We introduce a gradient iterative scheme with an optimal convergent factor for solving a generalized Sylvester matrix equation ∑i=1pAiXBi=F, where Ai,Bi and F are conformable rectangular matrices. The iterative scheme is derived from the gradients of the squared norm-errors of the associated subsystems for the equation. The convergence analysis reveals that the sequence of approximated solutions converge to the exact solution for any initial value if and only if the convergent factor is chosen properly in terms of the spectral radius of the associated iteration matrix. We also discuss the convergent rate and error estimations. Moreover, we determine the fastest convergent factor so that the associated iteration matrix has the smallest spectral radius. Furthermore, we provide numerical examples to illustrate the capability and efficiency of this method. Finally, we apply the proposed scheme to discretized equations for boundary value problems involving convection and diffusion.

中文翻译：

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求解广义 Sylvester 矩阵方程的具有最优收敛因子的梯度迭代法以及对扩散方程的应用

我们引入了具有最优收敛因子的梯度迭代方案，用于求解广义 Sylvester 矩阵方程 ∑i=1pAiXBi=F，其中 Ai、Bi 和 F 是一致的矩形矩阵。迭代方案是从方程相关子系统的平方范数误差的梯度导出的。收敛分析表明，当且仅当根据相关迭代矩阵的谱半径正确选择收敛因子时，近似解的序列收敛到任何初始值的精确解。我们还讨论了收敛率和误差估计。此外，我们确定最快收敛因子，以便相关的迭代矩阵具有最小的谱半径。此外，我们提供了数值例子来说明这种方法的能力和效率。最后，我们将所提出的方案应用于涉及对流和扩散的边值问题的离散方程。