On evaluating higher-order derivatives of the QR decomposition of tall matrices with full column rank in forward and reverse mode algorithmic differentiation

Abstract

We address the task of higher-order derivative evaluation of computer programs that contain QR decompositions of tall matrices with full column rank. 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.

Keywords:

• automatic differentiation
• null space
• univariate Taylor polynomial arithmetic

AMS Subject Classification :

• 65D25
• 12E05
• 58C15
• 65F15
• 65F25

Acknowledgements

The authors thank Bruce Christianson for his comments that helped to gain a deeper understanding of the matter and also to the anonymous reviewers who greatly helped to improve the readability of this work. This research was partially supported by the Bundesministerium für Bildung und Forschung (BMBF) within the project NOVOEXP (Numerische Optimierungsverfahren für die Parameterschätzung und den Entwurf optimaler Experimente unter Berücksichtigung von Unsicherheiten für die Modellvalidierung verfahrenstechnischer Prozesse der Chemie und Biotechnologie) (03GRPAL3), Humboldt-Universität zu Berlin.