Results 1 
9 of
9
Arbitrary Precision Real Arithmetic: Design and Algorithms
, 1996
"... this article the second representation mentioned above. We first recall the main properties of computable real numbers. We deduce from one definition, among the three definitions of this notion, a representation of these numbers as sequence of finite Badic numbers and then we describe algorithms fo ..."
Abstract

Cited by 20 (0 self)
 Add to MetaCart
this article the second representation mentioned above. We first recall the main properties of computable real numbers. We deduce from one definition, among the three definitions of this notion, a representation of these numbers as sequence of finite Badic numbers and then we describe algorithms for rational operations and transcendental functions for this representation. Finally we describe briefly the prototype written in Caml. 2. Computable real numbers
Static Analyses of FloatingPoint Operations
 In SAS’01, volume 2126 of LNCS
, 2001
"... Computers manipulate approximations of real numbers, called floatingpoint numbers. The calculations they make are accurate enough for most applications. Unfortunately, in some (catastrophic) situations, the floatingpoint operations lose so much precision that they quickly become irrelevant. In thi ..."
Abstract

Cited by 8 (0 self)
 Add to MetaCart
Computers manipulate approximations of real numbers, called floatingpoint numbers. The calculations they make are accurate enough for most applications. Unfortunately, in some (catastrophic) situations, the floatingpoint operations lose so much precision that they quickly become irrelevant. In this article, we review some of the problems one can encounter, focussing on the IEEE7541985 norm. We give a (sketch of a) semantics of its basic operations then abstract them (in the sense of abstract interpretation) to extract information about the possible loss of precision. The expected application is abstract debugging of software ranging from simple onboard systems (which use more and more ontheshelf microprocessors with floatingpoint units) to scientific codes. The abstract analysis is demonstrated on simple examples and compared with related work. 1
Instrumentation Of Fortran Programs For Automatic Roundoff Error Analysis And Performance Evaluation
, 1990
"... A pass to the Cedar Fortran preprocessor, cftn, has been developed which allows the user to instrument his source code in a variety of ways. By specifying different command line options and linking with different libraries, one can automatically generate a report of the program's use of the All ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
(Show Context)
A pass to the Cedar Fortran preprocessor, cftn, has been developed which allows the user to instrument his source code in a variety of ways. By specifying different command line options and linking with different libraries, one can automatically generate a report of the program's use of the Alliant FX/8's or Cedar's vector hardware, or apply several types of error analysis to obtain an indication of the numerical stability of the algorithm and its implementation. A library has been written with which the user can link to obtain a report of floating point operation counts, together with information regarding each type of operation or intrinsic function call, whether the operation is performed on scalar or array operands, and if so, the length and stride of the vector(s) involved. The package is called the op ct package, and has been integrated into the Faust programming environment under development at CSRD. A library for statistical roundoff error analysis has also been developed for ...
Dynamical strategies using Discrete Stochastic Arithmetic for approximation methods
"... Let us consider the converging sequence generated by successively dividing by two the step size used in an approximation method. With an appropriate stopping criterion, we show that in the last approximation obtained, the significant bits which are not affected by roundoff errors are in common with ..."
Abstract
 Add to MetaCart
(Show Context)
Let us consider the converging sequence generated by successively dividing by two the step size used in an approximation method. With an appropriate stopping criterion, we show that in the last approximation obtained, the significant bits which are not affected by roundoff errors are in common with the exact result, up to one. This strategy has been successfully applied to several composite quadrature methods. Other strategies, which are not based on “step halving”, are also proposed. For approximation methods of a relatively high order, these alternative strategies may sometimes be less costly. Key words: approximation methods, numerical validation, quadrature methods, trapezoidal rule, Simpson’s rule, GaussLegendre method, CESTAC method, Discrete Stochastic Arithmetic. 1
ESTIMATION OF ROUNDOFF ERRORS ON SEVERAL COMPUTERS ARCHITECTURES
"... Abstract: Numerical validation of computed results in scienti c computation is always an essential problem as well on sequential architecture as on parallel architecture. The probabilistic approach is the only one that allows to estimate the roundo error propagation of the oating point arithmetic o ..."
