Abstract
The need to sum floating-point numbers is ubiquitous in scientific computing. Standard recursive summation of $n$ summands, often implemented in a blocked form, has a backward error bound proportio...
Highlights
Summation is a key computational task at the heart of many numerical algorithms, most notably numerical linear algebra kernels involving inner products, such as matrix--vector or matrix--matrix multiplications, matrix factorizations, and the solution of linear systems
Backward error bounds for the basic summation algorithms are proportional to nu, where n is the number of summands and u the unit roundoff, and they grow linearly with n
Error bounds for recursive summation and for blocked summation with a moderate block size have the form O(nu) and so can readily exceed 1 in practice
Summary
Summation is a key computational task at the heart of many numerical algorithms, most notably numerical linear algebra kernels involving inner products, such as matrix--vector or matrix--matrix multiplications, matrix factorizations, and the solution of linear systems. MARY is increasingly common [2], [8], [12] For such large sizes and low precisions, error bounds of order nu exceed 1, and so do not provide any meaningful information. In this work we tackle this dilemma by introducing a new class of summation algorithms that excels in both performance and accuracy These algorithms achieve backward error bounds that do not grow with n (that is, the bounds are of the form cu + O(u2), where c is a moderate constant independent of n) and, at the same time, they can deliver a similar performance to optimized, standard algorithms by performing an arbitrarily low number of extra flops (e.g., less than a 1\% overhead) and by allowing efficient implementation on modern computers. This backward error bound grows linearly with n and is almost attainable [10, Prob. 4.2]
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