若干矩阵方程的数值求解算法和预处理技术研究

Matrix equations, saddle point problems and linear and nonlinear complementarity problems are widely derived from the fields of science and engineering applications. The corresponding solution problem of these equations is one of the key problems in scientific and engineering calculation. Therefore, it is of great theoretical significance and practical value to study the effective numerical solution and preconditioning technique of matrix equations. This project intends to solve the following three important problems: 1. The construction of numerical solutions and explicit solutions for several types of Sylvester equations. The solvability conditions of matrix equations, efficient and feasible numerical algorithms, and the form and properties of explicit solutions are studied. 2. Iterative methods and preconditioning techniques for large-scale sparse linear and nonlinear complementarity problems. Based on the modulus-based matrix splitting iteration method, the linear and nonlinear complementarity problems can be rewritten as the linear equations. The efficient solutions to the linear and nonlinear complementarity problems are designed by using relaxation techniques, parameter acceleration technique and matrix preconditioning strategy. 3. The numerical solution and preconditioning of saddle point problems with large-scale sparse structures. Based on the structural characteristics of the coefficient matrix of saddle point problem, efficient and stable iterative algorithms and preconditioning technique are constructed by combining the ideas of approximation, limit and extrapolation. This project will make the numerical algorithm theory and preconditioning technique of the above equations more systematic and in-depth. It is helpful to deepen the study of matrix equation theory and promote the development and application of related disciplines.

矩阵方程、鞍点问题及线性与非线性互补问题广泛来源于科学和工程应用领域。相应的求解问题是科学和工程计算的关键问题之一。因此研究矩阵方程的有效数值解法和预处理技术具有十分重要的理论意义和实际应用价值。本项目拟解决如下三个重要问题：1. Sylvester方程的数值求解及显式解的构建问题。研究其可解性条件、高效可行的数值算法、显式解的形式及性质。2. 线性及非线性互补问题的迭代法及预处理技术问题。基于模系矩阵分裂迭代方法，将线性和非线性互补问题等价转化为线性系统，拟利用松弛技术、参数加速技术及矩阵预处理 技术等设计求解线性和非线性互补问题的高效求解方法。3. 鞍点问题的迭代法及预处理技术问题。利用鞍点问题系数矩阵的结构特点，结合逼近、极限和外推等思想构造高效稳定的迭代法和预处理子。本项目将使上述方程的数值算法理论和预处理技术研究更为系统和深入。有助于深化矩阵方程理论的研究，促进相关学科的发展。

矩阵方程、鞍点问题及线性与非线性互补问题广泛来源于科学和工程应用领域。相应的求解问题是科学和工程计算的关键问题之一。因此研究矩阵方程的有效数值解法和预处理技术具有十分重要的理论意义和实际应用价值。本项目研究了如下重要问题：1. 针对线性、非线性及水平互补问题，基于模系矩阵分裂迭代方法，将互补问题等价转化为不动点格式，拟利用双松弛、参数、加速、两步扫描技术等设计了求解几类互补问题的高效求解方法，并对其进行理论分析和程序实现。2. 研究了耦合复共轭转置Sylvester方程的数值求解方法并分析了其收敛条件。3. 针对于鞍点问题，利用其系数矩阵的结构特点，结合逼近、极限和外推等思想构造了几类高效稳定的迭代法和预处理子。本项目使上述方程的数值算法理论和预处理技术研究更为系统和深入。有助于深化矩阵方程理论的研究，促进相关学科的发展。