  For the basic maximum likelihood estimating function of the two parameters
Weibull distribution, a simple proof on its global monotonicity is given to
ensure the existence and uniqueness of its solution. The boundary of the
function's first-order derivative is defined based on its scale-free property.
With a bounded derivative, the possible range of the root of this function can
be determined. A novel root-finding algorithm employing these established
results is proposed accordingly, its convergence is proved analytically as
well. Compared with other typical algorithms for this problem, the efficiency
of the proposed algorithm is also demonstrated by numerical experiments.
