Given a vector of floating-point numbers with exact sum s, we present an algorithm for calculating a faithful rounding of s into the set of floating-point numbers, i.e. one of the immediate floating-point neighbors of s. If the s is a floating-point number, we prove that this is the result of our algorithm. The algorithm adapts to the condition number of the sum, i.e. it is very fast for mildly conditioned sums with slowly increasing computing time proportional to the condition number. All statements are also true in the presence of underflow. Furthermore algorithms with K-fold accuracy are derived, where in that case the result is stored in a vector of K floating-point numbers. We also present an algorithm for rounding the sum s to the nearest floating-point number. Our algorithms are fast in terms of measured computing time because they neither require special operations such as access to mantissa or exponent, they contain no branch in the inner loop, nor do they require extra precision: The only operations used are standard floating-point addition, subtraction and multiplication in one working precision, for example double precision. Moreover, in contrast to other approaches, the algorithms are ideally suited for parallelization. We also sketch dot product algorithms with similar properties.

This talk is concerned with the orientation problems, which are the basic problems in computational geometry. The orientation method recognizes that a point is to the left, to the right or on a oriented line in case of 2dimensional space and a oriented plane in case of 3-dimensional space. These problems can be boiled down to the determinant predicate, i.e. whether the sign of the determinant is positive, negative or zero. Especially in this talk, we focus our mind on 2D / 3D orientation problems. It is known that the computation of the determinant can be transformed into that of a summation without rounding errors by applying so-called “error-free transformation ” to floating-point operations. The 2D orientation problem can be transformed into the computation of summation with 12 or 16 terms and the 3D orientation problem can be also transformed into the computation of summation with 96 or 192 terms. If one uses robust summation algorithms after this transformation, the sign of determinant can be guaranteed. For the orientation methods, Shewchuk developed an algorithm which uses his robust

In this paper, we present a recent algorithm given by Ogita, Rump and Oishi [39] for accurately computing the sum of n floating point numbers. They also give a computational error bound for the computed result. We apply this algorithm in computing determinant and more particularly in computing robust geometric predicates used in computational geometry. We improve existing results that use either a multiprecision libraries or extended large accumulators.

This paper discusses both the theoretical and statistical errors obtained by various well-known dot products, from the canonical to pairwise algorithms, and introduces a new and more general framework that we have named superblock which subsumes them and permits a practitioner to make trade-offs between computational performance, memory usage, and error behavior. We show that algorithms with lower error bounds tend to behave noticeably better in practice. Unlike many such error-reducing algorithms, superblock requires no additional floating point operations and should be implementable with little to no performance loss, making it suitable for use as a performance-critical building block of a linear algebra kernel.

algorithms using rounding to odd