Efficient Calculations of Faithfully Rounded l ₂ -Norms of n -Vectors

Abstract

In this paper, we present an efficient algorithm to compute the faithful rounding of the l 2-norm of a floatingpoint vector. This means that the result is accurate to within one bit of the underlying floating-point type. This algorithm does not generate overflows or underflows spuriously, but does so when the final result indeed calls for such a numerical exception to be raised. Moreover, the algorithm is well suited for parallel implementation and vectorization. The implementation runs up to 3 times faster than the netlib version on current processors.

Reproducible, Accurately Rounded and Efficient BLAS

“…Finally, we apply a square root that returns a faithfully rounded euclidean norm [7]. numbers that enclose the exact result).…”

Remark on Algorithm 539

“…Recently a new algorithm, FaithfulNorm, was proposed [Graillat et al 2015] for computing the l 2 norm of a vector with guaranteed accuracy to within a single bit of the floating point type being used (double precision in their implementation). This algorithm treats the data in a very similar way to the Blue and Kahan methods; three ranges are defined and elements whose absolute values fall outside the middle range are scaled prior to the accumulation of their squares.…”

“…The software package provided by the authors [Graillat et al 2015] contains a reference implementation of the double precision function (a single precision version is discussed in the article but no implementation is provided) which is written in C and requires the GNU MPFR Library [MPFR 2016] to be installed. Additionally two SIMD versions are provided for Intel and AMD processors using the x86 instruction set with and without Advanced Vector Extension (AVX) support.…”

“…In the worst case scenario use of any of the algorithms described in Sections 4.1 to 4.3 may lead to the last log 2 (n) bits of the binary result (or the last log 10 (n) digits of its decimal equivalent) being corrupted [Graillat et al 2015]. This means that all of these algorithms gradually lose accuracy as n, the number of elements in the input vector, increases.…”

Parallel Experiments with RARE-BLAS

“…The sum p 2 i can be correctly rounded using the previous dot product. Finally, we apply a square root that returns a faithfully rounded euclidean norm [13]. This does not compute a correctly rounded norm-2 but this faithful rounding is reproducible because it depends on a reproducible dot.…”