求解非线性方程组的元胞自动机方法及其全局收敛性证明

摘要

为了求得非线性方程组所有精确解,根据元胞自动机的特点构造了求解非线性方程组的全局收敛算法.在该算法中,将非线性方程组解的理论搜索空间划分为离散搜索空间,将离散搜索空间定义为元胞空间;离散搜索空间的每个点就是一个元胞,而一个元胞对应着非线性方程组的一个试探解;元胞的状态由其空间位置及位置修正量构成.将元胞空间划分为若干个非空子集,所有元胞的状态从一个非空子集转移到另一个非空子集的状态演化过程实现了元胞空间对理论搜索空间的搜索.在元胞状态演化过程中,元胞从一个状态转移到另一个状态的状态转移概率可以计算出来;元胞演化过程中的每个状态对应于有限Markov链上的一个状态.利用可归约随机矩阵的稳定性条件证明了该算法具有全局收敛性.仿真实例表明该算法是高效的.%To get all the accurate solutions to Nonlinear Systems of Equations (NSE), the algorithm with global convergence was constructed for solving NSE based on the characteristics of Cellular Automata (CA). In the algorithm, the theoretical search space of NSE was divided into the discrete space, the discrete space was defined as the cellular space; each point in the discrete space was a cell in the cellular space, and each cell was a trial solution of NSE; a cellular state consisted of position and increment of position. The cellular space was divided into many nonempty subsets, and states evolution of all cells from one nonempty subset to another realized the search of the cellular space on the theoretical search space. During evolution process of all cells, each cell's transition probability from one position to any another position could be simply calculated; each state of cells during evolution corresponded to a state of a finite Markov chain. The stability condition of a reducible stochastic matrix was used to prove the global convergence of the algorithm. The case study shows that the algorithm is efficient.