Derivatives of the full QR factorisation and of the factored-form and compact WY representations

TL;DR

This paper derives new formulas for the derivatives of the full QR factorisation, including the factored-form and compact WY representations, especially for tall matrices, enhancing automatic differentiation applications.

Contribution

It provides the first explicit derivatives for the full QR of tall matrices and the derivatives of the compact WY and factored-form representations of Q.

Findings

Derived formulas for the derivative of the full QR factorisation of tall matrices.

Extended automatic differentiation capabilities for QR-based computations.

Applications in nonlinear least squares and variable projection methods.

Abstract

QR factorisation plays an important role in matrix computations. Within the context of optimisation and of automatic differentiation of such computations, we need to compute the derivative of this factorisation. For tall matrices, however, existing results only cover the so-called thin case. We provide for the first time expressions for the derivative of the full QR factorisation of a tall matrix, in the usual case where the Q factor is a product of Householder reflections. These expressions are obtained based on novel results for the derivative of the compact WY representation of Q, which also yield the derivative of the factored-form representation of Q, both of which are useful on their own. These three results can be used directly in applications such as variable projection for solving separable non-linear least squares problems, and can also extend the current linear algebra capabilities of automatic differentiation frameworks.