following section we shall briefly revisit an adaptive prediction-type Newton algorithm from [@AmreinWihler:15].

Abstract Framework
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Let $ X, Y $ be two Banach spaces, with norms $\|\cdot\|_X$ and $\|\cdot\|_Y$, respectively. Given an open subset $\Xi\subset X$, and a (possibly nonlinear) operator ${\mathsf{F}}:\,\Xi\to Y$, we consider the [*nonlinear*]{} operator equation $$\label{eq:F=0}
{\mathsf{F}}(u)=0,$$ for some unknown zeros $u\in\Xi$. Supposing that the Fréchet derivative ${\mathsf{F}}'$ of ${\mathsf{F}}$ exists in $\Xi$ (or in a suitable subset), the classical Newton method for solving  starts from an initial guess $u_0\in\Xi$, and generates the iterates $$\label{eq:newton}
u_{n+1}=u_n+\delta_n,\qquad n\ge 0,$$ where the update $\delta_n\in X$ is implicitly given by the [*linear*]{} equation $${\mathsf{F}}'(u_n)\delta_n=-{\mathsf{F}}(u_n),\qquad n\ge 0.$$ Naturally, for this iteration to be well-defined, we need to assume that ${\mathsf{F}}'(u_n)$ is invertible for all $n\ge 0$, and that $\{u_n\}_{n\ge 0}\subset\Xi$.

An Adaptive Prediction Strategy {#sc:newton}
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In order to improve the reliability of the Newton method  in the case that the initial guess $u_0$ is relatively far away from a root $u_{\infty}\in\Xi$ of ${\mathsf{F}}$, ${\mathsf{F}}(u_\infty)=0$, introducing some damping in the Newton method is a well-known remedy. In that case  is rewritten as $$\label{eq:damped}
u_{n+1}=u_n-\Delta t_n{\mathsf{F}}'(u_n)^{-1}{\mathsf{F}}(u_n),\qquad n\ge 0,$$ where $\Delta t_n>0$, $n\ge 0$, is a damping parameter that may be 