基于复西尔维斯特矩阵方程的改进双阶尺度分裂数值计算方法
Improved Numerical Calculation Method for Second-Order Scale Splitting Based on the Complex Sylvester Matrix Equation
摘要: 连续时间的矩阵方程是矩阵方程中十分重要的一个类型，在矩形域上的椭圆边值问题的数值解法、线性统计、振动结构的共振控制、二次矩阵的特征值配置问题、受噪声影响的图像复原等问题中有重要地位。由于连续时间的矩阵方程广泛的应用背景，因此，对连续时间的Sylvester方程的数值解法的研究具有重要的理论和实际意义．改进的双步尺度分裂(MDSS)方法是研究一类大型复杂对称线性系统的一种有效方法。在本文中，我们将在双步尺度分裂(DSS)方法的基础上通过改进，得到MDSS方法，来求解复数域上的连续时间的西尔维斯特矩阵方程的近似解，证明了该迭代序列在任意初始条件的情况下都收敛于西尔维斯特矩阵方程的唯一解，并确定了其最优参数和相应的最优收敛因子。最后，给出了一个测试问题来说明该新技术的有效性。
Abstract: The continuous time matrix equation is a very important type of matrix equation, which plays an important role in numerical solutions of elliptic boundary value problems in rectangular domains, linear statistics, resonance control of vibration structures, eigenvalue assignment of quadratic matrices, image restoration affected by noise, and other problems. Due to the wide application background of continuous time matrix equations, the study of numerical solutions for the continuous time Sylvester equation has important theoretical and practical significance. The improved two-step scale splitting (MDSS) method is an effective method for studying a class of large complex symmetric linear systems. In this article, we will improve the two-step scale splitting (DSS) method and obtain the MDSS method to solve the approximate solution of the continuous time Sylvester matrix equation in the complex field. We prove that the iterative sequence converges to a unique solution of the Sylvester matrix equation under any initial condition, and determine its optimal parameters and corresponding optimal convergence factors. Finally, a testing question was presented to demonstrate the effectiveness of the new technology.
文章引用：杨创勋. 基于复西尔维斯特矩阵方程的改进双阶尺度分裂数值计算方法[J]. 人工智能与机器人研究, 2024, 13(2): 313-321. https://doi.org/10.12677/airr.2024.132033

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