Remark on Algorithm 539

Abstract

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Accurate Calculation of Euclidean Norms Using Double-word Arithmetic

“…This approach is unlikely to allow good performance on modern highly pipelined processors. All other approaches consist in scaling the terms a i , i.e., we multiply or divide them by one (or several) constant(s) such that computing sums of squares of the scaled values is over ow-free, and that under ow is either impossible or harmless (a good presentation, along with comparisons of existing Fortran codes can be found in [19]). A straightforward choice is to scale all values by the factor max |a i |, i.e., to evaluate the norm as…”

“…Higham [24, Pages 500 and 507] attributes to Hammarling a smart algorithm that consists in dynamically scaling the data. It was implemented in the LAPACK [19] package released by netlib 3 . We start from s 0 = 1 and t 0 = |x 0 |.…”

“…However, if BIG is nonempty, provided that the ratio maxmed/minmed is large enough, the value of S tiny has negligible in uence on the nal result. Graillat et al 's algorithm [18] and the variant of Blue's algorithm presented by Hanson and Hopkins in [19] take this into account and use two accumulators only: as soon as an element of BIG is met, we no longer need to accumulate elements of TINY.…”

“…• as soon as n ≥ 3, u ≤ 1/32, and 3(n − 1)u 2 < 1 (which is necessary for (19) to make sense), the bound ( 19) is larger than the bound ( 16);…”

“…Due to the importance of the topic, several solutions have been suggested and analyzed until recently for computing norms accurately and/or without spurious under ows and over ows [4,24,18,19,1]. We will present them in Section 2.1.…”