A Minimum Residual Based Gradient Iterative Method for a Class of Matrix Equations

In this paper, we present a minimum residual based gradient iterative method for solving a class of matrix equations including Sylvester matrix equations and general coupled matrix equations. The iterative method uses a negative gradient as steepest direction and seeks for an optimal step size to mi...

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Bibliographic Details
Published inActa Mathematicae Applicatae Sinica Vol. 40; no. 1; pp. 17 - 34
Main Author Zheng, Qing-qing
Format Journal Article
LanguageEnglish
Published Berlin/Heidelberg Springer Berlin Heidelberg 2024
Springer Nature B.V
EditionEnglish series
Subjects
Applications of Mathematics
Exact solutions
Iterative methods
Math Applications in Computer Science
Mathematical analysis
Mathematical and Computational Physics
Mathematics
Mathematics and Statistics
Theoretical
convergence analysis
15A24
Sylvester matrix equation
coupled matrix equation
minimum residual
15A06
65F15
gradient descent
93B30
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Summary:In this paper, we present a minimum residual based gradient iterative method for solving a class of matrix equations including Sylvester matrix equations and general coupled matrix equations. The iterative method uses a negative gradient as steepest direction and seeks for an optimal step size to minimize the residual norm of next iterate. It is shown that the iterative sequence converges unconditionally to the exact solution for any initial guess and that the norm of the residual matrix and error matrix decrease monotonically. Numerical tests are presented to show the efficiency of the proposed method and confirm the theoretical results.
ISSN:0168-9673
1618-3932
DOI:10.1007/s10255-024-1100-0