The Schur complements for $SDD_{1}$ matrices and their application to linear complementarity problems
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866910915181412352 |
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| author | Hu, Yang Liu, Jianzhou Zeng, Wenlong |
| author_facet | Hu, Yang Liu, Jianzhou Zeng, Wenlong |
| contents | In this paper we propose a new scaling method to study the Schur complements of $SDD_{1}$ matrices. Its core is related to the non-negative property of the inverse $M$-matrix, while numerically improving the Quotient formula. Based on the Schur complement and a novel norm splitting manner, we establish an upper bound for the infinity norm of the inverse of $SDD_{1}$ matrices, which depends solely on the original matrix entries. We apply the new bound to derive an error bound for linear complementarity problems of $B_{1}$-matrices. Additionally, new lower and upper bounds for the determinant of $SDD_{1}$ matrices are presented. Numerical experiments validate the effectiveness and superiority of our results. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_14308 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Schur complements for $SDD_{1}$ matrices and their application to linear complementarity problems Hu, Yang Liu, Jianzhou Zeng, Wenlong Numerical Analysis In this paper we propose a new scaling method to study the Schur complements of $SDD_{1}$ matrices. Its core is related to the non-negative property of the inverse $M$-matrix, while numerically improving the Quotient formula. Based on the Schur complement and a novel norm splitting manner, we establish an upper bound for the infinity norm of the inverse of $SDD_{1}$ matrices, which depends solely on the original matrix entries. We apply the new bound to derive an error bound for linear complementarity problems of $B_{1}$-matrices. Additionally, new lower and upper bounds for the determinant of $SDD_{1}$ matrices are presented. Numerical experiments validate the effectiveness and superiority of our results. |
| title | The Schur complements for $SDD_{1}$ matrices and their application to linear complementarity problems |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2504.14308 |