9 research outputs found
A Multiple-Precision Division Algorithm
The classical algorithm for multiple-precision division normalizes digits during each step and sometimes makes correction steps when the initial guess for the quotient digit turns out to be wrong. A method is presented that runs faster by skipping most of the intermediate normalization and recovers from wrong guesses without separate correction steps
Implementation of the Combined--Nonlinear Condensation Transformation
We discuss several applications of the recently proposed combined
nonlinear-condensation transformation (CNCT) for the evaluation of slowly
convergent, nonalternating series. These include certain statistical
distributions which are of importance in linguistics, statistical-mechanics
theory, and biophysics (statistical analysis of DNA sequences). We also discuss
applications of the transformation in experimental mathematics, and we briefly
expand on further applications in theoretical physics. Finally, we discuss a
related Mathematica program for the computation of Lerch's transcendent.Comment: 23 pages, 1 table, 1 figure (Comput. Phys. Commun., in press
Software needs in special functions
AbstractCurrently available software for special functions exhibits gaps and defects in comparison to the needs of moderm high-performance scientific computing and also, surprisingly, in comparison to what could be constructed from current algorithms. In this paper we expose some of these deficiencies and identify the related need for user-oriented testing software
Generation and evaluation of orthogonal polynomials in discrete Sobolev spaces II: numerical stability
AbstractIn this paper, we concern ourselves with the determination and evaluation of polynomials that are orthogonal with respect to a general discrete Sobolev inner product, that is, an ordinary inner product on the real line plus a finite sum of atomic inner products involving a finite number of derivatives. In a previous paper we provided a complete set of formulas to compute the coefficients of this recurrence. Here, we study the numerical stability of these algorithms for the generation and evaluation of a finite series of Sobolev orthogonal polynomials. Besides, we propose several techniques for reducing and controlling the rounding errors via theoretical running error bounds and a carefully chosen recurrence
Dynamics of a map with power-law tail
We analyze a one-dimensional piecewise continuous discrete model proposed
originally in studies on population ecology. The map is composed of a linear
part and a power-law decreasing piece, and has three parameters. The system
presents both regular and chaotic behavior. We study numerically and, in part,
analytically different bifurcation structures. Particularly interesting is the
description of the abrupt transition order-to-chaos mediated by an attractor
made of an infinite number of limit cycles with only a finite number of
different periods. It is shown that the power-law piece in the map is at the
origin of this type of bifurcation. The system exhibits interior crises and
crisis-induced intermittency.Comment: 28 pages, 17 figure
Contribution au calcul sur GPU: considérations arithmétiques et architecturales
L’optimisation du calcul passe par une gestion conjointe du matériel et du logiciel. Cette règle se trouve renforcée lorsque l’on aborde le domaine des architectures multicoeurs où les paramètres à considérer sont plus nombreux que sur une architecture superscalaire classique. Ces architectures offrent une grande variété d’unité de calcul, de format de représentation, de hiérarchie mémoire et de mécanismes de transfert de donnée.Dans ce mémoire, nous décrivons quelques-uns de nos résultats obtenus entre 2004 et 2013 au sein de l'équipe DALI de l'Université de Perpignan relatifs à l'amélioration de l’efficacité du calcul dans sa globalité, c'est-à-dire dans la suite d’opérations décrite au niveau algorithmique et exécutées par les éléments architecturaux, en nous concentrant sur les processeurs graphiques.Nous commençons par une description du fonctionnement de ce type d'architecture, en nous attardant sur le calcul flottant. Nous présentons ensuite des implémentations efficaces d'opérateurs arithmétiques utilisant des représentations non-conventionnelles comme l'arithmétique multiprécision, par intervalle, floue ou logarithmique. Nous continuerons avec nos contributions relatives aux éléments architecturaux associés au calcul à travers la simulation fonctionnelle, les bancs de registres, la gestion des branchements ou les opérateurs matériels spécialisés. Enfin, nous terminerons avec une analyse du comportement du calcul sur les GPU relatif à la régularité, à la consommation électrique, à la fiabilisation des calculs ainsi qu'à laprédictibilité
Analiza procesnih i računskih iteracija primenom savremenih računarskih aritmetika
The proposed theme relates to the field of application of modern computer arithmetics
in the analysis of process performance and computational iterations, where the concept of
the modern computer arithmetic applies to multiple-precision arithmetic and interval
arithmetic involved in the new standard IEEE 754 in 2008. Advance computer arithmetic,
first of all interval arithmetic and multi-precision arithmetic, are employed in the the
dissertation for the analysis of matrix models in the design iteration process software and
iterative numerical computations. These are important and current research topics from
the point at which the application is working in the world. Research in this area have led
to the development and analysis of algorithms to control the accuracy, optimality, rate
calculations and other performance aspects of various processes such as designing
software and hardware, designing industrial products, transportation optimization,
modeling systems, the implementation of numerical algorithms and others.
The first part of dissertation is devoted to engineering design and development of new
products, which are either industrial products, technical innovations, hardware or
software, often contain a very complex set of relationships among many coupled tasks.
Controlling, redesigning and identifying features of these tasks can be usefully performed
by a suitable model based on the design structure matrix in an iteration procedure. The
proposed interval matrix model of design iteration controls and predicts slow and rapid
convergence of iteration work on tasks within a project. A new model is based on Perron-
Frobenius theorem and interval linear algebra where intervals and interval matrices are
employed instead of real numbers and real matrices. In this way a more relaxed
quantitative estimation of tasks is achieved and the presence of undetermined quantities is
allowed to a certain extent. The presented model is demonstrated in the example of
simplified software development process. An additional contribution in this dissertation
is the ranking of tasks within a design mode using components of eigenvalue vector
corresponding to the spectral radius of design structure matrix.
The second part deals with computational efficiency of numerical iterative algorithms.
Computational cost of iterative procedures for the implementation of basic arithmetic
operations in multi-precision arithmetic are studied and later applied for the analysis of
computational efficiency of the iterative methods for solving nonlinear equations. A new
approach that deals with the weights of employed arithmetic operations, involved in the
realization of these algorithms, is proposed. This enables a precise ranking of considered
root-finding algorithms of different structure, especially in a regime of variable
(dynamic) precision of applied multi-precision arithmetic