Solving Block Low-Rank Matrix Eigenvalue Problems

• Item type: Trans(1)
• 公開日: 2022-07-28
• タイトル: Solving Block Low-Rank Matrix Eigenvalue Problems
• タイトル / 言語: en
• タイトル: Solving Block Low-Rank Matrix Eigenvalue Problems
• 言語: eng
• キーワード / 主題Scheme: Other
• キーワード / 主題: low-rank approximation, block low-rank matrices, hierarchical matrices, eigenvalue problem, block householder transformation, eigensolver
• 資源タイプ / 資源タイプ識別子: http://purl.org/coar/resource_type/c_6501
• 資源タイプ: journal article
• 著者所属: Research Institute for Value-Added-Information Generation (VAiG), Japan Agency for Marine-Earth Science and Technology (JAMSTEC)
• 著者所属(英): en
• 著者所属(英): Research Institute for Value-Added-Information Generation (VAiG), Japan Agency for Marine-Earth Science and Technology (JAMSTEC)
• 著者名: Akihiro, Ida
• 著者名(英): Akihiro, Ida
• 論文抄録 / 内容記述タイプ: Other
• 論文抄録 / 内容記述: To solve large-scale matrix eigenvalue problems (EVPs), a two-step tridiagonalization method using the block Householder transformation (HT) is often employed. Although the method based on dense matrix arithmetic requires a memory storage of O(N2) and an arithmetic operations of O(N3), in this study, these were reduced by approximating the method using block low-rank (BLR-) matrices. A special block HT for BLR-matrices and a two-step tridiagonalization method using it are proposed to solve an EVP with a real symmetric BLR-matrix. In the proposed block HT, block Householder vectors are also formed using BLR-matrices. It is demonstrated how the block size m in the BLR-matrix should be determined and confirmed that the memory and arithmetic complexities of the proposed method were O(N5/3) and O(N7/3), respectively, for typical cases when using an appropriate block size m ∝ N1/3. In numerical experiments of a string free vibration problem with known analytical solutions, for large eigenvalues, the calculated eigenvalues using the proposed method converge toward the analytical ones in accordance with the theoretical convergence curves. Owing to the reduced complexity, an EVP of a matrix was solved with about N = 300,000, which is significantly larger than the limit of the conventional method with dense matrices, within a reasonable amount of time on a CPU core. For the calculation time, the proposed method was faster than the conventional method when the matrix size N was larger than a few tens of thousands.; ------------------------------; This is a preprint of an article intended for publication Journal of; Information Processing(JIP). This preprint should not be cited. This; article should be cited as: Journal of Information Processing Vol.30(2022) (online); ------------------------------
• 論文抄録(英) / 内容記述タイプ: Other
• 論文抄録(英) / 内容記述: To solve large-scale matrix eigenvalue problems (EVPs), a two-step tridiagonalization method using the block Householder transformation (HT) is often employed. Although the method based on dense matrix arithmetic requires a memory storage of O(N2) and an arithmetic operations of O(N3), in this study, these were reduced by approximating the method using block low-rank (BLR-) matrices. A special block HT for BLR-matrices and a two-step tridiagonalization method using it are proposed to solve an EVP with a real symmetric BLR-matrix. In the proposed block HT, block Householder vectors are also formed using BLR-matrices. It is demonstrated how the block size m in the BLR-matrix should be determined and confirmed that the memory and arithmetic complexities of the proposed method were O(N5/3) and O(N7/3), respectively, for typical cases when using an appropriate block size m ∝ N1/3. In numerical experiments of a string free vibration problem with known analytical solutions, for large eigenvalues, the calculated eigenvalues using the proposed method converge toward the analytical ones in accordance with the theoretical convergence curves. Owing to the reduced complexity, an EVP of a matrix was solved with about N = 300,000, which is significantly larger than the limit of the conventional method with dense matrices, within a reasonable amount of time on a CPU core. For the calculation time, the proposed method was faster than the conventional method when the matrix size N was larger than a few tens of thousands.; ------------------------------; This is a preprint of an article intended for publication Journal of; Information Processing(JIP). This preprint should not be cited. This; article should be cited as: Journal of Information Processing Vol.30(2022) (online); ------------------------------
• 書誌レコードID / 収録物識別子タイプ: NCID
• 書誌レコードID / 収録物識別子: AA11833852
• 書誌情報: 情報処理学会論文誌コンピューティングシステム（ACS）; 巻 15, 号 1, 発行日 2022-07-28
• ISSN / 収録物識別子タイプ: ISSN
• ISSN / 収録物識別子: 1882-7829
• 出版者 / 言語: ja
• 出版者: 情報処理学会