报告题目:The upper-crossing/solution (US) algorithm for root-finding with strongly stable convergence
报告时间:2023-11-03 8:30-9:30
报 告 人 :田国梁 教授 南方科技大学
报告地点:老外楼概率统计系办公室
Abstract: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-thiteration 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
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.Thisis a joint work with Dr. XunjianLI.