On solution and perturbation estimates for the nonlinear matrix equation $$X-A^{*}e^{X}A=I$$ X - A ∗ e X A = I
Abstract This work incorporates an efficient inversion free iterative scheme into Newton’s method to solve Newton’s step regardless of the singularity of the Fr $${\acute{\text {e}}}$$ e ´ chet derivative. The proposed iterative scheme is constructed by extending the idea of the foundational form of...
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| Format: | Article |
| Language: | English |
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SpringerOpen
2022-09-01
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| Series: | Journal of the Egyptian Mathematical Society |
| Subjects: | |
| Online Access: | https://doi.org/10.1186/s42787-022-00152-z |
| _version_ | 1828175368750628864 |
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| author | Chacha S. Chacha |
| author_facet | Chacha S. Chacha |
| author_sort | Chacha S. Chacha |
| collection | DOAJ |
| description | Abstract This work incorporates an efficient inversion free iterative scheme into Newton’s method to solve Newton’s step regardless of the singularity of the Fr $${\acute{\text {e}}}$$ e ´ chet derivative. The proposed iterative scheme is constructed by extending the idea of the foundational form of the conjugate gradient method. Moreover, the resulting scheme is refined and employed to obtain a symmetric solution of the nonlinear matrix equation $$X-A^{*}e^{X}A=I.$$ X - A ∗ e X A = I . Furthermore, explicit expressions for the perturbation and residual bound estimates of the approximate positive definite solution are derived. Finally, five numerical case studies provided confirm both the preciseness of theoretical results and the effectiveness of the propounded iterative method. |
| first_indexed | 2024-04-12T04:26:24Z |
| format | Article |
| id | doaj.art-0f5064e9222e4fcab5bdcca216e7f42a |
| institution | Directory Open Access Journal |
| issn | 2090-9128 |
| language | English |
| last_indexed | 2024-04-12T04:26:24Z |
| publishDate | 2022-09-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Journal of the Egyptian Mathematical Society |
| spelling | doaj.art-0f5064e9222e4fcab5bdcca216e7f42a2022-12-22T03:48:03ZengSpringerOpenJournal of the Egyptian Mathematical Society2090-91282022-09-0130111810.1186/s42787-022-00152-zOn solution and perturbation estimates for the nonlinear matrix equation $$X-A^{*}e^{X}A=I$$ X - A ∗ e X A = IChacha S. Chacha0Department of Mathematics, Physics and Informatcs, Mkwawa University College of EducationAbstract This work incorporates an efficient inversion free iterative scheme into Newton’s method to solve Newton’s step regardless of the singularity of the Fr $${\acute{\text {e}}}$$ e ´ chet derivative. The proposed iterative scheme is constructed by extending the idea of the foundational form of the conjugate gradient method. Moreover, the resulting scheme is refined and employed to obtain a symmetric solution of the nonlinear matrix equation $$X-A^{*}e^{X}A=I.$$ X - A ∗ e X A = I . Furthermore, explicit expressions for the perturbation and residual bound estimates of the approximate positive definite solution are derived. Finally, five numerical case studies provided confirm both the preciseness of theoretical results and the effectiveness of the propounded iterative method.https://doi.org/10.1186/s42787-022-00152-zNewton’s methodIterative methodPerturbation estimateSymmetric solutionNonlinear matrix equation |
| spellingShingle | Chacha S. Chacha On solution and perturbation estimates for the nonlinear matrix equation $$X-A^{*}e^{X}A=I$$ X - A ∗ e X A = I Journal of the Egyptian Mathematical Society Newton’s method Iterative method Perturbation estimate Symmetric solution Nonlinear matrix equation |
| title | On solution and perturbation estimates for the nonlinear matrix equation $$X-A^{*}e^{X}A=I$$ X - A ∗ e X A = I |
| title_full | On solution and perturbation estimates for the nonlinear matrix equation $$X-A^{*}e^{X}A=I$$ X - A ∗ e X A = I |
| title_fullStr | On solution and perturbation estimates for the nonlinear matrix equation $$X-A^{*}e^{X}A=I$$ X - A ∗ e X A = I |
| title_full_unstemmed | On solution and perturbation estimates for the nonlinear matrix equation $$X-A^{*}e^{X}A=I$$ X - A ∗ e X A = I |
| title_short | On solution and perturbation estimates for the nonlinear matrix equation $$X-A^{*}e^{X}A=I$$ X - A ∗ e X A = I |
| title_sort | on solution and perturbation estimates for the nonlinear matrix equation x a e x a i x a ∗ e x a i |
| topic | Newton’s method Iterative method Perturbation estimate Symmetric solution Nonlinear matrix equation |
| url | https://doi.org/10.1186/s42787-022-00152-z |
| work_keys_str_mv | AT chachaschacha onsolutionandperturbationestimatesforthenonlinearmatrixequationxaexaixaexai |