摘要
§1.引言、点估计 Sieve Smale在1986年国际数学家大会上介绍了他在连续复杂性理论方面的开创性研究.从报告摘要[1]及背景论文[2]来看,他着重介绍了解方程的整体代价,其基础是[3]关于Newton迭代的点估计的工作. 设f是从Banach空间E到同型空间F的解析映照.对于点z_0∈E。
This paper analyses domain estimates of the Kantorovich theorem by the workof Smale on Newton's iterative point estimates. For example, if f is an analytic mapfrom one Banach space to another, we determine the best absolute constant α suchthat it satisfies the Kantorovich condition if α(z,f)≤α, where α is about 0.1221.By precise error estimation for the Kantorovich theorem, another absolute constantα about 0.1215 is determined, which can be used to judge whether z is an approxi-mate zero of f. By the Kantorovich condition it is meant that there is a nonnegativereal number h≤(1/2) such that if ‖w-z‖≤(1-(1-2h)^(1/2)·‖z'-z‖/h, then‖Df(z)^(-1)D^2f(w)‖·‖z'-z‖≤h,where z'=z-Df(z)^(-1)f(z). Under this condition, Ne-wton's iteration z_(n+1) = z_n-Df(z_n)^(-1)f(z_n) is defined for all n∈N_0 and ‖z_(n+1)-z_n‖≤q^(2n-1)‖z_1-z_0‖, where z_0 = z, q = (1-(1-2h)^(1/2))/(1+(1-2h)^(1/2)). We discover thatthere is a constant h_0 about 0.4650 such that the bigger h is, the more difficult it isto satisfy the codintion of the Kantorovich theorem for h_0≤h≤(1/2).
出处
《计算数学》
CSCD
北大核心
1990年第1期47-53,共7页
Mathematica Numerica Sinica
基金
国家自然科学基金
