解非线性方程组的一类算法

A Class of Methods for Solving Nonlinear Systems of Equations

§1.引言 求解线性方程组 a_i^Tx=b_i,i=1,2,…,n,(1.1)其中a_1,a_2,…,a_n线性无关. 设y^((1))为初值,U^((1))为任意非奇异n阶矩阵,我们用如下方法求解方程组(1.1). 先考虑前k-1个方程组成的亚定方程组 a_i^Tx=b_i,i=1,2,…,k-1.设{U^((k))}={a_1,a_2,…,a_(k-1)},这里{U^((k))}表示由U^((k))的列组成的子空间.显然,rank(U^((k)))=n-b+1.若y^((k))是相应的亚定方程的一个特解。

A class of algorithms for solving nonlinear systems of equations is proposed.Compared with the discrete Newton method, this class has the same quadratic co-nvergency, but its number of evaluations of the function is less than that of thediscrete Newton method per cycle. Some efficient algorithms can be obtained whenthe parameter is chosen by a different means. The nonlinear ABS algorithm, andthe Brown and Brent method are special cases of this class. The numerical exam-ples have showed its effectiveness.