Gradient-descent iterative algorithm for solving exact and weighted least-squares solutions of rectangular linear systems
Consider a linear system Ax=b where the coefficient matrix A is rectangular and of full-column rank. We propose an iterative algorithm for solving this linear system, based on gradient-descent optimization technique, aiming to produce a sequence of well-approximate least-squares solutions. Here, we...
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| Format: | Article |
| Language: | English |
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AIMS Press
2023-03-01
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| Series: | AIMS Mathematics |
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/math.2023596?viewType=HTML |
| _version_ | 1797856465314119680 |
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| author | Kanjanaporn Tansri Pattrawut Chansangiam |
| author_facet | Kanjanaporn Tansri Pattrawut Chansangiam |
| author_sort | Kanjanaporn Tansri |
| collection | DOAJ |
| description | Consider a linear system Ax=b where the coefficient matrix A is rectangular and of full-column rank. We propose an iterative algorithm for solving this linear system, based on gradient-descent optimization technique, aiming to produce a sequence of well-approximate least-squares solutions. Here, we consider least-squares solutions in a full generality, that is, we measure any related error through an arbitrary vector norm induced from weighted positive definite matrices W. It turns out that when the system has a unique solution, the proposed algorithm produces approximated solutions converging to the unique solution. When the system is inconsistent, the sequence of residual norms converges to the weighted least-squares error. Our work includes the usual least-squares solution when W=I. Numerical experiments are performed to validate the capability of the algorithm. Moreover, the performance of this algorithm is better than that of recent gradient-based iterative algorithms in both iteration numbers and computational time. |
| first_indexed | 2024-04-09T20:40:53Z |
| format | Article |
| id | doaj.art-1d00742ef064475095746d8cf8ffd5a4 |
| institution | Directory Open Access Journal |
| issn | 2473-6988 |
| language | English |
| last_indexed | 2024-04-09T20:40:53Z |
| publishDate | 2023-03-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | AIMS Mathematics |
| spelling | doaj.art-1d00742ef064475095746d8cf8ffd5a42023-03-30T01:29:18ZengAIMS PressAIMS Mathematics2473-69882023-03-0185117811179810.3934/math.2023596Gradient-descent iterative algorithm for solving exact and weighted least-squares solutions of rectangular linear systemsKanjanaporn Tansri0Pattrawut Chansangiam1Department of Mathematics, School of Science, King Mongkut's Institute of Technology Ladkrabang, Bangkok 10520, ThailandDepartment of Mathematics, School of Science, King Mongkut's Institute of Technology Ladkrabang, Bangkok 10520, ThailandConsider a linear system Ax=b where the coefficient matrix A is rectangular and of full-column rank. We propose an iterative algorithm for solving this linear system, based on gradient-descent optimization technique, aiming to produce a sequence of well-approximate least-squares solutions. Here, we consider least-squares solutions in a full generality, that is, we measure any related error through an arbitrary vector norm induced from weighted positive definite matrices W. It turns out that when the system has a unique solution, the proposed algorithm produces approximated solutions converging to the unique solution. When the system is inconsistent, the sequence of residual norms converges to the weighted least-squares error. Our work includes the usual least-squares solution when W=I. Numerical experiments are performed to validate the capability of the algorithm. Moreover, the performance of this algorithm is better than that of recent gradient-based iterative algorithms in both iteration numbers and computational time.https://www.aimspress.com/article/doi/10.3934/math.2023596?viewType=HTMLgradient-descentiterative methodleast-squares solutionweighted normconvergence analysis |
| spellingShingle | Kanjanaporn Tansri Pattrawut Chansangiam Gradient-descent iterative algorithm for solving exact and weighted least-squares solutions of rectangular linear systems AIMS Mathematics gradient-descent iterative method least-squares solution weighted norm convergence analysis |
| title | Gradient-descent iterative algorithm for solving exact and weighted least-squares solutions of rectangular linear systems |
| title_full | Gradient-descent iterative algorithm for solving exact and weighted least-squares solutions of rectangular linear systems |
| title_fullStr | Gradient-descent iterative algorithm for solving exact and weighted least-squares solutions of rectangular linear systems |
| title_full_unstemmed | Gradient-descent iterative algorithm for solving exact and weighted least-squares solutions of rectangular linear systems |
| title_short | Gradient-descent iterative algorithm for solving exact and weighted least-squares solutions of rectangular linear systems |
| title_sort | gradient descent iterative algorithm for solving exact and weighted least squares solutions of rectangular linear systems |
| topic | gradient-descent iterative method least-squares solution weighted norm convergence analysis |
| url | https://www.aimspress.com/article/doi/10.3934/math.2023596?viewType=HTML |
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