求解非线性方程组的混沌分形方法

混沌分形是动力系统普遍出现的一种现象,牛顿-拉夫森NR(Newton-Raphson)方法是重要的一维及多维迭代技术,其迭代本身对初始点非常敏感,该敏感区是牛顿-拉夫森法所构成的非线性离散动力系统Julia集,在Julia集中迭代函数会呈现出混沌分形现象,提出了一种寻找牛顿-拉夫森函数的Julia点的求解方法,利用非线性离散动力系统在其Julia集出现混沌分形现象的特点,提出了一种基于牛顿-拉夫森法的非线性方程组求解的新方法,计算实例表明了该方法的有效性和正确性.

Chaos & fractals method for solving systems of nonlonear equations

Chaos & Fractals is a universal phenomenon.Newton-Raphson method is an important technique for calculating one dimensional or multi-dimensional variable,Newton-Raphson method is a nonlinear discrete dynamic process that exhibits sensitive dependence on initial guess point,which shows a fractal nature.the sensitive area of Newton-Raphson function is called the Julia set that is the boundaries of basins of attractions(solutions)display the intricate fractal structures and chaos phenomena.By constructing repulsion two-cycle point optimization function and an optimization method to find Julia set point is proposed.A novel approach based on utilizing sensitive fractal areas to locate the Julia set point to find an the solutions of the nonlinear equations is proposed.The numerical simulation results show that the method is effective.