Differentiation via Logarithmic Expansions

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 arbitrary 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.