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"
}