On the Application of the Best Multiplication to Division in Affine Arithmetic
by
Shinya Miyajima
School of Science and Engineering, Waseda University, Japan
Coauthors: Masahide Kashiwagi

Numerical computation with guaranteed accuracy means numerical methods which calculate not only numerical results but also mathematically rigorous error evaluation of the results. Recently, a lot of algorithms with guaranteed accuracy for various numerical methods have been developed.

Most of those algorithms are realized by interval arithmetic. In interval arithmetic, a real number is represented as an interval [ lower, upper ] whose endpoints are machine representable numbers, and rounding errors of floating point arithmetic is able to be grasped by directed rounding technique.

On the other hand, interval arithmetic has an essential weak point that it causes unexpected overestimation of calculation results. The main reason of the phenomenon is that correlation between quantities are ignored. A great deal of effort has been put into finding a way of overcoming the problem.

In order to overcome this problem, affine arithmetic, an extension of standard interval arithmetic, was proposed in 1994. Because affine arithmetic is able to express correlation between quantities, affine arithmetic is able to overcome the problem and it often produces much tighter bounds than a bound standard interval arithmetic yields.

On the one hand in affine arithmetic, it is difficult to realize the efficient division and various dividing methods have been proposed, but on the other hand the best multiplication which is the theoretical limit of the multiplication is proposed.

The purpose of this paper is to propose a new dividing method which is composed of the combination of the nonlinear unary operation that a reciprocal number is calculated and the best multiplication. And this paper also shows the efficiency of the new dividing method by numerical examples.

Date received: April 25, 2003

Copyright © 2003 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 Mathematical Conference Abstracts. Document # cajy-32.