Skip to main content
SpringerLink
Log in

An Accelerated Jacobi-Gradient Iterative Algorithm to Solve the Matrix Equation \({\varvec{A}}\boldsymbol{ Z}-\overline{\boldsymbol{ Z}}{\varvec{B}}={\varvec{C}}\)

  • Research Paper
  • Published:
Iranian Journal of Science Aims and scope Submit manuscript

Abstract

In this paper, we present an accelerated Jacobi-gradient-based iterative AJGI algorithm to solve the matrix equation \(A{ Z}-\overline{ Z}B=C\), which is based on algorithms presented by Bayoumi (Appl Math Inf Sci, 2021). The iterative solution converges to the exact solution for any initial value under appropriate assumptions. Numerical examples are provided to support the suggested approach and verify its effectiveness and accuracy when compared to a recent one previously described in Bayoumi (Appl Math Inf Sci, 2021).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

Availability of Data and Materials

Because no datasets were collected or processed during the current study, and data sharing was not applicable to this paper.

References

  • Ameri A, Beik FPA (2018) Delayed over-relaxation in iterative schemes to solve rank deficient linear system of (matrix) equations. Filomat 32(9):3181–3198

    Article  MathSciNet  Google Scholar 

  • Bayoumi AME (2021) Jacobi-gradient iterative algorithm for solving the matrix equation \(AX-\overline{X}B=C\). Appl Math Inf Sci, (accepted)

  • Bayoumi AME (2023) Relaxed gradient-based iterative solutions to coupled Sylvester-conjugate transpose matrix equations of two unknowns. Eng Comput 40(9–10):2776–2793

    Article  Google Scholar 

  • Bayoumi AME, Ramadan MA (2018) An accelerated gradient-based iterative algorithm for solving extended Sylvester-conjugate matrix equations. Trans Inst Meas Control 40(1):341–347

    Article  Google Scholar 

  • Bayoumi AME, Ramadan MA (2022) R, S conjugate solution to coupled Sylvester complex matrix equations with conjugate of two unknowns. Automatika 63(3):454–462. https://doi.org/10.1080/00051144.2022.2052399

    Article  Google Scholar 

  • Bayoumi AME (2024) A shift-splitting Jacobi-gradient iterative algorithm for solving the matrix equation AV-VB=C. Opt Control Appl Method,1–10. https://doi.org/10.1002/oca.3112.

  • Beik FPA (2014) A modified iterative algorithm for the (Hermitian) reflexive solution of the generalized Sylvester matrix equation. Trans Inst Meas Control 36(6):815–827

    Article  Google Scholar 

  • Beik FPA, Salkuyeh DK (2017) A cyclic iterative approach and its modified version to solve coupled Sylvester-transpose matrix equations. Linear Multilinear Algebra 65(12):2406–2423

    Article  MathSciNet  Google Scholar 

  • Chen Z, Chen X (2022) Modification on the convergence results of the Sylvester matrix equation AX+XB=C. J Frankl Inst 359:3126–3147

    Article  MathSciNet  Google Scholar 

  • Ding F, Chen T (2005) Gradient based iterative algorithms for solving a class of matrix equations. IEEE Trans Automatic Control 50:1216–1221

    Article  MathSciNet  Google Scholar 

  • El-Shazly NM, Ramadan MA, El-Sharway MH (2022) A relaxed gradient based iterative algorithm of the generalized Sylvester-conjugate matrix equation over centro-symmetric and centro-Hermitian matrices. J Vib Control 29(19–20):1–17

    MathSciNet  Google Scholar 

  • Gu C, Fan WW (2008) Improved gradient iterative algorithms for solving Lyapunov matrix equations. J Shanghai Univ (engl Edit) 12(5):395–399

    Article  MathSciNet  Google Scholar 

  • Gu C, Xue H (2009) A shift-splitting hierarchical identification method for solving Lyapunov matrix equations. Linear Algebra Appl 430(5–6):1517–1530

    Article  MathSciNet  Google Scholar 

  • Li SK, Huang TZ (2011) A shift-splitting Jacobi-gradient algorithm for Lyapunov matrix equations arising from control theory. J Comput Anal Appl 13(7):1246–1257

    MathSciNet  Google Scholar 

  • Lv L, Jinbo C, Zhe Z, Baowen W, Lei Z (2021) A numerical solution of a class of periodic coupled matrix equations. J Frankl Inst 358(3):2039–2059

