The explicit formula for Gauss-Jordan elimination applied to flexible systems
Flexible systems are obtained from systems of linear equations by adding to the elements of the coefficient matrix and the right-hand side scalar neutrices, which are convex groups of (non-standard) real numbers. The neutrices model imprecisions, giving rise to calculation rules extending informal e...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
De Gruyter
2022-06-01
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| Series: | Special Matrices |
| Subjects: | |
| Online Access: | https://doi.org/10.1515/spma-2022-0168 |
| _version_ | 1818018066332647424 |
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| author | Tran Nam Van Justino Júlia Berg Imme van den |
| author_facet | Tran Nam Van Justino Júlia Berg Imme van den |
| author_sort | Tran Nam Van |
| collection | DOAJ |
| description | Flexible systems are obtained from systems of linear equations by adding to the elements of the coefficient matrix and the right-hand side scalar neutrices, which are convex groups of (non-standard) real numbers. The neutrices model imprecisions, giving rise to calculation rules extending informal error calculus. Stability conditions for flexible systems are given in terms of relative imprecision and size of determinants. We then apply the explicit formula for the elements of the successive intermediate matrices of the Gauss-Jordan elimination procedure to find the solution of flexible systems, keeping track of the error terms at every stage. The solution respects the original imprecisions in the right-hand side and is the same as the one given by Cramer’s rule. |
| first_indexed | 2024-04-14T07:35:19Z |
| format | Article |
| id | doaj.art-dd40d5dec3724b8cba60a88dafeda13d |
| institution | Directory Open Access Journal |
| issn | 2300-7451 |
| language | English |
| last_indexed | 2024-04-14T07:35:19Z |
| publishDate | 2022-06-01 |
| publisher | De Gruyter |
| record_format | Article |
| series | Special Matrices |
| spelling | doaj.art-dd40d5dec3724b8cba60a88dafeda13d2022-12-22T02:05:43ZengDe GruyterSpecial Matrices2300-74512022-06-0110136639310.1515/spma-2022-0168The explicit formula for Gauss-Jordan elimination applied to flexible systemsTran Nam Van0Justino Júlia1Berg Imme van den2Faculty of Applied Sciences, HCMC University of Technology and Education, Ho Chi Minh City, VietnamSetúbal School of Technology, Polytechnic Institute of Setúbal, Setúbal, PortugalResearch Center in Mathematics and Applications, University of Évora, Évora, PortugalFlexible systems are obtained from systems of linear equations by adding to the elements of the coefficient matrix and the right-hand side scalar neutrices, which are convex groups of (non-standard) real numbers. The neutrices model imprecisions, giving rise to calculation rules extending informal error calculus. Stability conditions for flexible systems are given in terms of relative imprecision and size of determinants. We then apply the explicit formula for the elements of the successive intermediate matrices of the Gauss-Jordan elimination procedure to find the solution of flexible systems, keeping track of the error terms at every stage. The solution respects the original imprecisions in the right-hand side and is the same as the one given by Cramer’s rule.https://doi.org/10.1515/spma-2022-0168gauss-jordan eliminationflexible systemsstabilityexternal numbers03h0515a0615b3365g99 |
| spellingShingle | Tran Nam Van Justino Júlia Berg Imme van den The explicit formula for Gauss-Jordan elimination applied to flexible systems Special Matrices gauss-jordan elimination flexible systems stability external numbers 03h05 15a06 15b33 65g99 |
| title | The explicit formula for Gauss-Jordan elimination applied to flexible systems |
| title_full | The explicit formula for Gauss-Jordan elimination applied to flexible systems |
| title_fullStr | The explicit formula for Gauss-Jordan elimination applied to flexible systems |
| title_full_unstemmed | The explicit formula for Gauss-Jordan elimination applied to flexible systems |
| title_short | The explicit formula for Gauss-Jordan elimination applied to flexible systems |
| title_sort | explicit formula for gauss jordan elimination applied to flexible systems |
| topic | gauss-jordan elimination flexible systems stability external numbers 03h05 15a06 15b33 65g99 |
| url | https://doi.org/10.1515/spma-2022-0168 |
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