Theoretical analysis and numerical solution to a vector equation $Ax-\|x\|_1x=b$
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866908952995823616 |
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| author | Wang, Yuezhi Kim, Gwi Soo Meng, Jie |
| author_facet | Wang, Yuezhi Kim, Gwi Soo Meng, Jie |
| contents | Theoretical and computational properties of a vector equation $Ax-\|x\|_1x=b$ are investigated, where $A$ is an invertible $M$-matrix and $b$ is a nonnegative vector. Existence and uniqueness of a nonnegative solution is proved. Fixed-point iterations, including a relaxed fixed-point iteration and Newton iteration, are proposed and analyzed.
A structure-preserving doubling algorithm is proved to be applicable in computing the required solution, the convergence is at least linear with rate 1/2. Numerical experiments are performed to demonstrate the effectiveness of the proposed algorithms. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_04971 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Theoretical analysis and numerical solution to a vector equation $Ax-\|x\|_1x=b$ Wang, Yuezhi Kim, Gwi Soo Meng, Jie Numerical Analysis Theoretical and computational properties of a vector equation $Ax-\|x\|_1x=b$ are investigated, where $A$ is an invertible $M$-matrix and $b$ is a nonnegative vector. Existence and uniqueness of a nonnegative solution is proved. Fixed-point iterations, including a relaxed fixed-point iteration and Newton iteration, are proposed and analyzed. A structure-preserving doubling algorithm is proved to be applicable in computing the required solution, the convergence is at least linear with rate 1/2. Numerical experiments are performed to demonstrate the effectiveness of the proposed algorithms. |
| title | Theoretical analysis and numerical solution to a vector equation $Ax-\|x\|_1x=b$ |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2507.04971 |