The upper-crossing/solution (US) algorithm for root-finding with strongly stable convergence

TL;DR

This work introduces the upper-crossing/solution (US) algorithm, a broadly applicable non-bracketing root-finding framework built around two steps: constructing a simple surrogate $U(\theta|\theta^{(t)})$ and solving the surrogate's equation $U(\theta|\theta^{(t)})=0$ to obtain $\theta^{(t+1)}$. The core contribution is a formal definition of $U$-functions, a convergence theory guaranteeing strongly stable convergence to a unique root $\theta^*$ independent of the starting value, and four practical $U$-function construction methods (FLB, SLUB, SLB, SUB) plus a fixed-block approach. The paper provides convergence-rate results (linear, quadratic, cubic) for these methods, an acceleration scheme for fast convergence, and extensive numerical experiments across quantile computation, MLEs for Yule–Simon and generalized Poisson distributions, and high-order polynomial roots. Overall, the US framework enables robust, fast, initial-value-insensitive root-finding with broad statistical applications. The combination of strong convergence guarantees and flexible surrogate design positions US as a powerful tool for equations with multiple roots and for repeated root-finding tasks in statistics and applied mathematics.

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 $t$-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.

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