报告题目：应用统计国家一流专业建设系列讲座——The upper-crossing/solution (US) algorithm for root-finding with strongly stable convergence(具有强稳定收敛性的求一元非线性方程之根的上穿求解算法)

主讲人：田国梁（南方科技大学）

时间：2022年11月22日（周二）10:00 a.m.

形式：线上讲座（腾讯会议）      

会议ID：888-695-182

主办单位：统计与数学学院

摘要：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  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 -function  [whose form depends on  being the -th iteration of  ] based on a new notion of so-called changing direction inequality, and the S-step solves the simple -equation  to obtain its explicit solution . The US algorithm holds two major advantages: (i) It strongly stably converges to the root ; 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 -function  such that an explicit solution to the -equation   is available. Based on the first-, second- and third-derivative of , three methods are given for constructing such -functions. We show various applications of the US algorithm in calculating quantile in continuous distributions, calculating exact -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.

主讲人简介：

田国梁博士曾在美国马里兰大学从事医学统计研究六年, 在香港大学统计与精算学系任副教授八年, 从2016年6月至今在南方科技大学统计与数据科学系任教授、博士生导师、副系主任。他目前的研究方向为EM/MM/US算法在统计中的应用、(0, 1) 区间上连续比例数据以及多元连续比例数据的统计分析、多元零膨胀计次数据分析, 在国外发表140篇SCI论文、出版3本英文专著、在科学出版社出版英文教材2本。他是四个国际统计期刊的副主编。主持国家自然科学基金面上项目二项、主持深圳市稳定支持面上项目一项、参加国家自然科学基金重点项目一项。