论文标题
用于根系探测的上层/溶液（US）算法，具有强稳定的收敛性
The upper-crossing/solution (US) algorithm for root-finding with strongly stable convergence
论文作者
论文摘要
在本文中，我们提出了一种新的且广泛的根发现方法，称为上跨/解决方案（US）算法，该方法属于非支架（或开放域）方法的类别。美国算法是迭代地寻求非线性方程$ g（θ）= 0 $的独特根$θ^{*} $的一般原则$ u（θ|θ^{（t）}）$ [其形式取决于$θ^{（t）} $是$ t $ - 基于更改方向的新概念的$θ^{*} $]的$ t $ - 迭代，而S-step则解决了简单的$ u $ -e $ u $ u $ u $ u $ u $ u $ u $ u $ u $ u $ u $ u $ u $ u $ u $ u $ u $ u $ u $ u $ u $ - $θ^{（T+1）} $。美国算法具有两个主要优点：（i）它强烈稳定地收敛到根$θ^{*} $; （ii）与牛顿的方法相反，它不取决于任何初始值。应用美国算法的关键步骤是构造一个简单的$ u $ -function $ u（θ|θ^{（t）}）$，以便对$ u $ equient $ u（θ|θ^{（t）}）的明确解决方案= 0 = 0 $可用。基于$ g（θ）$的第一，第二和第三个衍生物，给出了三种构建此类$ u $ functions的方法。我们显示了美国算法在连续分布中计算分位数中的各种应用程序，计算偏斜零分布的精确$ p $值，并找到一类连续/离散分布中参数的最大似然估计。还提供了美国算法的收敛速率和一些数值实验的分析。尤其是，由于具有强稳定的融合的特性，美国算法可能是解决具有多个根部的方程式的强大工具之一。
In this paper, we propose a new and broadly applicable root-finding method, called as the upper-crossing/solution (US) algorithm, which belongs to the category of non-bracketing (or open domain) methods. The US algorithm is a general principle for iteratively seeking the unique root $θ^{*}$ of a non-linear equation $g(θ)=0$ and its each iteration consists of two steps: an upper-crossing step (U-step) and a solution step (S-step), where the U-step finds an upper-crossing function or a $U$-function $U(θ|θ^{(t)})$ [whose form depends on $θ^{(t)}$ being the $t$-th iteration of $θ^{*}$] based on a new notion of so-called changing direction inequality, and the S-step solves the simple $U$-equation $U(θ|θ^{(t)}) =0$ to obtain its explicit solution $θ^{(t+1)}$. The US algorithm holds two major advantages: (i) It strongly stably converges to the root $θ^{*}$; and (ii) it does not depend on any initial values, in contrast to Newton's method. The key step for applying the US algorithm is to construct one simple $U$-function $U(θ|θ^{(t)})$ such that an explicit solution to the $U$-equation $U(θ|θ^{(t)}) =0$ is available. Based on the first-, second- and third-derivative of $g(θ)$, three methods are given for constructing such $U$-functions. We show various applications of the US algorithm in such as calculating quantile in continuous distributions, calculating exact $p$-values for skew null distributions, and finding maximum likelihood estimates of parameters in a class of continuous/discrete distributions. The analysis of the convergence rate of the US algorithm and some numerical experiments are also provided. Especially, because of the property of strongly stable convergence, the US algorithm could be one of the powerful tools for solving an equation with multiple roots.
