By Gutierrez J., Shpilrain V., Yu J.-T. (eds.)

ISBN-10: 0821834762

ISBN-13: 9780821834763

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5. Let f : D —τ R, DC R be a programmable function, that is, a function which may be described as an explicit expression without use of logical or conditional statements (such as "if ... ) in the following manner: Each function value f(x),x € D, can be written down as an expression m Inclusion Functions 31 (also denoted by f(x)) which is independent of the value of χ and such that this expression consists only of (1) the variable or arguments χ (or their components x i , . ). Let Y G 1(D) then the natural interval extension of / to Y is defined as that expression which is obtained from the expression f(x) by replacing each occurrence of the variable χ by the box Y, the arithmetic operations of R by the corresponding interval arithmetic operations, and each occurrence of a predeclared function g by the corresponding inclusion function G.

Ex tensive test series with randomly generated summands indicate that already 5% of the summands are in most cases sufficient to decide the sign of the sum after some preprocessing. (iv) If the summands are machine numbers, the computed sign is always the exact sign, since the computation is rounding-error-free. (v) No variable length fixed point accumulator is necessary in order to cover the field of possible partial sums. (vi) Exponential overflow and underflow control is not part of the algo rithm.

2 2 If, for instance, A = [-1, 2], then A - [-2, 4], A = [0, 4], and one obtains p(A) = [-36, 72] and P{A) = [0, 72], where, accidentally, P(A) is already the range of ρ over the box Α χ A χ A, that is, D (A) = P(A). These few examples show the importance of choosing a good formula for the evaluation of an inclusion for the range. Theoretical investigations in this area are found in [235, 14, 149, 217]. The selection of the form that results in the narrowest inclusion is a difficult problem since there is an infinite number of possible forms.

### Affine Algebraic Geometry by Gutierrez J., Shpilrain V., Yu J.-T. (eds.)

by David

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