Abstract
 Add to MetaCart
Abstract: Numerical validation of computed results in scienti c computation is always an essential problem as well on sequential architecture as on parallel architecture. The probabilistic approach is the only one that allows to estimate the roundo error propagation of the oating point arithmetic on computers. We begin by recalling the basics of the CESTAC method (Contr^ole et Estimation STochastique des Arrondis de Calculs). Then, the use of the CADNA software (Control of Accuracy and Debugging For Numerical Applications) is presented for numerical validation on sequential architecture. On parallel architecture, we present two solutions for the control of roundo errors. The rst one is the combination of CADNA and the PVM library. This solution allows to control roundo errors of parallel codes with the same architecture. It does not need more processors than the classical parallel code. The second solution is represented by the RAPP prototype. In this approach, the CESTAC method is directly parallelized. It works both on sequential and parallel programs. The essential di erence is that this solution requires more processors than the classical codes. These di erent approaches are tested on sequential and parallel programs of multiplication of matrices. 1
Numerical ‘health check ’ for scientific codes: the CADNA approach
"... Scientific computation has unavoidable approximations built into its very fabric. One important source of error that is difficult to detect and control is roundoff error propagation which originates from the use of finite precision arithmetic. We propose that there is a need to perform regular nume ..."
Abstract
 Add to MetaCart
(Show Context)
Scientific computation has unavoidable approximations built into its very fabric. One important source of error that is difficult to detect and control is roundoff error propagation which originates from the use of finite precision arithmetic. We propose that there is a need to perform regular numerical ‘health checks ’ on scientific codes in order to detect the cancerous effect of roundoff error propagation. This is particularly important in scientific codes that are built on legacy software. We advocate the use of the CADNA library as a suitable numerical screening tool. We present a case study to illustrate the practical use of CADNA in scientific codes that are of interest to the Computer Physics Communications readership. In doing so we hope to stimulate a greater awareness of roundoff error propagation and present a practical means by which it can be analyzed and managed.
PROGRAM SUMMARY
"... The CADNA library enables one to estimate roundoff error propagation using a probabilistic approach. With CADNA the numerical quality of any simulation program can be controlled. Furthermore by detecting all the instabilities which may occur at run time, a numerical debugging of the user code can b ..."
Abstract
 Add to MetaCart
The CADNA library enables one to estimate roundoff error propagation using a probabilistic approach. With CADNA the numerical quality of any simulation program can be controlled. Furthermore by detecting all the instabilities which may occur at run time, a numerical debugging of the user code can be performed. CADNA provides new numerical types on which roundoff errors can be estimated. Slight modifications are required to control a code with CADNA, mainly changes in variable declarations, input and output. This paper describes the features of the CADNA library and shows how to interpret the information it provides concerning roundoff error propagation in a code.
2 Stochastic Arithmetic as a Model of Granular Computing
, 2008
"... Numerical simulation is used more and more frequently in the analysis of physical phenomena. A simulation requires several phases. The first phase consists of constructing a physical model based on the results of experimenting with the phenomena. Next, the physical model is approximated by a mathema ..."
Abstract
 Add to MetaCart
Numerical simulation is used more and more frequently in the analysis of physical phenomena. A simulation requires several phases. The first phase consists of constructing a physical model based on the results of experimenting with the phenomena. Next, the physical model is approximated by a mathematical model. Generally, the
Arbitrary precision real arithmetic: design and algorithms Valerie MenissierMorain
"... We describe here a representation of computable real numbers and a set of algorithms for the elementary functions associated to this representation. A real number is represented as a sequence of nite Badic numbers and for each classical function (rational, algebraic or transcendental), we describe ..."
Abstract
 Add to MetaCart
(Show Context)
We describe here a representation of computable real numbers and a set of algorithms for the elementary functions associated to this representation. A real number is represented as a sequence of nite Badic numbers and for each classical function (rational, algebraic or transcendental), we describe how to produce a sequence representing the result of the application of this function to its arguments, according to the sequences representing these arguments. For each algorithm we prove that the resulting sequence is a valid representation of the exact real result. This arithmetic is the rst abritrary precision real arithmetic with mathematically proved algorithms. Resume Nous proposons une representation des nombres reels calculables ainsi que des algorithmes pour les fonctions elementaires usuelles pour cette representation. Un nombre reel est represente par une suite de nombres Badiques nis et pour chaque fonction classique (rationnelle, algebrique ou transcendante), nous montrons comment produire une suite representant le resultat a partir de suites representant les parametres. Pour chacun de ces algorithmes nous demontrons que la suite qui en resulte represente bien le resultat reel exact. Cette arithmetique est la premiere arithmetique reelle en precision arbitraire dotee d'un jeu complet d'algorithmes