical Analysis 65F15, 15A18, 65F10, 65F25 The spectral decomposition of a real skew-symmetric matrix $A$ can be mathematically transformed into a specific structured singular value decomposition (SVD) of $A$. Based on such equivalence, a skew-symmetric Lanczos bidiagonalization (SSLBD) method is proposed for the specific SVD problem that computes extreme singular values and the corresponding singular vectors of $A$, from which the eigenpairs of $A$ corresponding to the extreme conjugate eigenvalues in magnitude are recovered pairwise in real arithmetic. A number of convergence results on the method are established, and accuracy estimates for approximate singular triplets are given. In finite precision arithmetic, it is proven that the semi-orthogonality of each set of basis vectors and the semi-biorthogonality of two sets of basis vectors suffice to compute the singular values accurately. A commonly used efficient partial reorthogonalization strategy is adapted to maintaining the needed semi-orthogonality and semi-biorthogonality. For a practical purpose, an implicitly restarted SSLBD algorithm is developed with partial reorthogonalization. Numerical experiments illustrate the effectiveness and overall efficiency of the algorithm. |
| title | A skew-symmetric Lanczos bidiagonalization method for computing several largest eigenpairs of a large skew-symmetric matrix |
| topic | Numerical Analysis 65F15, 15A18, 65F10, 65F25 |
| url | https://arxiv.org/abs/2212.07574 |