The Mathematical-Function Computation Handbook

This highly comprehensive handbook provides a substantial advance in the computation of elementary and special functions of mathematics, extending the function coverage of major programming languages well beyond their international standards, including full support for decimal floating-point arithmetic. Written with clarity and focusing on the C language, the work pays extensive attention to little-understood aspects of floating-point and integer arithmetic, and to software portability, as well as to important historical architectures. It extends support to a future 256-bit, floating-point format offering 70 decimal digits of precision. Select Topics and Features: references an exceptionally useful, author-maintained MathCW website, containing source code for the book’s software, compiled libraries for numerous systems, pre-built C compilers, and other related materials; offers a unique approach to covering mathematicalfunction computation using decimal arithmetic; provides extremely versatile appendices for interfaces to numerous other languages: Ada, C#, C++, Fortran, Java, and Pascal; presupposes only basic familiarity with computer programming in a common language, as well as early level algebra; supplies a library that readily adapts for existing scripting languages, with minimal effort; supports both binary and decimal arithmetic, in up to 10 different floating-point formats; covers a significant portion (with highly accurate implementations) of the U.S National Institute of Standards and Technology’s 10-year project to codify mathematical functions. This highly practical text/reference is an invaluable tool for advanced undergraduates, recording many lessons of the intermingled history of computer hardw are and software, numerical algorithms, and mathematics. In addition, professional numerical analysts and others will find the handbook of real interest and utility because it builds on research by the mathematical software community over the last four decades.

The Mathematical-Function Computation Handbook: Programming Using the MathCW Portable Software Library

Some aspects of what a standard for the implementation of the mathematical functions could be are presented. Firstly, the need for such a standard is motivated. Then the proposed standard is given. The question of roundings constitutes an important part of this paper: three levels are proposed, ranging from a level relatively easy to attain (with fixed maximal relative error) up to the best quality one, with correct rounding on the whole domain of every function. We do not claim that we always suggest the right choices, or that we have thought about all relevant issues. The mere goal of this paper is to raise questions and to launch the discussion towards a standard.

/pdf/proposal-for-a-standardization-of-mathematical-function-16e7djbojx.pdf

Proposal for a Standardization of Mathematical Function Implementation in Floating-Point Arithmetic

This paper presents extensions to Lefevre's algorithm that computes a lower bound on the distance between a segment and a regular grid Zopf2. This algorithm and, in particular, the extensions are useful in the search for worst cases for the exact rounding of unary elementary functions or base-conversion functions. The proof that is presented is simpler and less technical than the original proof. This paper also gives benchmark results with various optimization parameters, explanations of these results, and an application to base conversion

/pdf/new-results-on-the-distance-between-a-segment-and-z-3rahjllalb.pdf

New Results on the Distance between a Segment and Z&#178;. Application to the Exact Rounding

Les systemes informatiques permettent d'evaluer des fonctions numeriques telles que f = exp, sin, arccos, etc. Cette these s'interesse au processus d'implementation de ces fonctions. Suivant la cible visee (logiciel ou materiel, faible ou grande precision), les problemes qui se posent sont differents, mais l'objectif est toujours d'obtenir l'implementation la plus efficace possible. Nous etudions d'abord, a travers un exemple, les problemes qui se posent dans le cas ou la precision est arbitraire. Lorsque, a l'inverse, la precision est connue d'avance, la fonction f est souvent remplacee par un polynome d'approximation p. Un tel polynome peut ensuite etre evalue tres efficacement en machine. En pratique, les coefficients de p doivent etre representables sur un nombre fini donne de bits. Nous proposons un ensemble d'algorithmes (certains sont heuristiques, d'autres rigoureux) pour trouver de tres bons polynomes d'approximation repondant a cette contrainte. Ces resultats s'etendent au cas ou la fonction d'approximation est une fraction rationnelle. Une fois p trouve, il faut prouver que l'erreur |p-f| n'excede pas un certain seuil. La nature particuliere de la fonction p-f (soustraction de deux fonctions tres proches) rend cette propriete difficile a prouver rigoureusement. Nous proposons un algorithme capable de contourner cette difficulte. Tous ces algorithmes ont ete integres au logiciel Sollya, developpe pendant la these. A l'origine concu pour faciliter l'implementation de fonctions, ce logiciel s'adresse a present a toute personne souhaitant faire des calculs numeriques dans un cadre completement fiable.

https://tel.archives-ouvertes.fr/tel-00460776/document

Évaluation efficace de fonctions numériques - Outils et exemples

We propose a new algorithm to find worst cases for correct rounding of an analytic function. We first reduce this problem to the real small value problem - i.e. for polynomials with real coefficients. Then we show that this second problem can be solved efficiently, by extending Coppersmith's work on the integer small value problem - for polynomials with integer coefficients - using lattice reduction (D. Coppersmith, 1996; 2001). For floating-point numbers with a mantissa less than N, and a polynomial approximation of degree d, our algorithm finds all worst cases at distance < N/sup -d2//(2d+1) from a machine number in time O(N/sup ((d+1)/(2d+1))+/spl epsiv//). For d=2, this improves on the O(N/sup 2/(3+/spl epsiv/)/) complexity from Lefevre's algorithm (V. Lefevre, 2000; V. Lefevre et al., 2001) to O(N/sup 3/(5+/spl epsiv/)/). We exhibit some new worst cases found using our algorithm, for double-extended and quadruple precision. For larger d, our algorithm can be used to check that there exist no worst cases at distance < N/sup -k/ in time O(N/sup (1/2)+O(1/k)/).

/pdf/worst-cases-and-lattice-reduction-1b8vlotd6e.pdf

Worst cases and lattice reduction

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Worst cases and lattice reduction