The Complex-Step Derivative Approximation on Matrix Lie Groups

TL;DR

This paper extends the complex-step derivative approximation to matrix Lie groups, enabling highly accurate numerical derivatives with a single evaluation, outperforming traditional methods in pose estimation applications.

Contribution

The paper introduces a novel extension of the complex-step derivative method to matrix Lie groups, maintaining machine precision accuracy for complex-valued functions.

Findings

Achieves analytical accuracy up to machine precision.

Outperforms central-difference schemes in accuracy.

Successfully applied to pose estimation problems.

Abstract

The complex-step derivative approximation is a numerical differentiation technique that can achieve analytical accuracy, to machine precision, with a single function evaluation. In this letter, the complex-step derivative approximation is extended to be compatible with elements of matrix Lie groups. As with the standard complex-step derivative, the method is still able to achieve analytical accuracy, up to machine precision, with a single function evaluation. Compared to a central-difference scheme, the proposed complex-step approach is shown to have superior accuracy. The approach is applied to two different pose estimation problems, and is able to recover the same results as an analytical method when available.