Skip to main content
Springer Nature Link
Log in

The use of homotopy analysis method for solving generalized Sylvester matrix equation with applications

  • Original Article
  • Published:
Engineering with Computers Aims and scope Submit manuscript

Abstract

In this research, we introduce and analyze homotopy analysis method (HAM) for solving approximately linear matrix equation \( \sum \limits \limits _{i=1}^{s}A_iXB_i+C=\mathbf{0} \), where \( A_i,\;B_i \;(i=1,\ldots ,s), \; C \in \mathbb {C}^{n \times n} \) and \(X\in \mathbb {C}^{n \times n} \) must be determined. In this method we consider a convergence control parameter \( \delta \), and then we determine the optimum value of \( \delta \) for obtaining fast convergence method. Moreover, we obtain the corresponding spectral radius of convergence factor of HAM method. Finally, we will apply this method to solve some test problems to support the theoretical results.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14

Similar content being viewed by others

Explore related subjects

Discover the latest articles and news from researchers in related subjects, suggested using machine learning.

References

  1. Abbasbandy S (2006) The application of homotopy analysis method to nonlinear equations arising in heat transfer. Phys Lett A 360:109–113

    MathSciNet  MATH  Google Scholar 

  2. Abbasbandy S (2006) Iterated He’s homotopy perturbation method for quadratic Riccati differential equation. Appl Math Comput 175:581–589

    MathSciNet  MATH  Google Scholar 

  3. Abbasbandy S (2006) Homotopy perturbation method for quadratic Riccati differential equation and comparison with Adomian’s decomposition method. Appl Math Comput 172:485–490

    MathSciNet  MATH  Google Scholar 

  4. Beléndez A, Beléndez T, Márquez A, Neipp C (2008) Application of He’s homotopy perturbation method to conservative truly nonlinear oscillators. Chaos Solitons Fractals 37:770–780

    MATH  Google Scholar 

  5. Bouhamidi A, Jbilou K (2007) Sylvester Tikhonov-regularization methods in image restoration. J Comput Appl Math 206(1):86–98

    MathSciNet  MATH  Google Scholar 

  6. Bouhamidi A, Jbilou K (2008) A note on the numerical approximate solutions for generalized Sylvester matrix equations with applications. Appl Math Comput 206:687–694

    MathSciNet  MATH  Google Scholar 

  7. Benner P (2008) Large-scale matrix equations of special type. Numer Linear Algebra Appl 15:747–754

    MathSciNet  Google Scholar 

  8. Bai Z-Z, Guo X-X, Xu S-F (2006) Alternately linearized implicit iteration methods for the minimal nonnegative solutions of nonsymmetric algebraic Riccati equations. Numer Linear Algebra Appl 13:655–674

    MathSciNet  MATH  Google Scholar 

  9. Beik FPA, Salkuyeh DK (2011) On the global Krylov subspace methods for solving general coupled matrix equations. Comput Math Appl 62:4605–4613

    MathSciNet  MATH  Google Scholar 

  10. Dilip ASA, Pillai HK (2016) Characterization of solutions of non-symmetric algebraic Riccati equations. Linear Algebra Appl 507:356–372

    MathSciNet  MATH  Google Scholar 

  11. Ding F, Liu XP, Ding J (2008) Iterative solutions of the generalized Sylvester matrix equations by using the hierarchical identification principle. Appl Math Comput 197(1):41–50

    MathSciNet  MATH  Google Scholar 

  12. Ding F, Zhang H (2014) Gradient-based iterative algorithm for a class of the coupled matrix equations related to control systems. IET Control Theory A 8(15):1588–1595

    MathSciNet  Google Scholar 

  13. Ding J, Liu Y, Ding F (2010) Iterative solutions to matrix equations of the form \( A_iXB_i = F_i \). Comput Math Appl 59(11):3500–3507

    MathSciNet  MATH  Google Scholar 

  14. Dehghan M, Hajarian M (2012) Two iterative algorithms for solving coupled matrix equations over reflexive and anti-reflexive matrices. Comput Appl Math 31:353–371

    MathSciNet  MATH  Google Scholar 

  15. Dehghan M, Hajarian M (2011) Analysis of an iterative algorithm to solve the generalized coupled Sylvester matrix equations. Appl Math Model 35(7):3285–3300

    MathSciNet  MATH  Google Scholar 

  16. Dehghan M, Shakeri F (2008) Solution of an integro-differential equation arising in oscillating magnetic fields using He’s homotopy perturbation method. Prog Electromagn Res 78:361–376

    Google Scholar 

  17. Dehghan M, Manafian J, Saadatmandi A (2010) Solving nonlinear fractional partial differential equations using the homotopy analysis method. Numer Methods Partial Differ Equ 26(2):448–479

