On the perturbation analysis of the maximal solution for the matrix equation X−∑i=1mAi∗X−1Ai+∑j=1nBj∗X−1Bj=I $$ X-\overset{m}{\sum \limits_{i=1}}{A}_i^{\ast}\kern0.1em {X}^{-1}\kern0.1em {A}_i+\sum \limits_{j=1}^n{B}_j^{\ast}\kern0.1em {X}^{-1}\kern0.1em {B}_j=I $$
Abstract In this paper, we study the perturbation estimate of the maximal solution for the matrix equation X−∑i=1mAi∗X−1Ai+∑j=1nBj∗X−1Bj=I $$ X-\overset{m}{\sum \limits_{i=1}}{A}_i^{\ast}\kern0.1em {X}^{-1}\kern0.1em {A}_i+\sum \limits_{j=1}^n{B}_j^{\ast}\kern0.1em {X}^{-1}\kern0.1em {B}_j=I $$ usin...
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
SpringerOpen
2020-01-01
|
| Series: | Journal of the Egyptian Mathematical Society |
| Subjects: | |
| Online Access: | https://doi.org/10.1186/s42787-019-0052-7 |
| _version_ | 1819056203096915968 |
|---|---|
| author | Mohamed A. Ramadan Naglaa M. El–Shazly |
| author_facet | Mohamed A. Ramadan Naglaa M. El–Shazly |
| author_sort | Mohamed A. Ramadan |
| collection | DOAJ |
| description | Abstract In this paper, we study the perturbation estimate of the maximal solution for the matrix equation X−∑i=1mAi∗X−1Ai+∑j=1nBj∗X−1Bj=I $$ X-\overset{m}{\sum \limits_{i=1}}{A}_i^{\ast}\kern0.1em {X}^{-1}\kern0.1em {A}_i+\sum \limits_{j=1}^n{B}_j^{\ast}\kern0.1em {X}^{-1}\kern0.1em {B}_j=I $$ using the differentiation of matrices. We derive the differential bound for this maximal solution. Moreover, we present a perturbation estimate and an error bound for this maximal solution. Finally, a numerical example is given to clarify the reliability of our obtained results. |
| first_indexed | 2024-12-21T13:19:40Z |
| format | Article |
| id | doaj.art-c57675eacbdd4c32b6d1821eb4d1a9ed |
| institution | Directory Open Access Journal |
| issn | 2090-9128 |
| language | English |
| last_indexed | 2024-12-21T13:19:40Z |
| publishDate | 2020-01-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Journal of the Egyptian Mathematical Society |
| spelling | doaj.art-c57675eacbdd4c32b6d1821eb4d1a9ed2022-12-21T19:02:37ZengSpringerOpenJournal of the Egyptian Mathematical Society2090-91282020-01-0128111310.1186/s42787-019-0052-7On the perturbation analysis of the maximal solution for the matrix equation X−∑i=1mAi∗X−1Ai+∑j=1nBj∗X−1Bj=I $$ X-\overset{m}{\sum \limits_{i=1}}{A}_i^{\ast}\kern0.1em {X}^{-1}\kern0.1em {A}_i+\sum \limits_{j=1}^n{B}_j^{\ast}\kern0.1em {X}^{-1}\kern0.1em {B}_j=I $$Mohamed A. Ramadan0Naglaa M. El–Shazly1Department of Mathematics and Computer Science, Faculty of Science, Menoufia UniversityDepartment of Mathematics and Computer Science, Faculty of Science, Menoufia UniversityAbstract In this paper, we study the perturbation estimate of the maximal solution for the matrix equation X−∑i=1mAi∗X−1Ai+∑j=1nBj∗X−1Bj=I $$ X-\overset{m}{\sum \limits_{i=1}}{A}_i^{\ast}\kern0.1em {X}^{-1}\kern0.1em {A}_i+\sum \limits_{j=1}^n{B}_j^{\ast}\kern0.1em {X}^{-1}\kern0.1em {B}_j=I $$ using the differentiation of matrices. We derive the differential bound for this maximal solution. Moreover, we present a perturbation estimate and an error bound for this maximal solution. Finally, a numerical example is given to clarify the reliability of our obtained results.https://doi.org/10.1186/s42787-019-0052-7Nonlinear matrix equationMaximal positive solutionIterationMatrix differentiationPerturbation bound |
