Publication: Towards a Universal Data Type for Scientific Computing
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Towards a Universal Data Type for Scientific Computing

- incollection -
 

Author(s)
Martin Berz

Published in
Automatic Differentiation of Algorithms: From Simulation to Optimization

Editor(s)
George Corliss, Christèle Faure, Andreas Griewank, Laurent Hascoët, Uwe Naumann

Year
2002

Publisher
Springer

Abstract
Modern scientific computing uses an abundance of data types. Besides floating point numbers, we routinely use intervals, univariate Taylor series, Taylor series with interval coefficients, and more recently multivariate Taylor series. Newer are Taylor models, which allow verified calculations like intervals, but largely avoid many of their limitations, including the cancellation effect, dimensionality curse, and low-order scaling of resulting width to domain width. Another more recent structure is the Levi-Civita numbers, which allow viewing many aspects of scientific computation as an application of arithmetic and analysis with infinitely small numbers, and which are useful for a variety of purposes including the assessment of differentiability at branch points. We propose new methods based on partially ordered Levi-Civita algebras that allow for a unification of all these various approaches into one single data type.

Cross-References
Corliss2002ADo

BibTeX
@INCOLLECTION{
         Berz2002TaU,
       author = "Martin Berz",
       title = "Towards a Universal Data Type for Scientific Computing",
       pages = "373--381",
       chapter = "45",
       crossref = "Corliss2002ADo",
       booktitle = "Automatic Differentiation of Algorithms: From Simulation to Optimization",
       year = "2002",
       editor = "George Corliss and Christ{\`e}le Faure and Andreas Griewank and Laurent
         Hasco{\"e}t and Uwe Naumann",
       series = "Computer and Information Science",
       publisher = "Springer",
       address = "New York, NY",
       abstract = "Modern scientific computing uses an abundance of data types. Besides floating point
         numbers, we routinely use intervals, univariate Taylor series, Taylor series with interval
         coefficients, and more recently multivariate Taylor series. Newer are Taylor models, which allow
         verified calculations like intervals, but largely avoid many of their limitations, including the
         cancellation effect, dimensionality curse, and low-order scaling of resulting width to domain width.
         Another more recent structure is the Levi-Civita numbers, which allow viewing many aspects of
         scientific computation as an application of arithmetic and analysis with infinitely small numbers,
         and which are useful for a variety of purposes including the assessment of differentiability at
         branch points. We propose new methods based on partially ordered Levi-Civita algebras that allow for
         a unification of all these various approaches into one single data type."
}


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