若干变形Newton迭代的点估计

POINT ESTIMATES ON DEFORMATION NEWTON'S ITERATIONS

引言 设E和F同是实的或同是复的Banach空间,f:E→F是一个非线性映照.由于方程 f(z)=0具有很强的概括性,所以用以求解这个方程的Newton迭代 z_(n+1)=z_n-Df(z_n)^(-1)f(z_n),?n∈N_0几乎成了经典应用数学的中心.

Let f:E→F be an analytic map from one Banach space to another. We call theiteration z_(n+1)=z_n—A_nf(z_n)^(-1),A_n∈N_0, a deformation Newton's iteration,if the mapA_n: F→E, is substituted for Df (z_n)^(-1) in a certain form. By means of Smale'spoint estimates for Newton's iteration, this paper studies the convergence of the fo-llowing three deformation Newton's iterations whose computational efficiencies arehigher than Newton's':(Ⅰ) A_n = Df(m[n/m])^(-1), for m N fixed;(Ⅱ) A_(n+1)=2A_n-A_nDf(z_(n+1)A_n;(Ⅲ) A_n=Df(ζ_n)^(-1),ζ_(n+1)=ζ_n-1/2A_nf(z_(n+1)). For m∈N fixed in (Ⅰ), A_0= Df(z_0)^(-1) in (Ⅱ) and ζ_0=z_0 in (Ⅲ), the resultsshow that convergence conditions of the above-mentioned three iterations are the sa-me as those of Newton's iteration.