Differentiation via Logarithmic Expansions

TL;DR

The paper introduces BLEND, a novel finite difference method using logarithmic expansions for highly precise derivative approximations, with efficient parallelization and dimension-independent complexity in vector cases.

Contribution

It presents the BLEND algorithm, a new logarithmic expansion-based finite difference method offering arbitrary precision and efficient computation, especially in high-dimensional vector settings.

Findings

Provides a formal logarithmic expansion framework for derivatives

Achieves arbitrarily high precision in derivative approximation

Complexity in vector cases is independent of dimension

Abstract

In this note, we introduce a new finite difference approximation called the Black-Box Logarithmic Expansion Numerical Derivative (BLEND) algorithm, which is based on a formal logarithmic expansion of the differentiation operator. BLEND capitalizes on parallelization and provides derivative approximations of arbitrarily precision, i.e., our analysis can be used to determine the number of terms in the series expansion to guarantee a specified number of decimal places of accuracy. Furthermore, in the vector setting, the complexity of the resulting directional derivative is independent of the dimension of the parameter.