• PDF / 396,241 Bytes
• 14 Pages / 439.37 x 666.142 pts Page_size
• 107 Downloads / 205 Views

DOWNLOAD

REPORT

---

Algorithm for min-range multiplication of affine forms Iwona Skalna

Received: 8 November 2011 / Accepted: 30 August 2012 © The Author(s) 2012. This article is published with open access at Springerlink.com

Abstract Affine arithmetic produces guaranteed enclosures for computed quantities, taking into account any uncertainties in the input data as well as round-off errors. Elementary operations on affine forms are redefined so they result in affine forms. Affine-linear operations result straightforwardly in affine forms. Non-linear operators, such as multiplication, must be approximated by affine forms. Choosing the appropriate approximation is a big challenge. The reason is that different approximations may be more accurate for specific purposes. This paper presents an efficient method for computing the minimum range (min-range) affine approximation of the product of arbitrary affine forms that do not contain zero properly. Numerical experiments are carried out to demonstrate the essential features of the proposed approach, especially its usefulness for bounding ranges of functions for global optimisation and for finding roots of functions. Keywords Affine arithmetic · Multiplication · Min-range approximation · Range bounding · Global optimisation · Roots of functions Notation xˆ xˆ , yˆ  x, [x] x x int(x)

an affine form the joint range of two affine forms xˆ and yˆ a closed interval the lower bound (left endpoint) of an interval x the upper bound (right endpoint) of an interval x the interior of an interval x

I. Skalna (B) AGH University of Science and Technology, Krakow, Poland e-mail: [email protected]

Numer Algor

xˇ = (x + x)/2 r(x) = (x − x)/2 [a, b ] xˆ , [xˆ ] x >= 0 (x