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

Numerical simulation of the dynamics of molecular markers involved in cell polarisation

A cell is polarised when it has developed a main axis of organisation through the reorganisation of its cytosqueleton and its intracellular organelles. Polarisation can occur spontaneously or be triggered by external signals, like gradients of signaling molecules ... In this work, we study mathematical models for cell polarisation. These models are based on nonlinear convection-diffusion equations. The nonlinearity in the transport term expresses the positive loop between the level of protein concentration localised in a small area of the cell membrane and the number of new proteins that will be convected to the same area. We perform numerical simulations and we illustrate that these models are rich enough to describe the apparition of a polarisome.Comment: 15 page

On-The-Fly Range Reduction

In several cases, the input argument of an elementary function evaluation is given bit-serially, most significant bit first. We suggest a solution for performing the first step of the evaluation (namely, the range reduction) on the fly: the computation is overlapped with the reception of the input bits. This algorithm can be used for the trigonometric functions sin, cos, tan as well as for the exponential function.Il arrive que l’oprande dont on doit calculer une fonction élémentaire soit disponible chiffre après chiffre, en série, en commençant par les poids forts. Nous proposons une solution permettant d’effectuer la première phase de l’évaluation(la réduction d’argument)au vol: le calcul et la réception des chiffres d’entré se recouvrent. Cet algorithme peut être utilisé pour les fonctions trigonométriques sin, cos, tan ainsi que pour l'exponentiell

Some notes on the possible under/overflow of the most common elementary functions

The purpose of this short note is not to describe when underflow or overflow must be signalled (it is quite clear that the rules are the same as for the basic arithmetic operations). We just want to show that for some of the most common functions and floating-point formats, in many cases, we can know in advance that the results will always lie in the range of the numbers that are representable by normal floating-point numbers, so that in these cases there is no need to worry about underflow or overflow. Note that when it is not the case, an implementation is still possible using a run-time test

Visualizing and Interpreting Feature Reuse of Pretrained CNNs for Histopathology

Reusing the parameters of networks pretrained on large scale datasets of natural images, such as ImageNet, is a common technique in the medical imaging domain. The large variability of objects and classes is, however, drastically reduced in most medical applications where images are dominated by repetitive patterns with, at times, subtle differences between the classes. This paper takes the example of finetuning a pretrained convolutional network on a histopathology task. Because of the reduced visual variability in this application domain, the network mostly learns to detect textures and simple patterns. As a result, the complex structures that maximize the channel activations of deep layers in the pretrained network are not present after finetuning. The learned features seem to be used by the network to spot atypical nuclei in the images, as shown by class activation maps. Finally, texture measures appear discriminative after finetuning, as shown by accurate Regression Concept Vectors

On the maximum relative error when computing integer powers by iterated multiplications in floating-point arithmetic

International audienceWe improve the usual relative error bound for the computation of x^n through iterated multiplications by x in binary floating-point arithmetic. The obtained error bound is only slightly better than the usual one, but it is simpler. We also discuss the more general problem of computing the product of n terms

Numerical simulation on a cell polarisation model: the polar case

20 pagesWhen it is polarised, a cell develops an asymmetric distribution of specific molecular markers, cytoskeleton and cell membrane shape. Polarisation can occur spontaneously or be triggered by external signals, like gradients of signalling molecules... In this work, we use the published models of cell polarisation and we set a numerical analysis for these models. They are based on nonlinear convection-diffusion equations and the nonlinearity in the transport term expresses the positive loop between the level of protein concentration localised in a small area of the cell membrane and the number of new proteins that will be convected to the same area. We perform numerical simulations and we illustrate that these models are rich enough to describe the apparition of a polarisome