Communications on Applied Mathematics and Computation ›› 2022, Vol. 4 ›› Issue (3): 1158-1174.doi: 10.1007/s42967-021-00152-3

• ORIGINAL PAPER • 上一篇    

On Iterative Algorithm and Perturbation Analysis for the Nonlinear Matrix Equation

Chacha Stephen Chacha   

  1. Department of Mathematics, Physics and Informatics, Mkwawa University College of Education, P.O. Box 2513, Iringa, Tanzania
  • 收稿日期:2020-11-26 修回日期:2021-05-19 出版日期:2022-09-20 发布日期:2022-07-04
  • 通讯作者: Chacha Stephen Chacha,E-mail:chchstephen@muce.ac.tz,chchstephen@yahoo.com E-mail:chchstephen@muce.ac.tz,chchstephen@yahoo.com
  • 基金资助:
    The author thanks three anonymous reviewers for their valuable observations, suggestions, and quite useful constructive comments.

On Iterative Algorithm and Perturbation Analysis for the Nonlinear Matrix Equation

Chacha Stephen Chacha   

  1. Department of Mathematics, Physics and Informatics, Mkwawa University College of Education, P.O. Box 2513, Iringa, Tanzania
  • Received:2020-11-26 Revised:2021-05-19 Online:2022-09-20 Published:2022-07-04
  • Contact: Chacha Stephen Chacha,E-mail:chchstephen@muce.ac.tz,chchstephen@yahoo.com E-mail:chchstephen@muce.ac.tz,chchstephen@yahoo.com
  • Supported by:
    The author thanks three anonymous reviewers for their valuable observations, suggestions, and quite useful constructive comments.

摘要: In this study, an iterative algorithm is proposed to solve the nonlinear matrix equation \begin{document}$ X+A^{*}{e}^{X}A=I_{n} $\end{document}. Explicit expressions for mixed and componentwise condition numbers with their upper bounds are derived to measure the sensitivity of the considered nonlinear matrix equation. Comparative analysis for the derived condition numbers and the proposed algorithm are presented. The proposed iterative algorithm reduces the number of iterations significantly when incorporated with exact line searches. Componentwise condition number seems more reliable to detect the sensitivity of the considered equation than mixed condition number as validated by numerical examples.

关键词: Mixed condition number, Componentwise condition number, Iterative algorithm, Perturbation analysis, Exact line search

Abstract: In this study, an iterative algorithm is proposed to solve the nonlinear matrix equation \begin{document}$ X+A^{*}{e}^{X}A=I_{n} $\end{document}. Explicit expressions for mixed and componentwise condition numbers with their upper bounds are derived to measure the sensitivity of the considered nonlinear matrix equation. Comparative analysis for the derived condition numbers and the proposed algorithm are presented. The proposed iterative algorithm reduces the number of iterations significantly when incorporated with exact line searches. Componentwise condition number seems more reliable to detect the sensitivity of the considered equation than mixed condition number as validated by numerical examples.

Key words: Mixed condition number, Componentwise condition number, Iterative algorithm, Perturbation analysis, Exact line search

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