# Gradient Iterative Method with Optimal Convergent Factor for Solving a Generalized Sylvester Matrix Equation with Applications to Diffusion Equations

^{*}

## Abstract

**:**

## 1. Introduction

**Method**

**1**

**.**Assume that the matrix Equation (1) has a unique solution X. Construct

**Method**

**2**

**.**Assume that the matrix Equation (1) has a unique solution X. For each $i=1,2,\dots ,p,$ construct,

## 2. Introducing a Gradient Iterative Method

**Method**

**3.**

## 3. Convergence Analysis

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

- (1).
- We have the following error estimates$$\begin{array}{cc}\hfill {\parallel X\left(k\right)-X\parallel}_{F}\phantom{\rule{0.222222em}{0ex}}& \le {\phantom{\rule{0.222222em}{0ex}}\rho \left[T\right]\phantom{\rule{0.277778em}{0ex}}\parallel X(k-1)-X\parallel}_{F},\hfill \end{array}$$$$\begin{array}{cc}\hfill {\parallel X\left(k\right)-X\parallel}_{F}\phantom{\rule{0.222222em}{0ex}}& \le \phantom{\rule{0.222222em}{0ex}}{\rho}^{k}\left[T\right]\phantom{\rule{0.277778em}{0ex}}{\parallel X\left(0\right)-X\parallel}_{F}.\hfill \end{array}$$Moreover, the asymptotic convergence rate of Method 3 is governed by $\rho \left[T\right]$ in (11).
- (2).
- Let $\epsilon >0$ be a satisfactory error. We have ${\parallel X\left(k\right)-X\parallel}_{F}<\u03f5$ after the k-th iteration for any $k\in \mathbb{N}$ that satisfies$$k>\frac{log\u03f5-log{\parallel X\left(0\right)-X\parallel}_{F}}{log\rho \left(T\right)}.$$

**Proof.**

**Theorem**

**3.**

**Proof.**

## 4. The GIO Method for the Sylvester Equation

**Method**

**4.**

**Corollary**

**1.**

- (i)
- The approximate solutions generated by Method 4 converge to the exact solution for any initial value $X\left(0\right)$ if and only if$$0<\tau \phantom{\rule{0.222222em}{0ex}}<\phantom{\rule{0.222222em}{0ex}}\frac{2}{{\parallel Q\parallel}_{2}^{2}}.$$In this case, the spectral radius of the associated iteration matrix $S={I}_{np}-\tau {Q}^{T}Q$ is given by$$\rho \left[S\right]=max\left\{\right|1-\tau {\lambda}_{max}\left({Q}^{T}Q\right)|,|1-\tau {\lambda}_{min}\left({Q}^{T}Q\right)\left|\right\}.$$
- (ii)
- The asymptotic convergence rate of Method 4 is governed by $\rho \left[S\right]$ in (20).
- (iii)
- The optimal value of $\tau >0$ for which Method 4 has the fastest asymptotic convergence rate is determined by$${\tau}_{opt}\phantom{\rule{0.222222em}{0ex}}=\phantom{\rule{0.222222em}{0ex}}\frac{2}{{\lambda}_{max}\left({Q}^{T}Q\right)+{\lambda}_{min}\left({Q}^{T}Q\right)}.$$

