On Floating-Point Summation

Espelid, Terje O.

doi:10.1137/1037130

Cited by 24 publications

(3 citation statements)

References 7 publications

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“…. , x n the classical analysis bounds the absolute error by γ ℓ n i=1 |x i |, where ℓ is the height of the binary tree underlying the evaluation order [1,3]. For pairwise summation this tree has the minimum possible height, namely, ℓ = ⌈log 2 n⌉, but in this case the actual error can be larger than ℓu n i=1 |x i |.…”

Section: Preliminariesmentioning

confidence: 99%

Improved Backward Error Bounds for LU and Cholesky Factorizations

Rump¹,

Jeannerod²

2014

SIAM J. Matrix Anal. & Appl.

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Assuming standard floating-point arithmetic (in base β, precision p) and barring underflow and overflow, classical rounding error analysis of the LU or Cholesky factorization of an n × n matrix A provides backward error bounds of the form |∆A| γn| L|| U | or |∆A| γ n+1 | R T || R|. Here, L, U , and R denote the computed factors, and γn is the usual fraction nu/(1−nu) = nu+O(u 2) with u the unit roundoff. Similarly, when solving an n × n triangular system T x = b by substitution, the computed solution x satisfies (T + ∆T) x = b with |∆T | γn|T |. All these error bounds contain quadratic terms in u and limit n to satisfy either nu < 1 or (n + 1)u < 1. We show in this paper that the constants γn and γ n+1 can be replaced by nu and (n + 1)u, respectively, and that the restrictions on n can be removed. To get these new bounds the main ingredient is a general framework for bounding expressions of the form |ρ − s|, where s is the exact sum of a floating-point number and n − 1 real numbers, and where ρ is a real number approximating the computed sum s. By instantiating this framework with suitable values of ρ, we obtain improved versions of the well-known Lemma 8.4 from [N. J. Higham, Accuracy and Stability of Numerical Algorithms, SIAM, 2002] (used for analyzing triangular system solving and LU factorization) and of its Cholesky variant. All our results hold for rounding to nearest with any tie-breaking strategy and no matter what the order of summation.

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Section: Preliminariesmentioning

confidence: 99%

Improved Backward Error Bounds for LU and Cholesky Factorizations

Rump¹,

Jeannerod²

2014

SIAM J. Matrix Anal. & Appl.

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“…The inexactness of floating point arithmetic makes it hard to compute this sign exactly in some cases. There are already several references about the computation sums of floating point numbers [3,4,5,6,7,8,10,12,13,15,16,17], with applications in computations geometry. The idea of decomposing products in sums is also as old as [2,11], and the fourth chapter of [14] presents an algorithm for obtaining the signs of such sums.…”

Section: Introductionmentioning

confidence: 99%

Computing the exact sign of sums of products with floating point arithmetic

Mascarenhas¹

2021

Preprint

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In computational geometry, the construction of essential primitives like convex hulls, Voronoi diagrams and Delaunay triangulations require the evaluation of the signs of determinants, which are sums of products. The same signs are needed for the exact solution of linear programming problems and systems of linear inequalities. Computing these signs exactly with inexact floating point arithmetic is challenging, and we present yet another algorithm for this task. Our algorithm is efficient and uses only of floating point arithmetic, which is much faster than exact arithmetic. We prove that the algorithm is correct and provide efficient and tested C++ code for it.

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“…Motwani, Panigrahy, and Xu [14] showed an O( √ n) time approximation scheme for computing the sum of n nonnegative elements. There is a long history of research for the accuracy of summation of floating point numbers (for examples, see [10,2,1,4,5,6,8,11,12,13,15,16]). The efforts were mainly spent on finding algorithms with small rounding errors.…”

Section: Introductionmentioning

confidence: 99%

On the Complexity of Approximate Sum of Sorted List

Fu¹

2013

Frontiers in Algorithmics and Algorithmic Aspects in Information and Management

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We consider the complexity for computing the approximate sum a1 + a2 + · · · + an of a sorted list of numbers a1 ≤ a2 ≤ · · · ≤ an. We show an algorithm that computes an (1 + ǫ)-approximation for the sum of a sorted list of nonnegative numbers in an O( 1 ǫ min(log n, log( xmax x min )) · (log 1 ǫ + log log n)) time, where xmax and xmin are the largest and the least positive elements of the input list, respectively. We prove a lower bound Ω(min(log n, log( xmax x min )) time for every O(1)-approximation algorithm for the sum of a sorted list of nonnegative elements. We also show that there is no sublinear time approximation algorithm for the sum of a sorted list that contains at least one negative number.

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On Floating-Point Summation

Abstract: In this paper we focus on some general error analysis results in floating-point summation. We emphasize analysis useful from both a scientific and a teaching point of view.

Cited by 24 publications

References 7 publications

Improved Backward Error Bounds for LU and Cholesky Factorizations

Improved Backward Error Bounds for LU and Cholesky Factorizations

Computing the exact sign of sums of products with floating point arithmetic

On the Complexity of Approximate Sum of Sorted List

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