Efficient minimization of numerical summation errors

Ming Yang Kao, Jie Wang

Research output: Chapter in Book/Report/Conference proceedingConference contribution

1 Scopus citations

Abstract

Given a multiset X = {x1,⋯, xn} of real numbers, the floating-point set summation (FPS) problem asks for Sn = x1 + ··· + xn, and the floating point prefix set summation problem (FPPS) asks for Sk = x1 + ··· + Xk for all k = 1,⋯, n. Let E*k denote the minimum worst-case error over all possible orderings of evaluating Sk-We prove that if X has both positive and negative numbers, it is NP-hard to compute Sn with the worst-case error equal to En. We then give the first known polynomial-time approximation algorithm for computing Sn that has a provably small error for arbitrary X. Our algorithm incurs a worstcase error at most 2([log(n - 1)] + 1)E*n.1 After X is sorted, it runs in O(n) time, yielding an O(n2)-time approximation algorithm for computing Sk for all k = 1,⋯, n such that the worst-case error for each Sk is less than 2[log(k - 1)1 + 1)E*k. For the case where X is either all positive or all negative, we give another approximation algorithm for computing Sn with a worst-case error at most [log log n]E*n. Even for unsorted X, this algorithm runs in 0(n) time. Previously, the best linear-time approximation algorithm had a worst-case error at most flog n] En, while E n was known to be attainable in O(n log n) time using Huffman coding. Consequently, FPPS is solvable in O(n2) time such that the worst-case error for each Sk is the minimum. To improve this quadratic time bound in practice, we design two on-line algorithms that calculate the next Sk by taking advantage of the current S k and thus reduce redundant computation.

Original languageEnglish (US)
Title of host publicationAutomata, Languages and Programming - 25th International Colloquium, ICALP 1998, Proceedings
Pages375-386
Number of pages12
StatePublished - Dec 1 1998
Event25th International Colloquium on Automata, Languages and Programming, ICALP 1998 - Aalborg, Denmark
Duration: Jul 13 1998Jul 17 1998

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume1443 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Other

Other25th International Colloquium on Automata, Languages and Programming, ICALP 1998
CountryDenmark
CityAalborg
Period7/13/987/17/98

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computer Science(all)

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  • Cite this

    Kao, M. Y., & Wang, J. (1998). Efficient minimization of numerical summation errors. In Automata, Languages and Programming - 25th International Colloquium, ICALP 1998, Proceedings (pp. 375-386). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 1443 LNCS).