# Piecewise linear secant approximation via Algorithmic Piecewise Differentiation

## TL;DR

This paper introduces a method for locally approximating piecewise differentiable functions with a bilinear order error using only two sample points, and applies it to develop a generalized Newton's method with proven convergence properties.

## Contribution

It presents a novel approach for piecewise linear approximation of functions using minimal samples and derives a stable implementation for the method.

## Key findings

- Discrepancy between function and model is bilinear order O(‖x−x̂‖‖x−x̌‖)
- Generalized Newton's method based on piecewise linearization converges under certain conditions
- Formulas for numerically stable implementation of the approximation method

## Abstract

It is shown how piecewise differentiable functions $F: \mathbb R^n \mapsto \mathbb R^m $ that are defined by evaluation programs can be approximated locally by a piecewise linear model based on a pair of sample points $\check x$ and $\hat x$. We show that the discrepancy between function and model at any point $x$ is of the bilinear order $O(\|x-\check x\| \|x-\hat x\|)$. This is a little surprising since $x \in \mathbb R^n$ may vary over the whole Euclidean space, and we utilize only two function samples $\check F=F(\check x)$ and $\hat F=F(\hat x)$, as well as the intermediates computed during their evaluation. As an application of the piecewise linearization procedure we devise a generalized Newton's method based on successive piecewise linearization and prove for it sufficient conditions for convergence and convergence rates equaling those of semismooth Newton. We conclude with the derivation of formulas for the numerically stable implementation of the aforedeveloped piecewise linearization methods.