Convergence properties of Jacobi gradient-based iteration algorithm for the complex conjugate and transpose Sylvester matrix equations

In this paper, inspired by the modified relaxed gradient-based iterative (MRGI) al-gorithm proposed by Huang et al. ( Numer. Algorithms 97(4), 1955–2009 (2024)), a modifiedJacobi gradient iterative (MJGI) algorithm is developed to solve the complex conjugate trans-pose Sylvester matrix equation through updates and modifications. Specifically, we replace theoriginal full matrices by extracting the diagonal parts of the matrices Ai and Bi, while proposinga progressive update mechanism with hybrid historical iterative values. This mechanism dynam-ically integrates the newly computed subblocks with the global historical values Y (l) during theupdating process. By retaining the memory effect of historical information, we enhance the algo-rithmic accuracy. Furthermore, we establish an w-parameterized convergence factor optimizationframework that significantly accelerates the convergence rate. Theoretical analysis, grounded inthe real representation of matrices and Kronecker product operations, establishes the conver-gence conditions of the algorithm and determines explicit optimization ranges for the relaxationfactor and step size parameters. The numerical example demonstrates that the MJGI algorithmoutperforms the MRGI algorithm in terms of iteration counts and CPU running time, especiallyunder high-precision requirements (residual threshold τ ≤ 10−6), with an improvement in it-eration efficiency, defined as the number of iterations required to achieve a specified residualthreshold, by more than 76. 31%.