We describe a paradigm for numerical computing, based on exact computation. This emerging paradigm has many advantages compared to the standard paradigm which is based on fixed-precision. We first survey the literature on multiprecision number packages, a prerequisite for exact computation. Next we survey some recent applications of this paradigm. Finally, we outline some basic theory and techniques in this paradigm. 1 This paper will appear as a chapter in the 2nd edition of Computing in Euclidean Geometry, edited by D.-Z. Du and F.K. Hwang, published by World Scientific Press, 1994. 1 1 Two Numerical Computing Paradigms Computation has always been intimately associated with numbers: computability theory was early on formulated as a theory of computable numbers, the first computers have been number crunchers and the original mass-produced computers were pocket calculators. Although one's first exposure to computers today is likely to be some non-numerical application, numeri...

Algorithms for summation and dot product of floating point numbers are presented which are fast in terms of measured computing time. We show that the computed results are as accurate as if computed in twice or K-fold working precision, K 3. For twice the working precision our algorithms for summation and dot product are some 40 % faster than the corresponding XBLAS routines while sharing similar error estimates. Our algorithms are widely applicable because they require only addition, subtraction and multiplication of floating point numbers in the same working precision as the given data. Higher precision is unnecessary, algorithms are straight loops without branch, and no access to mantissa or exponent is necessary.

Abstract: The result of a simple oating-point computation can be in great error, even though no error is signaled, no coding mistakes are in the program, and the computer hardware is functioning correctly. This paper proposes a set of instructions appropriate for a general purpose microprocessor that can be used to improve the credibility and accuracy of numerical computations. Such instructions provide direct hardware support for monitoring events which may threaten computational integrity, implementing oating-point data types of arbitrary precision, and repeating calculations with greater precision. These useful features are obtained by the e cient implementation of high radix on-line arithmetic. The prevalence of super-scalar and VLIW processors makes this approach especially attractive.

Language independent arithmetic — Part 2: Elementary numerical functions Technologies de l’information — Arithmétique indépendante de langage — Partie 2: Fonctions numériques élémentaires FINAL COMMITTEE DRAFT

Abstract. From 06.01. to 11.01.2008, the Dagstuhl Seminar 08021 Numerical Validation in Current Hardware Architectures was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The rst section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available.

... In this document, we present a new rounding standard for transcendental functions and vector arithmetic. Some suggestions are also introduced for analysis of the cancellation phenomenon. This includes a standard measure for signaling catastrophic cancellation of a single operation or a compound accumulation. Finally, the new functionalities leading to a hardware cost-efficient dot product are defined. With low extra-cost, the accuracy monitoring does not slow pipelined computation of accurate results, and it is able to signal erroneous results. This allows the user to surely rely on guaranteed results. We specifically describe the mechanism to detect catastrophic cancellation.