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Finite iterative algorithm for the symmetric periodic least squares solutions of a class of periodic Sylvester matrix equations

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Abstract

The present work proposes a finite iterative algorithm to find the least squares solutions of periodic matrix equations over symmetric ξ-periodic matrices. By this algorithm, for any initial symmetric ξ-periodic matrices, the solution group can be obtained in finite iterative steps in the absence of round-off errors, and the solution group with least Frobenius norm can be obtained by choosing a special kind of initial matrices. Furthermore, in the solution set of the above problem, the unique optimal approximation solution group to a given matrix group in the Frobenius norm can be derived by finding the least Frobenius norm symmetric ξ-periodic solution of a new corresponding minimum Frobenius norm problem. Finally, numerical examples are provided to illustrate the efficiency of the proposed algorithm and testify the conclusions suggested in this paper.

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Acknowledgments

Supported by National Science Foundation of China (41725017, 41590864) and National Basic Research Program of China under grant number 2014CB845906. It is also partially supported by the Strategic Priority Research Program (B) of the Chinese Academy of Sciences (XDB18010202) and Fujian Natural Science Foundation (2016J01005).

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Authors and Affiliations

  1. College of Mathematics and Informatics & FJKLMAA, Fujian Normal University, Fuzhou, 350117, People’s Republic of China

    Baohua Huang & Changfeng Ma

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  1. Baohua Huang
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  2. Changfeng Ma
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Correspondence to Changfeng Ma.

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Huang, B., Ma, C. Finite iterative algorithm for the symmetric periodic least squares solutions of a class of periodic Sylvester matrix equations. Numer Algor 81, 377–406 (2019). https://doi.org/10.1007/s11075-018-0553-8

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