    Article  MathSciNet  Google Scholar 

  • Lv L, Jinbo C, Lei Z, Fengrui Z (2022) Gradient-based neural networks for solving periodic Sylvester matrix equations. J Frankl Inst 359(18):10849–10866

    Article  MathSciNet  Google Scholar 

  • Niu Q, Wang X, Lu LZ (2011) A relaxed gradient-based algorithm for solving Sylvester equations. Asian J Control 13(3):461–464

    Article  MathSciNet  Google Scholar 

  • Ramadan MA, Bayoumi AME (2018) A modified gradient-based algorithm for solving extended Sylvester-conjugate matrix equations. Asian J Control 20(1):228–235

    Article  MathSciNet  Google Scholar 

  • Ramadan MA, El-Danaf TS, Bayoumi AME (2014) A relaxed gradient-based algorithm for solving extended Sylvester-conjugate matrix equations. Asian J Control 16(5):1334–1341

    Article  MathSciNet  Google Scholar 

  • Rao N, Raiz M, Mursaleen MA, Mishra VN (2023) Approximation properties of extended beta-type Szász-Mirakjan operators. Iran J Sci 47:1771–1781

    Article  MathSciNet  Google Scholar 

  • Salkuyeh DK, Beik FPA (2014) On the gradient-based algorithm for solving the general coupled matrix equations. Trans Inst Meas Control 36(3):375–381

    Article  Google Scholar 

  • Salkuyeh DK, Beik FPA (2015) Minimum norm least-squares solution to general complex coupled linear matrix equations via iteration. Filomat 29(6):1389–1407

    Article  MathSciNet  Google Scholar 

  • Sasaki N, Chansangiam P (1831) Modified Jacobi-gradient iterative method for generalized Sylvester matrix equation. Symmetry 2020(12):1–15

    Google Scholar 

  • Tian Z, Tian M, Gu C, Hao X (2017) An accelerated Jacobi-gradient based iterative algorithm for solving Sylvester matrix equations. Filomat 31(8):2381–2390

    Article  MathSciNet  Google Scholar 

  • Wang W, Song C (2023) A novel iterative method for solving the coupled Sylvester conjugate matrix equations and its application in antilinear system. J Appl Anal Comput 13(1):249–274

    MathSciNet  Google Scholar 

  • Wang X, Dai L, Liao D (2012) A modified gradient-based algorithm for solving Sylvester equations. Appl Math Comput 218:5620–5628

    MathSciNet  Google Scholar 

  • Wang W, Qu G, Song C (2023) Cyclic gradient based iterative algorithm for a class of generalized coupled Sylvester-conjugate matrix equations. J Frankl Inst 360:7206–7229

    Article  MathSciNet  Google Scholar 

  • Wu AG, Zeng X, Duan GR, Wu WJ (2010) Iterative solutions to the extended Sylvester-conjugate matrix equation. Appl Math Comput 217(1):130–142

    MathSciNet  Google Scholar 

  • Xie YJ, Ma CF (2016) The accelerated gradient based iterative algorithm for solving a class of generalized Sylvester-transpose matrix equation. Appl Math Comput 273:1257–1269

    MathSciNet  Google Scholar 

Download references

Acknowledgements

Not applicable.

Funding

The author declares that no funds, grants, or other support were received during the preparation of this manuscript.

Author information

Authors and Affiliations

  1. Department of Mathematics, Faculty of Education, Ain Shams University, Cairo, Egypt

    Ahmed M. E. Bayoumi

Authors
  1. Ahmed M. E. Bayoumi
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

Ahmed M. E. Bayoumi provided conceptualization, resources, methodology, formal analysis, validation, writing—original draft preparation, visualization and reviewing, writing the algorithm, numerical examples, writing the draft and the last version of the manuscript.

Corresponding author

Correspondence to Ahmed M. E. Bayoumi.

Ethics declarations

Conflict of interest

There are no conflicts of interest declared by the authors.

Ethical Approval and Consent to Participate

Not applicable.

Consent for Publication

Not applicable.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bayoumi, A.M.E. An Accelerated Jacobi-Gradient Iterative Algorithm to Solve the Matrix Equation \({\varvec{A}}\boldsymbol{ Z}-\overline{\boldsymbol{ Z}}{\varvec{B}}={\varvec{C}}\). Iran J Sci (2024). https://doi.org/10.1007/s40995-024-01629-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s40995-024-01629-5

Keywords

JEL Classification

Search

Navigation