| spellingShingle | Mohamed A. Ramadan Naglaa M. El–Shazly On the perturbation analysis of the maximal solution for the matrix equation X−∑i=1mAi∗X−1Ai+∑j=1nBj∗X−1Bj=I $$ X-\overset{m}{\sum \limits_{i=1}}{A}_i^{\ast}\kern0.1em {X}^{-1}\kern0.1em {A}_i+\sum \limits_{j=1}^n{B}_j^{\ast}\kern0.1em {X}^{-1}\kern0.1em {B}_j=I $$ Journal of the Egyptian Mathematical Society Nonlinear matrix equation Maximal positive solution Iteration Matrix differentiation Perturbation bound |
| title | On the perturbation analysis of the maximal solution for the matrix equation X−∑i=1mAi∗X−1Ai+∑j=1nBj∗X−1Bj=I $$ X-\overset{m}{\sum \limits_{i=1}}{A}_i^{\ast}\kern0.1em {X}^{-1}\kern0.1em {A}_i+\sum \limits_{j=1}^n{B}_j^{\ast}\kern0.1em {X}^{-1}\kern0.1em {B}_j=I $$ |
| title_full | On the perturbation analysis of the maximal solution for the matrix equation X−∑i=1mAi∗X−1Ai+∑j=1nBj∗X−1Bj=I $$ X-\overset{m}{\sum \limits_{i=1}}{A}_i^{\ast}\kern0.1em {X}^{-1}\kern0.1em {A}_i+\sum \limits_{j=1}^n{B}_j^{\ast}\kern0.1em {X}^{-1}\kern0.1em {B}_j=I $$ |
| title_fullStr | On the perturbation analysis of the maximal solution for the matrix equation X−∑i=1mAi∗X−1Ai+∑j=1nBj∗X−1Bj=I $$ X-\overset{m}{\sum \limits_{i=1}}{A}_i^{\ast}\kern0.1em {X}^{-1}\kern0.1em {A}_i+\sum \limits_{j=1}^n{B}_j^{\ast}\kern0.1em {X}^{-1}\kern0.1em {B}_j=I $$ |
| title_full_unstemmed | On the perturbation analysis of the maximal solution for the matrix equation X−∑i=1mAi∗X−1Ai+∑j=1nBj∗X−1Bj=I $$ X-\overset{m}{\sum \limits_{i=1}}{A}_i^{\ast}\kern0.1em {X}^{-1}\kern0.1em {A}_i+\sum \limits_{j=1}^n{B}_j^{\ast}\kern0.1em {X}^{-1}\kern0.1em {B}_j=I $$ |
| title_short | On the perturbation analysis of the maximal solution for the matrix equation X−∑i=1mAi∗X−1Ai+∑j=1nBj∗X−1Bj=I $$ X-\overset{m}{\sum \limits_{i=1}}{A}_i^{\ast}\kern0.1em {X}^{-1}\kern0.1em {A}_i+\sum \limits_{j=1}^n{B}_j^{\ast}\kern0.1em {X}^{-1}\kern0.1em {B}_j=I $$ |
| title_sort | on the perturbation analysis of the maximal solution for the matrix equation x ∑i 1mai∗x 1ai ∑j 1nbj∗x 1bj i x overset m sum limits i 1 a i ast kern0 1em x 1 kern0 1em a i sum limits j 1 n b j ast kern0 1em x 1 kern0 1em b j i |
| topic | Nonlinear matrix equation Maximal positive solution Iteration Matrix differentiation Perturbation bound |
| url | https://doi.org/10.1186/s42787-019-0052-7 |
| work_keys_str_mv | AT mohamedaramadan ontheperturbationanalysisofthemaximalsolutionforthematrixequationxi1maix1aij1nbjx1bjixoversetmsumlimitsi1aiastkern01emx1kern01emaisumlimitsj1nbjastkern01emx1kern01embji AT naglaamelshazly ontheperturbationanalysisofthemaximalsolutionforthematrixequationxi1maix1aij1nbjx1bjixoversetmsumlimitsi1aiastkern01emx1kern01emaisumlimitsj1nbjastkern01emx1kern01embji |