Error Analysis of Krylov Subspace approximation Based on IDR($s$) Method for Matrix Function Bilinear Forms
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866908706835267584 |
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| author | Xue, Qianqian Yue, Xiaoqiang Gu, Xian-Ming |
| author_facet | Xue, Qianqian Yue, Xiaoqiang Gu, Xian-Ming |
| contents | The bilinear form of a matrix function, namely $\mathbf{u}^\top f(A) \mathbf{v}$, appears in many scientific computing problems, where $\mathbf{u}, \mathbf{v} \in \mathbb{R}^n$, $A \in \mathbb{R}^{n \times n}$, and $f(z)$ is a given analytic function. The Induced Dimension Reduction IDR($s$) method was originally proposed to solve a large-scale linear system, and effectively reduces the complexity and storage requirement by dimension reduction techniques while maintaining the numerical stability of the algorithm. In fact, the IDR($s$) method can generate an interesting Hessenberg decomposition, our study just applies this fact to establish the numerical algorithm and a posteriori error estimate for the bilinear form of a matrix function $\mathbf{u}^{\top} f(A) \mathbf{v}$. Through the error analysis of the IDR($s$) algorithm, the corresponding error expansion is derived, and it is verified that the leading term of the error expansion serves as a reliable posteriori error estimate. Based on this, in this paper, a corresponding stopping criterion is proposed. Numerical examples are reported to support our theoretical findings and show the utility of our proposed method and its stopping criterion over the traditional Arnoldi-based method. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_08563 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Error Analysis of Krylov Subspace approximation Based on IDR($s$) Method for Matrix Function Bilinear Forms Xue, Qianqian Yue, Xiaoqiang Gu, Xian-Ming Numerical Analysis The bilinear form of a matrix function, namely $\mathbf{u}^\top f(A) \mathbf{v}$, appears in many scientific computing problems, where $\mathbf{u}, \mathbf{v} \in \mathbb{R}^n$, $A \in \mathbb{R}^{n \times n}$, and $f(z)$ is a given analytic function. The Induced Dimension Reduction IDR($s$) method was originally proposed to solve a large-scale linear system, and effectively reduces the complexity and storage requirement by dimension reduction techniques while maintaining the numerical stability of the algorithm. In fact, the IDR($s$) method can generate an interesting Hessenberg decomposition, our study just applies this fact to establish the numerical algorithm and a posteriori error estimate for the bilinear form of a matrix function $\mathbf{u}^{\top} f(A) \mathbf{v}$. Through the error analysis of the IDR($s$) algorithm, the corresponding error expansion is derived, and it is verified that the leading term of the error expansion serves as a reliable posteriori error estimate. Based on this, in this paper, a corresponding stopping criterion is proposed. Numerical examples are reported to support our theoretical findings and show the utility of our proposed method and its stopping criterion over the traditional Arnoldi-based method. |
| title | Error Analysis of Krylov Subspace approximation Based on IDR($s$) Method for Matrix Function Bilinear Forms |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2509.08563 |