Towards a Universal Data Type for Scientific Computing
AbstractModern 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.
M. Berz, in: "Automatic Differentiation: From Simulation to Optimization", G. Corliss, C. Faure, A. Griewank, L. Hascoet, U. Naumann (Eds.) (2001) Springer
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