**Remark**

**1.**

## 5. Numerical Examples for Generalized Sylvester Matrix Equation

**Example**

**1.**

**Example**

**2.**

**Example**

**3.**

## 6. An Application to Discretization of the Convection-Diffusion Equation

**Method**

**5.**

**Example**

**4.**

**Example**

**5.**

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Benner, P. Factorized solution of Sylvester equations with application in control. In Theory Networks and System; International Symposium of Mathematics: Berlin, Germany, 2014. [Google Scholar]
- Tsui, C.C. On robust observer compensator design. Automatica
**1988**, 24, 687–692. [Google Scholar] [CrossRef] - Van Dooren, P. Reduce order observer: A new algorithm and proof. Syst. Control Lett.
**1984**, 4, 243–251. [Google Scholar] [CrossRef] - Bartels, R.; Stewart, G. Solution of the matrix equation AX + XB = C. Circuits Syst. Signal Process.
**1994**, 13, 820–826. [Google Scholar] [CrossRef] - Sadeghi, A. A new approach for computing the solution of Sylvester matrix equations. J. Interpolat. Approx. Sci. Comput.
**2016**, 2, 66–76. [Google Scholar] [CrossRef] [Green Version] - Li, S.-Y.; Shen, H.-L.; Shao, X.-H. PHSS iterative method for solving generalized Lyapunov equation. Mathematics
**2019**, 7, 38. [Google Scholar] [CrossRef] [Green Version] - Shen, H.-L.; Li, Y.-R.; Shao, X.-H. The four-parameter PSS method for solving the Sylvester equation. Mathematics
**2019**, 7, 105. [Google Scholar] [CrossRef] [Green Version] - Ding, F.; Chen, T. Hierarchical gradient-based identification methods for multivariable discrete time systems. Automatica
**2005**, 41, 397–402. [Google Scholar] [CrossRef] - Jonsson, I.; Kagstrom, B. Recursive blocked algorithms for solving triangular system Part I: One-side and coupled Sylvester-type matrix equation. ACM Trans. Math. Softw.
**2002**, 28, 392–415. [Google Scholar] [CrossRef] - Zhang, H.M.; Ding, F. A property of the eigenvalues of the symmetric positive definite matrix and the iterative algorithm for coupled Sylvester matrix equations. J. Frankl. Inst.
**2014**, 351, 340–357. [Google Scholar] [CrossRef] - Wang, X.; Li, Y.; Dai, L. On the Hermitian and skew-Hermitian splitting iteration methods for the linear matrix equation AXB = C. Comput. Math. Appl.
**2013**, 65, 657–664. [Google Scholar] [CrossRef] - Zhu, M.Z.; Zhang, G.F. A class of iteration methods based on the HSS for Toeplitz system of weakly nonlinear equation. Comput. Appl. Math.
**2015**, 290, 433–444. [Google Scholar] [CrossRef] - Bai, Z.Z. On Hermitian and skew-Hermitian splitting iteration method for continuous Sylvester equation. J. Comput. Math.
**2011**, 29, 185–198. [Google Scholar] [CrossRef] [Green Version] - Zheng, Q.Q.; Ma, C.F. On normal and skew-Hermitian splitting iteration methods for large sparse continuous Sylvester equation. J. Comp. Appl. Math.
**2014**, 268, 145–154. [Google Scholar] [CrossRef] - Ding, F.; Chen, T. Gradient based iterative algorithms for solving a class of matrix equation. IEEE Trans. Autom. Control
**2005**, 50, 1216–1221. [Google Scholar] [CrossRef] - Ding, F.; Chen, T. Iterative least square solutions of coupled Sylvester matrix equation. Syst. Control Lett.
**2005**, 54, 95–107. [Google Scholar] [CrossRef] - Ding, F.; Liu, X.P.; Ding, J. Iterative solution of the generalized Sylvester matrix equations by using the hierarchical identification principle. Appl. Math. Comput.
**2008**, 197, 41–50. [Google Scholar] [CrossRef] - Nui, Q.; Wang, X.; Lu, L.-Z. A relaxed gradient based iterative algorithms for solving Sylvester equation. Asian J. Cont.
**2011**, 13, 461–464. [Google Scholar] - Xie, Y.; Ma, C.F. The accelerated gradient based iterative algorithm for solving a class of generalized Sylvester-transpose matrix equation. Appl. Math. Comput.
**2016**, 273, 1257–1269. [Google Scholar] [CrossRef] - Fan, W.; Gu, C.; Tian, Z. Jacobi-gradient iterative algorithms for Sylvester matrix equations. In Linear Algebra Society Topics; Shanghai University: Shanghai, China, 2007; pp. 16–20. [Google Scholar]
- Li, S.K.; Huang, T.Z. A shift-splitting Jacobi-gradient algorithm for Lyapunov matrix equation arising form control theory. J. Comput. Anal. Appl.
**2011**, 13, 1246–1257. [Google Scholar] - Tian, Z.; Tian, M.; Gu, C.; Hao, X. An accelerated Jacobi-gradient based iterative algorithm for solving Sylvester matrix equation. Filomat
**2017**, 31, 2381–2390. [Google Scholar] [CrossRef]

Method | IT | CT |
---|---|---|

Direct | - | 3.1380 |

GIO | 161 | 0.0413 |

GI ($\tau =0.0001$) | 3061 | 0.2508 |

GI ($\tau =0.00003$) | 10,204 | 0.8994 |

Method | IT | CT |
---|---|---|

Direct | - | 53.4063 |

GIO | 389 | 0.5439 |

GI ($\tau =0.00005$) | 19,314 | 28.0245 |

GI ($\tau =0.000001$) | 96,557 | 148.4039 |

Method | GIO | GI | LS | RGI | MGI | JGI | AJGI | Direct |
---|---|---|---|---|---|---|---|---|

IT | 18 | 33 | 167 | 70 | 25 | - | 51 | - |

CT | 0.000273 | 0.000589 | 0.0114 | 0.0012 | 0.000789 | - | 0.0014 | 0.1704 |

Method | IT | CT | Error |
---|---|---|---|

Direct | - | 2.085 | 0 |

GIO | 100 | 0.0113 | 0.0199 |

GI | 100 | 0.0281 | 0.0648 |

LS | 100 | 0.0469 | 1.6574 |

RGI | 100 | 0.0324 | 0.1417 |

MGI | 100 | 0.0313 | 0.0397 |

JGI | 100 | 0.2813 | 0.7698 |

AJGI | 100 | 0.0938 | 0.0307 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Boonruangkan, N.; Chansangiam, P.
Gradient Iterative Method with Optimal Convergent Factor for Solving a Generalized Sylvester Matrix Equation with Applications to Diffusion Equations. *Symmetry* **2020**, *12*, 1732.
https://doi.org/10.3390/sym12101732

**AMA Style**

Boonruangkan N, Chansangiam P.
Gradient Iterative Method with Optimal Convergent Factor for Solving a Generalized Sylvester Matrix Equation with Applications to Diffusion Equations. *Symmetry*. 2020; 12(10):1732.
https://doi.org/10.3390/sym12101732

**Chicago/Turabian Style**

Boonruangkan, Nunthakarn, and Pattrawut Chansangiam.
2020. "Gradient Iterative Method with Optimal Convergent Factor for Solving a Generalized Sylvester Matrix Equation with Applications to Diffusion Equations" *Symmetry* 12, no. 10: 1732.
https://doi.org/10.3390/sym12101732