  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 $\theta^{*}$ of a
non-linear equation $g(\theta)=0$ 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 $U(\theta|\theta^{(t)})$
[whose form depends on $\theta^{(t)}$ being the $t$-th iteration of
$\theta^{*}$] based on a new notion of so-called changing direction inequality,
and the S-step solves the simple $U$-equation $U(\theta|\theta^{(t)}) =0$ to
obtain its explicit solution $\theta^{(t+1)}$. The US algorithm holds two major
advantages: (i) It strongly stably converges to the root $\theta^{*}$; 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
$U(\theta|\theta^{(t)})$ such that an explicit solution to the $U$-equation
$U(\theta|\theta^{(t)}) =0$ is available. Based on the first-, second- and
third-derivative of $g(\theta)$, 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.
