# Towards an Efficient Implementation of Interval Arithmetic

### Ioannis Z. Emiris and Richard J. Fateman

###
EECS Department

University of California, Berkeley

Technical Report No. UCB/CSD-92-693

July 1992

### http://www.eecs.berkeley.edu/Pubs/TechRpts/1992/CSD-92-693.pdf

Interval mathematics is well studied and offers a solution to several important problems arising in computer arithmetic, such as error bounding and sign tests for constant expressions. However, modern computer algebra systems have a limited arsenal of routines dealing with intervals which, moreover, produce unnecessarily crude answers. This work attempts to fill in this gap, by proposing efficient yet simple algorithms for function evaluation and equation solving. The main contributions are, first, an evaluation procedure that outputs tight bounds for a large class of continuous differentiable functions and, second, an adaptation of Newton's iteration. Both improve upon the routines of Maple V and Mathematica 2.0 for functions with complex or real interval coefficients and arguments.

BibTeX citation:

@techreport{Emiris:CSD-92-693, Author = {Emiris, Ioannis Z. and Fateman, Richard J.}, Title = {Towards an Efficient Implementation of Interval Arithmetic}, Institution = {EECS Department, University of California, Berkeley}, Year = {1992}, Month = {Jul}, URL = {http://www.eecs.berkeley.edu/Pubs/TechRpts/1992/6247.html}, Number = {UCB/CSD-92-693}, Abstract = {Interval mathematics is well studied and offers a solution to several important problems arising in computer arithmetic, such as error bounding and sign tests for constant expressions. However, modern computer algebra systems have a limited arsenal of routines dealing with intervals which, moreover, produce unnecessarily crude answers. This work attempts to fill in this gap, by proposing efficient yet simple algorithms for function evaluation and equation solving. The main contributions are, first, an evaluation procedure that outputs tight bounds for a large class of continuous differentiable functions and, second, an adaptation of Newton's iteration. Both improve upon the routines of Maple V and Mathematica 2.0 for functions with complex or real interval coefficients and arguments.} }

EndNote citation:

%0 Report %A Emiris, Ioannis Z. %A Fateman, Richard J. %T Towards an Efficient Implementation of Interval Arithmetic %I EECS Department, University of California, Berkeley %D 1992 %@ UCB/CSD-92-693 %U http://www.eecs.berkeley.edu/Pubs/TechRpts/1992/6247.html %F Emiris:CSD-92-693