  In many iterative optimization methods, fixed-point theory enables the
analysis of the convergence rate via the contraction factor associated with the
linear approximation of the fixed-point operator. While this factor
characterizes the asymptotic linear rate of convergence, it does not explain
the non-linear behavior of these algorithms in the non-asymptotic regime. In
this letter, we take into account the effect of the first-order approximation
error and present a closed-form bound on the convergence in terms of the number
of iterations required for the distance between the iterate and the limit point
to reach an arbitrarily small fraction of the initial distance. Our bound
includes two terms: one corresponds to the number of iterations required for
the linearized version of the fixed-point operator and the other corresponds to
the overhead associated with the approximation error. With a focus on the
convergence in the scalar case, the tightness of the proposed bound is proven
for positively quadratic first-order difference equations.
