一个寻求解析函数零点的单纯同伦算法及复杂性分析

§1.引言 求解一维实函数的零点,二分法为我们提供了一种有效的整体解法。通常,对于复变函数不仅有实零点,还有复零点,那么能否用二分法的思想来求解复变函数的零点呢?与二分法对应的一个概念是幅角原理,对于直接利用这个原理来确定复函数在某有界区域内零点的问题,虽然作过大量的尝试,但成功者甚少,譬如,Delves-Lyness在2]中构造的算法,由于反复运算而导致计算效率非常低。D.H.Lehmer对上述原理作了进一

AN ALGORITHM FOR FINDING ALL ZEROS OF ANALYTIC FUNCTIONS AND ITS COMPLEXITY ANALYSIS

This paper presents a global algorithm for numerically Obtaining all zeros ofan analytic function in a compact region over the complex plane by use of the complex degreereported in paper 6]. A great deal of numerical examples as well as the conve?enceanalysis imply that this simple algorithm is safe and efficient, and is of moderate computa-tional cost. Furthermore, when this algorithm is,used to determine all zeros of an algebraicpolynomial, a triangular polynomial; or an exponential polynomial, the safe grid size of thetriangulation involved in the algorithm can be easily estimated. Our results also show thealgorithm proceeds well with the 2-dimensional compact manifold having boundaries.