DOI number:10.1007/s11075-025-02277-5

Journal:Numerical Algorithms

Key Words:Conjugate and transpose Sylvester matrix equations ; modified Jacobi gradient iterative ; Convergence properties; Optimal convergence factor

Abstract:In this paper, inspired by the modified relaxed gradient-based iterative (MRGI) algorithm proposed by Huang et al. ( Numer. Algorithms 97(4), 1955–2009 (2024)), an modified Jacobi gradient iterative (MJGI) algorithm is developed for solving the complex conjugate transpose Sylvester matrix equation through updates and modifications. Specifically, we replace the original full matrices by extracting the diagonal parts of matrices $A_i$ and $B_i$ ($i = 1, 2, 3, 4$), while proposing a progressive update mechanism with hybrid historical iterative values. This mechanism dynamically integrates the newly computed sub-blocks (e.g. $Y_i(l+1)$) with the global historical values $Y(l)$ during the updating process.By retaining the memory effect of historical information, we enhance the algorithmic accuracy. Furthermore, we establish an $\omega$-parameterized convergence factor optimization framework that significantly accelerates the convergence rate. Theoretical analysis, grounded in the real representation of matrices and Kronecker product operations, establishes the convergence conditions of the algorithm and determines explicit optimization ranges for the relaxation factor and step size parameters. The numerical example demonstrates that the MJGI algorithm outperforms the MRGI algorithm in terms of iteration counts and CPU running time, especially under high-precision requirements (residual threshold $\tau \leq 10^{-6}$), with an improvement in iteration efficiency, defined as the number of iterations required to achieve a specified residual threshold, by more than 76. 31\%.

First Author:Jiating He

Indexed by:Journal paper

Correspondence Author:Xuesong Chen

Translation or Not:no

Date of Publication:2026-04-30

Included Journals:SCI