    MathSciNet  MATH  Google Scholar 

  18. Dehghan M, Salehi R (2011) The use of homotopy analysis method to solve the time-dependent nonlinear Eikonal partial differential equation. Z Naturforsch A 66(5):259–271

    Google Scholar 

  19. Dehghan M, Shirilord A (2019) A generalized modified Hermitian and skew-Hermitian splitting (GMHSS) method for solving complex Sylvester matrix equation. Appl Math Comput 348:632–651

    MathSciNet  MATH  Google Scholar 

  20. Dehghan M, Shirilord A (2019) Solving complex Sylvester matrix equation by accelerated double-step scale splitting (ADSS) method. Eng Comput. https://doi.org/10.1007/s00366-019-00838-6

    Article  MATH  Google Scholar 

  21. Dehghan M, Shirilord A (2020) HSS-like method for solving complex nonlinear Yang-Baxter matrix equation. Eng Comput. https://doi.org/10.1007/s00366-020-00947-7

    Article  MATH  Google Scholar 

  22. He J-H (1999) Homotopy perturbation technique. Comput Methods Appl Mech Eng 178:257–262

    MathSciNet  MATH  Google Scholar 

  23. He J-H (2003) Homotopy perturbation method: a new non-linear analytical technique. Appl Math Comput 135:73–79

    MathSciNet  MATH  Google Scholar 

  24. He J-H (2005) Application of homotopy perturbation method to nonlinear wave equations. Chaos Solitons Fractals 26:695–700

    MATH  Google Scholar 

  25. He J-H (2004) The homotopy perturbation method for nonlinear oscillators with discontinuities. Appl Math Comput 151:287–292

    MathSciNet  MATH  Google Scholar 

  26. He J-H (2004) Comparison of homotopy perturbation method and homotopy analysis method. Appl Math Comput 156:527–539

    MathSciNet  MATH  Google Scholar 

  27. Hayat T, Khan M, Ayub M (2004) Couette and Poiseuille flows of an Oldroyd 6-constant fluid with magnetic field. J Math Anal Appl 298:225–244

    MathSciNet  MATH  Google Scholar 

  28. Hayat T, Khan M, Asghar S (2004) Homotopy analysis of MHD flows of an Oldroyd 8-constant fluid. Acta Mech 168:213–232

    MATH  Google Scholar 

  29. Hajarian M (2014) Extending LSQR methods to solve the generalized Sylvester-transpose and periodic Sylvester matrix equations. Math Methods Appl Sci 37:2017–2028

    MathSciNet  MATH  Google Scholar 

  30. He ZH (2019) Some new results on a system of Sylvester-type quaternion matrix equations. Linear Multilinear A. https://doi.org/10.1080/03081087.2019.1704213

    Article  Google Scholar 

  31. He ZH, Wang QW, Zhang Y (2018) A system of quaternary coupled Sylvester-type real quaternion matrix equations. Automatica 87:25–31

    MathSciNet  MATH  Google Scholar 

  32. He ZH, Wang QW, Zhang Y (2019) A simultaneous decomposition for seven matrices with applications. J Comput Appl Math 349:93–113

    MathSciNet  MATH  Google Scholar 

  33. Keramati B (2009) An approach to the solution of linear system of equations by He’s homotopy perturbation method. Chaos Solitons Fractals 41:152–156

    MathSciNet  MATH  Google Scholar 

  34. Liu HK (2011) Application of homotopy perturbation methods for solving systems of linear equations. Appl Math Comput 217:5259–5264

    MathSciNet  MATH  Google Scholar 

  35. Li J-F, Li W, Huang R (2016) An efficient method for solving a matrix least squares problem over a matrix inequality constraint. Comput Optim Appl 63:393–423

    MathSciNet  MATH  Google Scholar 

  36. Lia L, Wang Q-W, Shen S-Q (2015) On positive definite solutions of the nonlinear matrix equations \( X\pm A^*X^qA=Q \). Appl Math Comput 271:556–566

    MathSciNet  Google Scholar 

  37. Liao SJ (2003) Beyond perturbation: introduction to the homotopy analysis method. CRC Press, Chapman & Hall, Boca Raton

    Google Scholar 

  38. Liao SJ (2004) On the homotopy anaylsis method for nonlinear problems. Appl Math Comput 147:499–513

    MathSciNet  MATH  Google Scholar 

  39. Liao SJ (2005) Comparison between the homotopy analysis method and homotopy perturbation method. Appl Math Comput 169:1186–1194

    MathSciNet  MATH  Google Scholar 

  40. Liao SJ (2005) A new branch of solutions of boundary-layer flows over an impermeable stretched plate. Int J Heat Mass Transf 48:2529–2539

