Efficient minimization of numerical summation errors

Abstract

Given a multiset X={x ₁,... x _n } of real numbers, the floating-point set summation (FPS) problem asks for S _n = x ₁+...+ x _n , and the floating point prefix set summation problem (FPPS) asks for S _k =x ₁+...+x _k for all k = 1, ... n. Let E ^*_k denote the minimum worst-case error over all possible orderings of evaluating S _k . We prove that if X has both positive and negative numbers, it is NP-hard to compute S _n with the worst-case error equal to E ^*_n . We then give the first known polynomial-time approximation algorithm for computing S_n that has a provably small error for arbitrary X. Our algorithm incurs a worstcase error at most 2([log(n−1)]+1) E ^*_n . After X is sorted, it runs in O(n) time, yielding an O(n ²)-time approximation algorithm for computing S _k for all k = 1, ..., n such that the worst-case error for each S _k is less than 2(⌈ log(k−1)⌋+ 1)E ^*_k .

For the case where X is either all positive or all negative, we give another approximation algorithm for computing S _n with a worst-case error at most ⌈ log log n⌋E ^*_n . Even for unsorted X, this algorithm runs in O(n) time. Previously, the best linear-time approximation algorithm had a worst-case error at most ⌈ log n⌋E ^*_n , while E ^*_n was known to be attainable in O(n log n) time using Huffman coding. Consequently, FPPS is solvable in O(n ²) time such that the worst-case error for each S _k is the minimum. To improve this quadratic time bound in practice, we design two on-line algorithms that calculate the next S _k by taking advantage of the current Sk and thus reduce redundant computation.

Supported in part by NSF Grant CCR-9531028.

Supported in part by NSF Grant CCR-9424164.