Fixed-point iteration with a residual stop and finite budget

Let $(X,d)$ be a metric space (Metric space: $d(x,y)=0$ iff $x=y$, symmetry, and the triangle inequality; pseudometric and ultrametric), let $G:X\to X$ be a specified map, and choose an initial point $x_0\in X$, a residual tolerance $\epsilon \ge 0$, and an iteration budget $N\in {\mathbb{N}}_0$. A fixed point of $G$ is a point $x$ with $G(x)=x$. The procedure returns a point and a status as follows. For $k=0,1,\dots ,N$, compute the fixed-point residual $r_k:=d(G(x_k),x_k)$. If $r_k\le \epsilon$, return $(x_k,\mathtt{\text{residual-tolerance-met}},r_k,k)$. Otherwise, if $k=N$, return $(x_k,\mathtt{\text{iteration-budget-exhausted}},r_k,k)$. Otherwise set $x_{k+1}:=G(x_k)$ and continue.

A positive residual tolerance reports an approximate fixed point only; this definition asserts no convergence or error bound.