On the Efficiency of Newton's Method in Approximating All Zeros of a System of Complex Polynomials
On the Efficiency of Newton's Method in Approximating All Zeros of a System of Complex Polynomials
复制标题
论牛顿法逼近复多项式系全零点的效率
DOI：
10.1287/moor.12.1.121
复制
发表时间：
1987
期刊：
影响因子：
--
通讯作者：
J. Renegar
中科院分区：
文献类型：
--
作者：
J. Renegar
本文研究了一种基于牛顿法的算法逼近多项式组f =（f1，f2，…，f n）：（C-openface）n（右箭头）（C-openface）n . f的零点w的成功近似y的标准包括以下内容：给定（ω）> 0，y在w的距离（ω）内;应用于f并在y处开始的牛顿方法导致二次收敛到w ;给定（ω）> 0，|f i（y）|i = 1，2，...，n，其中||是（C-开放面）上的欧几里得范数。它示出，概率，每个零的f成功地近似在一个确定的步骤数。
This paper studies the efficiency of an algorithm based on Newton's method is approximating all zeros of a system of polynomials f = ( f 1 , f 2 , ..., f n ): (C-openface) n (rightarrow) (C-openface) n . The criteria for a successful approximation y of a zero w of f include the following: given (epsilon) > 0, y is within distance (epsilon) of w ; Newton's method applied to f and initiated at y results in quadratic convergence to w ; given (epsilon) > 0, | f i ( y )| i = 1, 2, ..., n , where | | is the Euclidean norm on (C-openface). It is shown that, probabilistically, each zero of f is successfully approximated within a determined number of steps.