Publication: Automatic Differentiation Through the Use of Hyper-Dual Numbers for Second Derivatives
Introduction
Applications
Tools
Research Groups
Workshops
Publications
   List Publications
   Advanced Search
   Info
   Add Publications
My Account
About

Automatic Differentiation Through the Use of Hyper-Dual Numbers for Second Derivatives

- incollection -
 

Author(s)
Jeffrey A. Fike , Juan J. Alonso

Published in
Recent Advances in Algorithmic Differentiation

Editor(s)
Shaun Forth, Paul Hovland, Eric Phipps, Jean Utke, Andrea Walther

Year
2012

Publisher
Springer

Abstract
Automatic Differentiation techniques are typically derived based on the chain rule of differentiation. Other methods can be derived based on the inherent mathematical properties of generalized complex numbers that enable first-derivative information to be carried in the non-real part of the number. These methods are capable of producing effectively exact derivative values. However, when second-derivative information is desired, generalized complex numbers are not sufficient. Higher-dimensional extensions of generalized complex numbers, with multiple non-real parts, can produce accurate second-derivative information provided that multiplication is commutative. One particular number system is developed, termed hyper-dual numbers, which produces exact first- and second-derivative information. The accuracy of these calculations is demonstrated on an unstructured, parallel, unsteady Reynolds-Averaged Navier-Stokes solver.

Cross-References
Forth2012RAi

AD Theory and Techniques
Hessian

BibTeX
@INCOLLECTION{
         Fike2012ADT,
       title = "Automatic Differentiation Through the Use of Hyper-Dual Numbers for Second
         Derivatives",
       doi = "10.1007/978-3-642-30023-3_15",
       author = "Jeffrey A. Fike and Juan J. Alonso",
       abstract = "Automatic Differentiation techniques are typically derived based on the chain rule
         of differentiation. Other methods can be derived based on the inherent mathematical properties of
         generalized complex numbers that enable first-derivative information to be carried in the non-real
         part of the number. These methods are capable of producing effectively exact derivative values.
         However, when second-derivative information is desired, generalized complex numbers are not
         sufficient. Higher-dimensional extensions of generalized complex numbers, with multiple non-real
         parts, can produce accurate second-derivative information provided that multiplication is
         commutative. One particular number system is developed, termed hyper-dual numbers, which produces
         exact first- and second-derivative information. The accuracy of these calculations is demonstrated
         on an unstructured, parallel, unsteady Reynolds-Averaged Navier-Stokes solver.",
       pages = "163--173",
       crossref = "Forth2012RAi",
       booktitle = "Recent Advances in Algorithmic Differentiation",
       series = "Lecture Notes in Computational Science and Engineering",
       publisher = "Springer",
       address = "Berlin",
       volume = "87",
       editor = "Shaun Forth and Paul Hovland and Eric Phipps and Jean Utke and Andrea Walther",
       isbn = "978-3-540-68935-5",
       issn = "1439-7358",
       year = "2012",
       ad_theotech = "Hessian"
}


back
  

Contact:
autodiff.org
Username:
Password:
(lost password)