Affine Arithmetic: Concepts and Applications

Algorithm for min-range multiplication of affine forms

“…Assuming that all ε j ( j = i) are fixed to +1 or −1 according to (6) or (8), i.e. the respective side is reached, the function f * becomes a second degree polynomial in ε i :…”

“…. , n), the noise symbols ε j and ε j are set to +1 or −1 according to formula (8) or (6) and the respective sums a x , a y and a x , a y (see formula (11)) are recalculated. After the inner loop is finished, the latter are used to compute the coefficients a, b , c and a, b , c of the respective second degree polynomials (9).…”

“…In addition, it takes into account correlations between computed and input quantities, and is, therefore, able to provide much tighter bounds on computed quantities than standard interval arithmetic. In affine arithmetic [2,[4][5][6] an unknown ideal quantity x is represented by an affine formx = x 0 + x 1 ε 1 + . .…”

“…Given a non-affine function of two variables z = f (x, y) and two affine formsx andŷ representing x and y, an affine formẑ representing z must be computed. It is desirable thatẑ is consistent withx andŷ and that it preserves the information provided by them as much as possible [6]. It can be easily seen that z = f (x,ŷ) is a function of the noise symbols ε i :…”

“…This means that there are n + 1 degrees of freedom for the choice of an affine approximation f a . In fact, two basic approaches to compute affine approximation are used the most frequently [6]. The first one is to minimise the approximation error z k (Chebyshev approximation), the second one is to minimise the range [x ·ŷ] (minimum range or shortly min-range approximation).…”