On solution and perturbation estimates for the nonlinear matrix equation X − AeXA = I
Ali, A. (2022). On solution and perturbation estimates for the nonlinear matrix equation X − AeXA = I. EKB Journal Management System, 30(1), 1-18. doi: 10.1186/s42787-022-00152-z
Ayyaz Ali. "On solution and perturbation estimates for the nonlinear matrix equation X − AeXA = I". EKB Journal Management System, 30, 1, 2022, 1-18. doi: 10.1186/s42787-022-00152-z
Ali, A. (2022). 'On solution and perturbation estimates for the nonlinear matrix equation X − AeXA = I', EKB Journal Management System, 30(1), pp. 1-18. doi: 10.1186/s42787-022-00152-z
Ali, A. On solution and perturbation estimates for the nonlinear matrix equation X − AeXA = I. EKB Journal Management System, 2022; 30(1): 1-18. doi: 10.1186/s42787-022-00152-z

On solution and perturbation estimates for the nonlinear matrix equation X − AeXA = I

Journal of the Egyptian Mathematical Society
Volume 30, Issue 1, 2022, Page 1-18  XML PDF (2.66 MB)
Document Type: Original Article
DOI: 10.1186/s42787-022-00152-z
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Author
Ayyaz Ali
Department of Mathematics, Faculty of Sciences, HITEC University, Taxila, Pakistan
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´echet 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∗eXA = 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.
Keywords
Newton’s method; Iterative method; Perturbation estimate; Symmetric solution; Nonlinear matrix equation
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