We present an efficient and reliable method for computing the unit–in–the–last–place (ulp) of a double precision floating–point number, taking advantage of the standard binary representation for floating– point numbers defined by IEEE Std 754–1985. The ulp is necessary to perform software rounding for robust rounded interval arithmetic (RIA) operations. Hardware rounding, using two of the stand...

The numerical construction of polynomials in the product representation (as used for instance in variants of the multiboson technique) can become problematic if rounding errors induce an imprecise or even unstable evaluation of the polynomial. We give criteria to quantify the e ects of these rounding errors on the computation of polynomials approximating the function 1=s. We consider polynomial...

For a given sequence a = (a1, . . . , an) of numbers, a global rounding is an integer sequence b = (b1, . . . , bn) such that the rounding error |∑i∈I (ai − bi )| is less than one in all intervals I ⊆ {1, . . . , n}. We give a simple characterization of the set of global roundings of a. This allows to compute optimal roundings in time O(n logn) and generate a global rounding uniformly at random...

Consider an analytic map from a neighborhood of 0 in a vector space to a Euclidean space. Suppose that this map takes all germs of vector lines to germs of circles. Such map is called rounding. Two roundings are equivalent if they take the same lines to the same circles. We prove that any rounding whose differential at 0 has rank at least 2 is equivalent to a fractional quadratic rounding. The ...

It is important to understand the consequences of using the ROUND function in the SAS System. By default, numbers with a trailing 5 are rounded away from zero (i.e., 0.15 to 0.2 and -0.25 to -0.3), neglecting any numerical precision errors that a computer’s hardware limitations may introduce. In some instances, this type of rounding can lead to an overestimation of the true mean of a set of num...

Exact rounding is provided for elementary oating-point arithmetic operations (e.g. in the IEEE standard). Many authors have felt that it should be provided for other operations, in particular for geometric constructions. We show how one may round modular representation of numbers to the closest f.p. rep-resentable number, and demonstrate how it can be applied to a variety of geometric construct...

Various forms of dependent rounding are useful when handling a mixture of “hard” (e.g., matroid) constraints and “soft” (packing) constraints. We consider a few classes of such problems that arise in facility location, where one aims for small additive violations of the packing constraints, and where we require substantial “near-independence” properties among the variables being rounded. While ...

In search of key-performance predictors in sailing, we examined to what degree visual search, movement behaviour and boat control contribute to skilled performance while rounding the windward mark. To this end, we analysed 62 windward mark roundings sailed without opponents and 40 windward mark roundings sailed with opponents while competing in small regattas. Across conditions, results reveale...

The paper describes and studies an iterative algorithm for finding small values of a set of linear forms over vectors of integers. The algorithm uses a linear recurrence relation to generate a vector sequence, the basic idea being to choose the integral coefficients in the recurrence relation in such a way that the linear forms take small values, subject to the requirement that the integers sho...

There are two fundamental approximation algorithm design techniques based on linear programming: (a) LP-relaxation and rounding, and (b) the primal-dual method. In this lecture note, we will discuss the former. The idea of LP-relaxation and rounding is quite simple. We first formulate an optimization problem as an integer program (IP), which is like a linear program (LP) with integer variables....