几类非线性矩阵方程的高精度保结构加倍算法研究

Nonlinear matrix equations arise from a wide range of important applications, such as optimal control theory, applied probability, Markov-modulated fluid queue theory and vibration analysis of fast trains, etc. It is important to research numerical methods for solving the nonlinear matrix equations rapidly, stably and efficiently, which is one of the core problems of scientific calculation and computational mathematics. Moreover, current applications also require the methods have the ability of providing high accuracy solutions. The structure-preserving doubling algorithms are impressive for their iterative speed and computational accuracy. This projects devotes to studying highly accurate structure-preserving doubling algorithms for several kinds of nonlinear matrix equations, which include the quadratic matrix equation derived from quasi birth and death process, the matrix polynomial equation from the numerical solution of M/G/1 type Markov chains, the M-matrix quadratic matrix equation from damping vibration system and the nonlinear matrix equation reduced from preconditioned hyperbolic quadratic eigenvalue problem; analyzing their convergence; providing theoretical basis for highly accurate implementation; and reporting numerical results. The results for this project not only provide several efficient structure-preserving doubling algorithms for solving these special types of nonlinear matrix equations, but also put some insights on further extensions to the general cases and give some alternatives to the researchers from application fields.

非线性矩阵方程在最优控制理论、应用概率、Markov调制流排队理论、高速列车振动分析等领域有广泛应用, 研究其快速、稳定且高效的数值解法是数值代数和数值优化的重要问题,也是科学计算的核心问题之一. 保结构加倍算法往往在迭代速度和计算精度上具有优势. 本项目针对几类非线性矩阵方程, 即源于拟生灭过程的二次矩阵方程、源于M/G/1型Markov链的矩阵多项式方程、源于过阻尼振动系统的M-矩阵二次矩阵方程和对双曲二次特征值问题预处理后得到的非线性矩阵方程, 研究求解它们的高精度保结构加倍算法, 分析算法的收敛性、收敛率; 结合最优化技术, 研究算法中参数的最优选取及算法的预处理、加速技术; 分析算法高精度实现的理论依据, 并用数值实验验证. 本项目研究结果不仅为几类有实际应用背景的非线性矩阵方程设计高精度数值解法, 同时也进一步完善高精度保结构加倍算法的理论基础, 具有重要的理论和现实意义.

紧密围绕项目申请书的研究内容,我们做了如下几个方面的研究工作:. 1.提出了求解源于拟生灭过程的二次矩阵方程的极小非负解的(SF1)-型高精度保结构加倍算法,证明了算法在非临界情形时的全局单调二次收敛性及在临界情形时的全局单调线性收敛性,线性收敛因子为1/2.通过五个数值例子验证了高精度保结构加倍算法计算得到该二次矩阵方程的高元素相对精度的极小非负解的能力.. 2.提出了求解源于过阻尼振动系统的M-矩阵二次矩阵方程的(SF1)-型保结构加倍算法,证明了算法的全局单调二次收敛性,并通过数值实验验证了所提出的算法的可行性及有效性.. 3.提出了求解绝对值方程组的精确和非精确Douglas–Rachford分裂迭代法、修正不动点迭代法、无逆动力学模型和惯性无逆动力学模型等,分析了算法的收敛性,并用数值实验验证了算法的有效性.特别地,我们所提出的精确和非精确Douglas–Rachford分裂迭代法适用于绝对值方程组解存在但不唯一的情形,这个性质是很多其他方法不具备的.. 总之,我们在计划研究内容的各个方面开展工作,顺利完成了本项目的大部分研究内容,并为今后的研究打下坚实的基础.