BibTeX
@INCOLLECTION{
         Makino1996RDA,
       author = "Kyoko Makino and Martin Berz",
       editor = "Martin Berz and Christian Bischof and George Corliss and Andreas Griewank",
       title = "Remainder Differential Algebras and their Applications",
       booktitle = "Computational Differentiation: Techniques, Applications, and Tools",
       pages = "63--74",
       publisher = "SIAM",
       address = "Philadelphia, PA",
       key = "Makino1996RDA",
       crossref = "Berz1996CDT",
       abstract = "In many practical problems in which derivatives are calculated, their basic purpose
         is to be used in the modeling of a functional dependence, often based on a Taylor expansion to first
         or higher orders. While the practical computation of such derivatives is greatly facilitated and in
         many cases is possible only through the use of forward or reverse computational differentiation,
         there is usually no direct information regarding the accuracy of the functional model based on the
         Taylor expansion. We show how, in parallel to the accumulation of derivatives, error bounds of all
         functional dependencies can be carried along the computation. The additional effort is minor, and
         the resulting bounds are usually rather sharp, in particular at higher orders. This Remainder
         Differential Algebraic Method is more straightforward and can yield tighter bounds than the mere
         interval bounding of the Taylor remainder's $(n+1)$st order derivative obtained via forward
         differentiation. The method can be applied to various numerical problems: Here we focus on global
         optimization, where blow-up can often be substantially reduced compared with interval methods, in
         particular for the cases of complicated functions or many variables. This problem is at the core of
         many questions of nonlinear dynamics and can help facilitate a detailed, quantitative
         understanding.",
       keywords = "Remainder Differential Algebras, Differential Algebras, error bound, interval
         method, high-order derivatives, Taylor polynomial, Taylor remainder, beam physics, COSY INFINITY,
         Fortran precompiler.",
       referred = "[Berz1996CIa], [Berz2002TaU], [Jerrell1996EoE], [Makino2002NAo].",
       year = "1996",
       ad_tools = "COSY INFINITY"
}