Towards a Universal Data Type for Scientific Computing
by
M. Berz

There is an abundance of data types in use for modern scientific computing; besides floating point numbers, already established are 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 are the Levi-Civita numbers, which allow to view 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. Besides utilizing the strengths of each of these approaches, in addition the method is able to provide rigorous estimates for the asymptotics of all high-order derivatives. Questions of implementation and efficiency are also addressed, and it is shown that the cost of the method for each sub-task compares favorably with the separate methods above.

http://homepages.feis.herts.ac.uk/~comqun/AD2000/Ext_Abstracts/Berz.txt

Date received: February 11, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cads-70.