An iterative algorithm for the reflexive solution of the general coupled matrix equations
1Geomathematics Key Laboratory of Sichuan Province, College of Management Science, Chengdu University of Technology, Chengdu 610059, China.
Thescientificworldjournal
|December 11, 2013
Summary
This study introduces an iterative algorithm to solve general coupled matrix equations, finding reflexive solutions efficiently. The method determines the least Frobenius norm reflexive solution and optimal approximations within finite steps.
Area of Science:
- Control theory
- System theory
- Numerical analysis
Background:
- General coupled matrix equations, including Sylvester types, are crucial in control and system theory.
- Solving these equations for reflexive matrix solutions is a significant challenge.
Purpose of the Study:
- To develop an iterative algorithm for solving general coupled matrix equations over reflexive matrices.
- To derive the least Frobenius norm reflexive solution and optimal approximation solutions.
Main Methods:
- Construction of a novel iterative algorithm.
- Finite iterative steps for solution determination (assuming no round-off errors).
- Selection of an appropriate initial matrix for least Frobenius norm solution.
Main Results:
- The algorithm converges to the reflexive solution in finite steps when equations are consistent.
- The least Frobenius norm reflexive solution is obtainable.
- The unique optimal approximation reflexive solution in Frobenius norm is derived.
Conclusions:
- The proposed iterative algorithm is effective for solving general coupled matrix equations over reflexive matrices.
- The method provides a robust way to find both exact and approximate reflexive solutions.
- Demonstrated effectiveness through a numerical example.
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