Convergence analysis of a gradient iterative algorithm with optimal convergence factor for a generalized Sylvester-transpose matrix equation

Nunthakarn Boonruangkan; Pattrawut Chansangiam; Nunthakarn Boonruangkan; Pattrawut Chansangiam

doi:10.3934/math.2021492

AIMS Mathematics

2021, Volume 6, Issue 8: 8477-8496. doi: 10.3934/math.2021492

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Research article

Convergence analysis of a gradient iterative algorithm with optimal convergence factor for a generalized Sylvester-transpose matrix equation

Department of Mathematics, Faculty of Science, King Mongkut's Institute of Technology Ladkrabang, Bangkok 10520, Thailand

Received: 01 April 2021 Accepted: 27 May 2021 Published: 03 June 2021
MSC : 15A12, 15A60, 46A22, 65F45

Consider a generalized Sylvester-transpose matrix equation with rectangular coefficient matrices. Based on gradients and hierarchical identification principle, we derive an iterative algorithm to produce a sequence of approximated solutions with a reasonable stopping rule concerning a relative norm-error. A convergence analysis via Banach fixed-point theorem reveals the sequence converges to a unique solution of the matrix equation for any given initial matrix if and only if the convergence factor is chosen appropriately in a certain range. The performance of algorithm is theoretically analysed through the convergence rate and error estimations. The optimal convergence factor is chosen to attain the fastest asymptotic behaviour. Finally, numerical experiments are provided to illustrate the capability and efficiency of the proposed algorithm, compared to recent gradient-based iterative algorithms.
- generalized Sylvester-transpose matrix equation,
- gradient,
- Kronecker product,
- matrix norm,
- Banach fixed-point theorem
Citation: Nunthakarn Boonruangkan, Pattrawut Chansangiam. Convergence analysis of a gradient iterative algorithm with optimal convergence factor for a generalized Sylvester-transpose matrix equation[J]. AIMS Mathematics, 2021, 6(8): 8477-8496. doi: 10.3934/math.2021492

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Abstract

Consider a generalized Sylvester-transpose matrix equation with rectangular coefficient matrices. Based on gradients and hierarchical identification principle, we derive an iterative algorithm to produce a sequence of approximated solutions with a reasonable stopping rule concerning a relative norm-error. A convergence analysis via Banach fixed-point theorem reveals the sequence converges to a unique solution of the matrix equation for any given initial matrix if and only if the convergence factor is chosen appropriately in a certain range. The performance of algorithm is theoretically analysed through the convergence rate and error estimations. The optimal convergence factor is chosen to attain the fastest asymptotic behaviour. Finally, numerical experiments are provided to illustrate the capability and efficiency of the proposed algorithm, compared to recent gradient-based iterative algorithms.

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