Sylvester矩阵方程的数值求解方法及预处理技术研究

发布时间：2018-05-12 09:58

本文选题：Sylvester矩阵方程 + HSS迭代法　；参考：《南昌大学》2017年硕士论文

【摘要】：矩阵方程的高效求解是计算数学学科中一个极其重要的问题。在理学、工学等科学和工程技术计算领域中,求解矩阵方程有着很广泛地应用,比如散射光成像、磁场数据的处理、结构动力学、处理相关的数字信号、处理相关的数字图像、估计流体力学、石油数据的处理、地动数据的处理、数值模拟天气预报和核爆、控制论系统、量子化学和涡流问题、神经网络,以及偏微分方程数值解等。所以,设计数值求解相关矩阵方程的有效方法是一个紧要的,并且具备实际意义和应用价值的课题。本文主要考虑的是用两种不同的迭代解法去求解连续型Sylvester矩阵方程:(1)修正的广义的PSS,即MGPSS迭代法。为了更加有效地求解连续的Sylvester方程组,在PSS迭代算法和广义的PSS迭代算法的研究基础上,我们提出了MGPSS迭代法。证明了对于系数矩阵A和B满足一定条件的Sylvester方程,MGPSS迭代是无条件收敛的。而且在迭代数IT和运行时间CPU方面,数值实验结果也表明了MGPSS迭代法更加有效;(2)预条件的PSS,即PPSS迭代法。本文提出了一种预条件的正定和反埃尔米特(PPSS)迭代法用于求解Sylvester矩阵方程,并在理论上证明了其收敛性;也建立了一种不精确的PPSS(IPPSS)迭代,并给出了几个数值例子,验证了PPSS迭代比之前存在的方法都有效。本文共分为四章,组织如下:第一章主要是介绍了矩阵方程,特别是连续的Sylvester矩阵方程,针对其研究背景、研究近况进行了详细的介绍,也给出了相关的理论知识。第二章基于对广义的PSS迭代算法的研究,提出了修正的广义的PSS,即MGPSS迭代法,得到了求解连续型Sylvester方程AX(10)XB(28)C的MGPSS迭代算法,并证明了该算法是无条件收敛的。最后,给出的数值例子也证实了MGPSS迭代法更加有效。第三章通过研究,为了更加有效地求解Sylvester矩阵方程,提出了一种预条件的正定和反埃尔米特(PPSS)迭代算法,并在理论上证明了其收敛性;同时建立了一种不精确的PPSS(IPPSS)迭代,并给出了几个数值例子,验证了PPSS迭代比之前存在的方法都有效。第四章总结全文,并展望了以后的研究工作。
[Abstract]:The efficient solution of matrix equations is an extremely important problem in computational mathematics. In the fields of science and engineering, such as science and engineering, solving matrix equations is widely used, such as scattering light imaging, magnetic field data processing, structural dynamics, processing of related digital signals, processing of related digital images. Estimation of fluid dynamics, processing of petroleum data, processing of geodynamic data, numerical simulation of weather forecasting and nuclear explosion, cybernetics systems, quantum chemistry and eddy current problems, neural networks, and numerical solutions of partial differential equations, etc. Therefore, it is very important to design an effective method for solving the correlation matrix equation, which is of practical significance and practical value. The main consideration of this paper is to use two different iterative methods to solve the continuous Sylvester matrix equation: 1) modified generalized PSS, that is, the MGPSS iterative method. In order to solve the continuous Sylvester equations more effectively, we propose a MGPSS iterative method based on the PSS iterative algorithm and the generalized PSS iterative algorithm. It is proved that the MGPSS iteration is unconditionally convergent for the Sylvester equations whose coefficient matrices A and B satisfy some conditions. Moreover, in terms of iteration number IT and runtime CPU, the numerical results also show that the MGPSS iteration method is more effective than the PPSS iteration method. In this paper, a preconditioned positive definite and anti Hermitt Sylvester iterative method is proposed to solve the Sylvester matrix equation, and its convergence is proved in theory, and an imprecise PPSS IPPSS iteration is also established, and several numerical examples are given. It is proved that the PPSS iteration is more effective than the previous methods. This paper is divided into four chapters. The first chapter mainly introduces the matrix equation, especially the continuous Sylvester matrix equation. According to its research background, the recent research situation is introduced in detail, and the relevant theoretical knowledge is also given. In the second chapter, based on the study of generalized PSS iterative algorithm, a modified generalized MGPSS iterative method is proposed. The MGPSS iterative algorithm for solving continuous Sylvester equation AX(10)XB(28)C is obtained, and it is proved that the algorithm is unconditionally convergent. Finally, a numerical example is given to prove that the MGPSS iterative method is more effective. In chapter 3, in order to solve the Sylvester matrix equation more effectively, a preconditioned positive definite and anti-Hermitian PPSS iterative algorithm is proposed, and its convergence is proved theoretically, and an imprecise PPSS iteration is established. Several numerical examples are given to verify that the PPSS iteration is more efficient than the previous methods. The fourth chapter summarizes the full text and looks forward to the future research work.
【学位授予单位】：南昌大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O241.6

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