1. Modified Jacobi-Gradient Iterative Method for Generalized Sylvester Matrix Equation.
- Author
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Sasaki, Nopparut and Chansangiam, Pattrawut
- Subjects
- *
SYLVESTER matrix equations , *HAMILTON-Jacobi equations , *KRONECKER products , *ERROR rates , *MATRIX norms - Abstract
We propose a new iterative method for solving a generalized Sylvester matrix equation A 1 X A 2 + A 3 X A 4 = E with given square matrices A 1 , A 2 , A 3 , A 4 and an unknown rectangular matrix X. The method aims to construct a sequence of approximated solutions converging to the exact solution, no matter the initial value is. We decompose the coefficient matrices to be the sum of its diagonal part and others. The recursive formula for the iteration is derived from the gradients of quadratic norm-error functions, together with the hierarchical identification principle. We find equivalent conditions on a convergent factor, relied on eigenvalues of the associated iteration matrix, so that the method is applicable as desired. The convergence rate and error estimation of the method are governed by the spectral norm of the related iteration matrix. Furthermore, we illustrate numerical examples of the proposed method to show its capability and efficacy, compared to recent gradient-based iterative methods. [ABSTRACT FROM AUTHOR]
- Published
- 2020
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