Modified and accelerated relaxed gradient-based iterative algorithms for the complex conjugate and transpose matrix equations

In this paper, by applying the updated technique to the relaxed gradient-based iterative algorithm proposed by Wang et al. (J. Appl. Math. Comput. 67 , 317–341 2021 ), we develop the modified relaxed gradient-based iterative algorithm for the complex conjugate and transpose Sylvester matrix equation...

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Bibliographic Details
Published in:Numerical algorithms Vol. 97; no. 4; pp. 1955 - 2009
Main Authors: Huang, Zhengge, Cui, Jingjing
Format: Journal Article
Language:English
Published: New York Springer US 01.12.2024
Springer Nature B.V
Subjects:
Algebra
Algorithms
Computational mathematics
Computer Science
Conjugates
Convergence
Iterative algorithms
Iterative methods
Linear systems
Methods
Numeric Computing
Numerical Analysis
Original Paper
System theory
Theory of Computation
Modified relaxed gradient-based iterative algorithm
Optimal convergence factor
Complex conjugate and transpose matrix equation
65F10
Real representation
Accelerated relaxed gradient-based iterative algorithm
Convergence
ISSN:1017-1398, 1572-9265
Online Access:Get full text
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Summary:In this paper, by applying the updated technique to the relaxed gradient-based iterative algorithm proposed by Wang et al. (J. Appl. Math. Comput. 67 , 317–341 2021 ), we develop the modified relaxed gradient-based iterative algorithm for the complex conjugate and transpose Sylvester matrix equations. Compared with the relaxed gradient-based iterative algorithm, the modified relaxed gradient-based iterative algorithm can make full use of the latest information to compute the next result and lead to a faster convergence rate. Furthermore, in order to reduce the computation cost of each iteration of the relaxed gradient-based iterative algorithm, by replacing the coefficient matrices by their diagonal parts, we construct the accelerated relaxed gradient-based iterative algorithm. By utilizing the properties of the real representation of a complex matrix, matrix norms, and techniques of inequalities, we prove that two proposed iterative algorithms are convergent under proper restrictions. And the quasi-optimal convergence factor of the accelerated relaxed gradient-based iterative algorithm is also derived. Also, some numerical examples are given to show the effectiveness and the superiorities of the proposed algorithms. Lastly, the application in time-varying linear system of the proposed algorithms is presented.
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ISSN:1017-1398
1572-9265
DOI:10.1007/s11075-024-01775-2