Positive Definite Solutions of the Matrix Equation Xr-∑i=1mAi∗X-δiAi=I
We investigate the nonlinear matrix equation Xr-∑i=1mAi∗X-δiAi=I, where r is a positive integer and δi∈(0,1], for i=1,2,…,m. We establish necessary and sufficient conditions for the existence of positive definite solutions of this equation. A sufficient condition for the equation to have a unique po...
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| Format: | Article |
| Language: | English |
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Wiley
2015-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2015/473965 |
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| author | Asmaa M. Al-Dubiban |
| author_facet | Asmaa M. Al-Dubiban |
| author_sort | Asmaa M. Al-Dubiban |
| collection | DOAJ |
| description | We investigate the nonlinear matrix equation Xr-∑i=1mAi∗X-δiAi=I, where r is a positive integer and δi∈(0,1], for i=1,2,…,m. We establish necessary and sufficient conditions for the existence of positive definite solutions of this equation. A sufficient condition for the equation to have a unique positive definite solution is established. An iterative algorithm is provided to compute the positive definite solutions for the equation and error estimate. Finally, some numerical examples are given to show the effectiveness and convergence of this algorithm. |
| format | Article |
| id | doaj-art-b86b8132faab44e389270c19eb9cf916 |
| institution | Kabale University |
| issn | 1085-3375 1687-0409 |
| language | English |
| publishDate | 2015-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Abstract and Applied Analysis |
| spelling | doaj-art-b86b8132faab44e389270c19eb9cf9162025-02-03T05:59:13ZengWileyAbstract and Applied Analysis1085-33751687-04092015-01-01201510.1155/2015/473965473965Positive Definite Solutions of the Matrix Equation Xr-∑i=1mAi∗X-δiAi=IAsmaa M. Al-Dubiban0Faculty of Science and Arts, Qassim University, P.O. Box 1162, Buraydah 51431, Saudi ArabiaWe investigate the nonlinear matrix equation Xr-∑i=1mAi∗X-δiAi=I, where r is a positive integer and δi∈(0,1], for i=1,2,…,m. We establish necessary and sufficient conditions for the existence of positive definite solutions of this equation. A sufficient condition for the equation to have a unique positive definite solution is established. An iterative algorithm is provided to compute the positive definite solutions for the equation and error estimate. Finally, some numerical examples are given to show the effectiveness and convergence of this algorithm.http://dx.doi.org/10.1155/2015/473965 |
| spellingShingle | Asmaa M. Al-Dubiban Positive Definite Solutions of the Matrix Equation Xr-∑i=1mAi∗X-δiAi=I Abstract and Applied Analysis |
| title | Positive Definite Solutions of the Matrix Equation Xr-∑i=1mAi∗X-δiAi=I |
| title_full | Positive Definite Solutions of the Matrix Equation Xr-∑i=1mAi∗X-δiAi=I |
| title_fullStr | Positive Definite Solutions of the Matrix Equation Xr-∑i=1mAi∗X-δiAi=I |
| title_full_unstemmed | Positive Definite Solutions of the Matrix Equation Xr-∑i=1mAi∗X-δiAi=I |
| title_short | Positive Definite Solutions of the Matrix Equation Xr-∑i=1mAi∗X-δiAi=I |
| title_sort | positive definite solutions of the matrix equation xr ∑i 1mai∗x δiai i |
| url | http://dx.doi.org/10.1155/2015/473965 |
| work_keys_str_mv | AT asmaamaldubiban positivedefinitesolutionsofthematrixequationxri1maixdiaii |