函数因子法——非线性方程组求解中处理奇异问题的一种新方法

§1.引言 非线性方程组 F(x)=0,F:D?R~n→R~n (1.1)嵌入参数t,构成同伦H:0,T]×D?R~(n 1)→R~n,使得 H(0,x~0)=0,H(T,x)=F(x),(1.2)这里T可以是有限的或 ∞,当T为 ∞时以极限过程代替求值.若 H(t.x)=0(1.3)存在连续解x(t):0,T]→D,则非线性方程组(1.1)的解x~*=x(T).若(1.3)的解

FUNCTION-FACTOR METHOD A new method of passing the singularities which arise in the continuation methods for solving systems of nonlinear equations.

When we use continuation methods for solving systems of nonlinear equationsF(x) =0, the partial derivative Hx of homotopy H(t, x) may be singular on the homo-topy curve x(t). This paper presents a method in which the function F is multipliedby a matrix factor to modify the homotopy H for passing the singularities. The dia-gonal matrix factors consisting of exponential function are considered in the case ofthe general homotopy F(x) --b (t)·F(x~0) =0. Some convergent results have been obtainedin the sequence forms. This paper also gives two approximation solutions of the dif-ferential equations and an iterative process for linear factor of exponential function.