Drastic changes in the molecular absorption at redshift z=0.89 toward the quasar PKS 1830-211

A 12 year-long monitoring of the absorption caused by a z=0.89 spiral galaxy on the line of sight to the radio-loud gravitationally lensed quasar PKS 1830-211 reveals spectacular changes in the HCO+ and HCN (2-1) line profiles. The depth of the absorption toward the quasar NE image increased by a factor of ~3 in 1998-1999 and subsequently decreased by a factor >=6 between 2003 and 2006. These changes were echoed by similar variations in the absorption line wings toward the SW image. Most likely, these variations result from a motion of the quasar images with respect to the foreground galaxy, which could be due to a sporadic ejection of bright plasmons by the background quasar. VLBA observations have shown that the separation between the NE and SW images changed in 1997 by as much as 0.2 mas within a few months. Assuming that motions of similar amplitude occurred in 1999 and 2003, we argue that the clouds responsible for the NE absorption and the broad wings of the SW absorption should be sparse and have characteristic sizes of 0.5-1 pc.Comment: accepted for publication in A&

Finding the "truncated" polynomial that is closest to a function

When implementing regular enough functions (e.g., elementary or special functions) on a computing system, we frequently use polynomial approximations. In most cases, the polynomial that best approximates (for a given distance and in a given interval) a function has coefficients that are not exactly representable with a finite number of bits. And yet, the polynomial approximations that are actually implemented do have coefficients that are represented with a finite - and sometimes small - number of bits: this is due to the finiteness of the floating-point representations (for software implementations), and to the need to have small, hence fast and/or inexpensive, multipliers (for hardware implementations). We then have to consider polynomial approximations for which the degree-$i$ coefficient has at most $m_i$ fractional bits (in other words, it is a rational number with denominator $2^{m_i}$). We provide a general method for finding the best polynomial approximation under this constraint. Then, we suggest refinements than can be used to accelerate our method.Comment: 14 pages, 1 figur

Computing Integer Powers in Floating-Point Arithmetic

We introduce two algorithms for accurately evaluating powers to a positive integer in floating-point arithmetic, assuming a fused multiply-add (fma) instruction is available. We show that our log-time algorithm always produce faithfully-rounded results, discuss the possibility of getting correctly rounded results, and show that results correctly rounded in double precision can be obtained if extended-precision is available with the possibility to round into double precision (with a single rounding).Comment: Laboratoire LIP : CNRS/ENS Lyon/INRIA/Universit\'e Lyon

On the error of Computing ab + cd using Cornea, Harrison and Tang's method

International audienceIn their book, Scientific Computing on the Itanium, Cornea et al. [2002] introduce an accurate algorithm for evaluating expressions of the form ab + cd in binary floating-point arithmetic, assuming an FMA instruction is available. They show that if p is the precision of the floating-point format and if u = 2^{−p}, the relative error of the result is of order u. We improve their proof to show that the relative error is bounded by 2u + 7u^2 + 6u^3. Furthermore, by building an example for which the relative error is asymptotically (as p → ∞ or, equivalently, as u → 0) equivalent to 2u, we show that our error bound is asymptotically optimal

Generating function approximations at compile time

ISBN : 12-4244-0785-0 ISSN: 1058-6393International audienceUsually, the mathematical functions used in a numerical programs are decomposed into elementary functions (such as sine, cosine, exponential, logarithm...), and for each of these functions, we use a program from a library. This may have some drawbacks: first in frequent cases, it is a compound function (e.g. log(1 + exp(−x))) that is needed, so that directly building a polynomial or rational approximation for that function (instead of decomposing it) would result in a faster and/or more accurate calculation. Also, at compile-time, we might have some information (e.g., on the range of the input value) that could help to simplify the program. We investigate the possibility of directly building accurate approximations at compile-time

Vers des primitives propres en arithmétique des ordinateurs

La norme IEEE-754 consacrée à l'arithmétique virgule flottante spécifie le comportement des quatre opérations arithmétiques. Une spécification des fonctions élémentaires devrait voir le jour dans les années à venir. On s'intéresse dans cet article aux avantages que l'on peut tirer d'un système dont les «primitives numériques» sont complètement spécifiées