    MATH  Google Scholar 

  41. Liao SJ (1999) An explicit, totally analytic approximation of Blasius viscous flow problems. Int J Nonlinear Mech 34:759–778

    MATH  Google Scholar 

  42. Liao SJ (1992) The proposed homotopy analysis technique for the solution of nonlinear problems. PhD dissertation, Shanghai Jiao Tong University

  43. Liao SJ (2010) An optimal homotopy-analysis approach for strongly nonlinear differential equations. Commun Nonlinear Sci Numer Simul 15:2003–2016

    MathSciNet  MATH  Google Scholar 

  44. Liao SJ (2012) The homotopy analysis method in nonlinear differential equations. Higher Education Press & Springer, Beijing

    MATH  Google Scholar 

  45. Liang S, Jeffrey DJ (2009) Comparison of homotopy analysis method and homotopy perturbation method through an evolution equation. Commun Nonlinear Sci Numer Simul 14:4057–4064

    MathSciNet  MATH  Google Scholar 

  46. Noor MA, Noor KI, Khan S, Waseem M (2013) Modified homotopy perturbation method for solving system of linear equations. J Assoc Arab Univ Basic Appl Sci 13:35–37

    Google Scholar 

  47. Robbé M, Sadkane M (2002) A convergence analysis of GMRES and FOM methods for Sylvester equations. Numer Algorithms 30:71–89

    MathSciNet  MATH  Google Scholar 

  48. Saadatmandi A, Dehghan M, Eftekhari A (2009) Application of He’s homotopy perturbation method for non-linear system of second-order boundary value problems. Nonlinear Anal Real World Appl 10:1912–1922

    MathSciNet  MATH  Google Scholar 

  49. Sadeghi A (2015) A new approximation to the solution of the linear matrix equation \( AXB = C \). J Linear Topol Algebra 4:305–315

    MathSciNet  MATH  Google Scholar 

  50. Sadeghi A (2016) A new approach for computing the solution of Sylvester matrix equation. J Interpolat Approx Sci Comput 2:66–76

    MathSciNet  Google Scholar 

  51. Sajid M, Hayat T (2008) Comparison of HAM and HPM methods in nonlinear heat conduction and convection equations. Nonlinear Anal Real World Appl 9:2296–2301

    MathSciNet  MATH  Google Scholar 

  52. Shakeri F, Dehghan M (2008) Solution of the delay differential equations via homotopy perturbation method. Math Comput Model 48:486–498

    MathSciNet  MATH  Google Scholar 

  53. Tan Y, Abbasbandy S (2008) Homotopy analysis method for quadratic Riccati differential equation. Commun Nonlinear Sci Numer Simul 13:539–546

    MATH  Google Scholar 

  54. Wang QW, He ZH, Zhang Y (2019) Constrained two-sided coupled Sylvester-type quaternion matrix equations. Automatica 101:207–213

    MathSciNet  MATH  Google Scholar 

  55. Wang QW (2005) The general solution to a system of real quaternion matrix equations. Comput Math Appl 49:665–675

    MathSciNet  MATH  Google Scholar 

  56. Xu G-P, Wei M-S, Zheng D-S (1998) On solutions of matrix equation \( AXB + CYD = F \). Linear Algebra Appl 279:93–109

    MathSciNet  MATH  Google Scholar 

  57. Yusufoglu E (2009) An improvement to homotopy perturbation method for solving system of linear equations. Comput Math Appl 58:2231–2235

    MathSciNet  MATH  Google Scholar 

  58. Zhou B, Duan GR (2006) A new solution to the generalized Sylvester matrix equation \( AV-EVF=BW \). Syst Control Lett 55:193–198

    MATH  Google Scholar 

  59. Zhou B, Duan GR (2008) On the generalized Sylvester mapping and matrix equations. Syst Control Lett 57:200–208

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors extend their appreciation to reviewers for their valuable suggestions to revise this paper.

Author information

Authors and Affiliations

  1. Department of Applied Mathematics, Faculty of Mathematics and Computer Sciences, Amirkabir University of Technology, No. 424, Hafez Ave., Tehran, 15914, Iran

    Mehdi Dehghan & Akbar Shirilord

Authors
  1. Mehdi Dehghan
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Akbar Shirilord
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Mehdi Dehghan.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Dehghan, M., Shirilord, A. The use of homotopy analysis method for solving generalized Sylvester matrix equation with applications. Engineering with Computers 38, 2699–2716 (2022). https://doi.org/10.1007/s00366-020-01219-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00366-020-01219-0

Keywords

AMS subject classification

Profiles

  1. Mehdi Dehghan View author profile