##
**Least squares solution with the minimum-norm to general matrix equations via iteration.**
*(English)*
Zbl 1186.65047

Two iterative algorithm are presented to solve the minimal norm least squares solution to a general linear matrix equations including the well-known Sylvester matrix equation and Lyapunov matrix equation as special cases. The first algorithm is based on the gradient based searching principle and the other one can be viewed as its dual. Necessary and sufficient conditions for the step sizes in these two algorithms are proposed to guarantee the convergence of the algorithms for arbitrary initial conditions.

Reviewer: Rózsa Horvàth-Bokor (Budapest)

### MSC:

65F20 | Numerical solutions to overdetermined systems, pseudoinverses |

65F30 | Other matrix algorithms (MSC2010) |

15A24 | Matrix equations and identities |

### Keywords:

iterative algorithm; minimal norm least squares solution; optimal step size; convergence; general linear matrix equations; Sylvester matrix equation; Lyapunov matrix equation
PDF
BibTeX
XML
Cite

\textit{Z.-Y. Li} et al., Appl. Math. Comput. 215, No. 10, 3547--3562 (2010; Zbl 1186.65047)

Full Text:
DOI

### References:

[1] | Bhattacharyya, S.P.; De Souza, E., Pole assignment via sylvester’s equation, Systems and control letters, 1, 261-263, (1972) · Zbl 0473.93037 |

[2] | R. Byers, N. Rhee, Cyclic Schur and Hessenberg-Schur numerical methods for solving periodic Lyapunov and Sylvester equations. Technical Report, Dept. of Mathematics, Univ. of Missouri at Kansas City, 1995. |

[3] | Byers, R., Solving the algebraic Riccati equation with the matrix sign function, Linear algebra and its applications, 85, 267-279, (1987) · Zbl 0611.65027 |

[4] | Chen, J.L.; Chen, X.H., Special matrices, (2002), Tsinghua University Press, (in Chinese) |

[5] | Desouza, E.; Bhattacharyya, S.P., Controllability, observability and the solution of \(\mathit{AX} - \mathit{XB} = C\), Linear algebra and its applications, 39, 167-188, (1981) |

[6] | Ding, F.; Chen, T., Iterative least squares solutions of coupled Sylvester matrix equations, Systems and control letters, 54, 2, 95-107, (2005) · Zbl 1129.65306 |

[7] | Ding, F.; Chen, T., Gradient based iterative algorithms for solving a class of matrix equations, IEEE transactions on automatic control, 50, 8, 1216-1221, (2005) · Zbl 1365.65083 |

[8] | Ding, F.; Chen, T., On iterative solutions of general coupled matrix equations, SIAM journal on control and optimization, 44, 6, 2269-2284, (2006) · Zbl 1115.65035 |

[9] | Ding, F.; Liu, P.X.; Ding, J., Iterative solutions of the generalized Sylvester matrix equations by using the hierarchical identification principle, Applied mathematics and computation, 197, 1, 41-50, (2008) · Zbl 1143.65035 |

[10] | Doyle, J.C.; Glover, K.; Khargonekar, P.P.; Francis, B.A., State-space solutions to standard \(\mathcal{H}_2\) and \(\mathcal{H}_\infty\) control problems, IEEE transactions on automatic control, 34, 8, 831-847, (1989) · Zbl 0698.93031 |

[11] | Duan, G.R., Solutions to matrix equation \(\mathit{AV} + \mathit{BW} = \mathit{VF}\) and their application to eigenstructure assignment in linear systems, IEEE transactions on automatic control, 38, 2, 276-280, (1993) |

[12] | Duan, G.R., On the solution to Sylvester matrix equation \(\mathit{AV} + \mathit{BW} = \mathit{EVF}\), IEEE transactions on automatic control, 41, 4, 612-614, (1996) |

[13] | Huang, G.X.; Yin, F.; Guo, K., An iterative method for the skew-symmetric solution and the optimal approximate solution of the matrix equation \(\mathit{AXB} = C\), Journal of computational and applied mathematics, 212, 2, 231-244, (2008) · Zbl 1146.65036 |

[14] | Kailath, T., Linear systems, (1980), Prentice-Hall Englewood Cliffs, NJ · Zbl 0458.93025 |

[15] | Z. Li, Y. Wang, Iterative algorithm for minimal norm least squares solution to general linear matrix equations, International Journal of Computer Mathematics, doi:10.1080/00207160802684459. · Zbl 1203.65080 |

[16] | cman, Adem Kılı; Abdel Aziz Al Zhour, Zeyad, Vector least-squares solutions for coupled singular matrix equations, Journal of computational and applied mathematics, 206, 1051-1069, (2007) · Zbl 1132.65034 |

[17] | Piao, F.; Zhang, Q.; Wang, Z., The solution to matrix equation \(\mathit{AX} + X^{\operatorname{T}} C = B\), Journal of the franklin institute, 344, 8, 1056-1062, (2007) · Zbl 1171.15015 |

[18] | Peng, X.Y.; Hu, X.Y.; Zhang, L., The reflexive and anti-reflexive solutions of the matrix equation \(A^{\operatorname{H}} \mathit{XB} = C\), Journal of computational and applied mathematics, 200, 2, 749-760, (2007) · Zbl 1115.15014 |

[19] | Peng, Z.H.; Hu, X.Y.; Zhang, L., An effective algorithm for the least-squares reflexive solution of the matrix equation \(A_1 \mathit{XB}_1 = C_1, A_2 \mathit{XB}_2 = C_2\), Applied mathematics and computation, 181, 988-999, (2006) |

[20] | Qiu, Y.Y.; Zhang, Z.Y.; Lu, J.F., Matrix iterative solutions to the least squares problem of \(\mathit{BXA}^{\operatorname{T}} = F\) with some linear constraints, Applied mathematics and computation, 185, 284-300, (2007) |

[21] | Wang, M.H.; Cheng, X.H.; Wei, M.H., Iterative algorithms for solving the matrix equation \(\mathit{AXB} + \mathit{CX}^{\operatorname{T}} D = E\), Applied mathematics and computation, 187, 2, 622-629, (2007) |

[22] | Wang, Q.; Lam, J.; Wei, Y.; Chen, T., Iterative solutions of coupled discrete Markovian jump Lyapunov equations, Computers and mathematics with applications, 55, 4, 843-850, (2008) · Zbl 1139.60334 |

[23] | Zhou, K.; Doyle, J.; Glover, K., Robust and optimal control, (1996), Prentice-Hall |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.