Piecewise linear secant approximation via algorithmic piecewise differentiation

Abstract

It is shown how piecewise differentiable functions F : R n → R m that are defined by evaluation programs can be approximated locally by a piecewise linear model based on a pair of sample pointsx andx. We show that the discrepancy between function and model at any point x is of the bilinear order O( x −x x −x ). 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.

Integrating Lipschitzian dynamical systems using piecewise algorithmic differentiation

“…F is, by assumption, piecewise composite differentiable and thus locally Lipschitz continuous. Moreover, by [GSHR,Prop. 4.2] we know that the piecewise linearization is Lipschitz continuous.…”

“…4.2] we know that the piecewise linearization is Lipschitz continuous. Consequently, with [GSHR,Prop. 4.2] there exists a ball B ρ (0) about the base point x 0 = 0, such that for all x ∈ B ρ (0) there exists a β F > 0, such that it holds…”

“…and the latter expression, in contrast to the construction of the classical trapezoidal rule, by its piecewise linearization, where equality holds as consequence of [GSHR,Prop. 4.2]…”

“…7]. Additionally, a division-free implementation and thus numerically stable implementation is discussed in [GSHR,Section 6].…”

Feature Representation and Similarity Measure Based on Covariance Sequence for Multivariate Time Series

“…There exist many algorithms and related methods used to reduce the dimensionality of univariate time series, such as piecewise linear approximation [7], [8], piecewise aggregate approximation [7], [8], symbolic aggregated approximation [9], [10], polynomial representation [11] and…”