Low complexity k-dimensional centered forms

Abstract

--ZusammenfassungLow Complexity k-dimensional Centered Forms. A new method for computing the centered form in k dimensions for polynomials and rational functions is presented. T~le method is not nearly as computationally intensive as the method proposed in Ratschek-Schroder [-7] in that it avoids the calculation of the partial derivatives of the functions. The method is also easier to implement than the slope method proposed by . Some numer.ic~l 'results are given.

Enhanced Interval Analysis for Phase Stability: Cubic Equation of State Models

“…There have been considerable e orts made toward developing systematic methods for most sharply bounding the range of a given real function over an interval e.g., Ratschek and Rokne, 1984;Rokne, 1986;Neumaier, 1990. Among these are the use of centered forms of the function, such a s the mean value form, and the use of slope forms.…”

The number of centered forms for a polynomial

“…We continue with xj2 --= Xl and get p2(xl) = c2 +c2x2xl, p2(cl) = c2 +c2x~cl X1X2+CIC2X2+ClC2+I)) is the kernel of the centered form. If X = (X~, Xz)el 2 such that mid(Xi) = % i = 1,2 (the midpoints of the intervals) then p(X) would provide an outer estimate for the range of p(x) over X (see also [9]). …”

“…Assuming that tl/r~ and tHr J include a common variable product v = {tl/ri, t~/ 2-N(vz). This number has to be subtracted from the total number (9).…”

“…Here we focus on the developments that resulted in a computationally efficient form in [9]. This form is now summarized by the following algorithm where we employ the notation of multi-indices (see for example [8]).…”

Interval Taylor forms

“…~ e-t~/Zd co (15) using an implementation of the pseudo-Pascal code given in the last section in FORTRAN. As an example of the calculation procedure some symbolic calculations are given in Table 3.…”