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**Gradient based iterative algorithm for solving coupled matrix equations.**
*(English)*
Zbl 1159.93323

Summary: This paper is concerned with iterative methods for solving a class of coupled matrix equations including the well-known coupled Markovian jump Lyapunov matrix equations as special cases. The proposed method is developed from an optimization point of view and contains the well-known Jacobi iteration, Gauss-Seidel iteration and some recently reported iterative algorithms by using the hierarchical identification principle, as special cases. We have provided analytically a necessary and sufficient condition for the convergence of the proposed iterative algorithm. Simultaneously, the optimal step size such that the convergence rate of the algorithm is maximized is also established in explicit form. The proposed approach requires less computation and is numerically reliable as only matrix manipulation is required. Some other existing results require either matrix inversion or special matrix products. Numerical examples show the effectiveness of the proposed algorithm.

### MSC:

93B40 | Computational methods in systems theory (MSC2010) |

93C05 | Linear systems in control theory |

15A18 | Eigenvalues, singular values, and eigenvectors |

60J75 | Jump processes (MSC2010) |

### Keywords:

gradient based iteration; coupled matrix equations; coupled Markovian jump Lyapunov matrix equations; maximal convergence rate; numerical solutions; Sylvester equations### Software:

KELLEY
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\textit{B. Zhou} et al., Syst. Control Lett. 58, No. 5, 327--333 (2009; Zbl 1159.93323)

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### References:

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