Higher-order derivatives of the QR and of the real symmetric eigenvalue decomposition in forward and reverse mode algorithmic differentiation

TL;DR

This paper develops methods for efficiently computing higher-order derivatives of programs involving QR and eigenvalue decompositions using combined forward/reverse mode algorithmic differentiation, with explicit algorithms and illustrative examples.

Contribution

It introduces a novel approach combining Taylor polynomial arithmetic and matrix calculus for higher-order derivatives in AD, specifically for QR and eigenvalue decompositions.

Findings

Explicit algorithms for higher-order derivatives are derived.

The approach effectively combines forward and reverse mode AD.

Illustrative examples demonstrate the method's applicability.

Abstract

We address the task of higher-order derivative evaluation of computer programs that contain QR decompositions and real symmetric eigenvalue decompositions. The approach is a combination of univariate Taylor polynomial arithmetic and matrix calculus in the (combined) forward/reverse mode of Algorithmic Differentiation (AD). Explicit algorithms are derived and presented in an accessible form. The approach is illustrated via examples.