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The Exact Computation Paradigm
, 1994
"... 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 fixedprecision. We first survey the literature on multiprecision number packages, a prerequisite for exact computation. Next ..."
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Cited by 95 (10 self)
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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 fixedprecision. 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 massproduced computers were pocket calculators. Although one's first exposure to computers today is likely to be some nonnumerical application, numeri...
Towards Exact Geometric Computation
, 1994
"... Exact computation is assumed in most algorithms in computational geometry. In practice, implementors perform computation in some fixedprecision model, usually the machine floatingpoint arithmetic. Such implementations have many wellknown problems, here informally called "robustness issues". To rec ..."
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Cited by 90 (6 self)
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Exact computation is assumed in most algorithms in computational geometry. In practice, implementors perform computation in some fixedprecision model, usually the machine floatingpoint arithmetic. Such implementations have many wellknown problems, here informally called "robustness issues". To reconcile theory and practice, authors have suggested that theoretical algorithms ought to be redesigned to become robust under fixedprecision arithmetic. We suggest that in many cases, implementors should make robustness a nonissue by computing exactly. The advantages of exact computation are too many to ignore. Many of the presumed difficulties of exact computation are partly surmountable and partly inherent with the robustness goal. This paper formulates the theoretical framework for exact computation based on algebraic numbers. We then examine the practical support needed to make the exact approach a viable alternative. It turns out that the exact computation paradigm encomp...
Recent Progress in Exact Geometric Computation
 English Today
, 2001
"... Computational geometry has produced an impressive wealth of efficient algorithms. ..."
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Cited by 19 (6 self)
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Computational geometry has produced an impressive wealth of efficient algorithms.
A Basis for Implementing Exact Geometric Algorithms (Extended Abstract)
, 1993
"... Our ultimate goal is to develop exact geometric computation as an viable alternative to the usual computing paradigm based on fixedprecision arithmetic. Use of exact computation has numerous advantages; in particular, it will abolish the nonrobustness issues that has so far defied satisfactory so ..."
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Cited by 15 (7 self)
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Our ultimate goal is to develop exact geometric computation as an viable alternative to the usual computing paradigm based on fixedprecision arithmetic. Use of exact computation has numerous advantages; in particular, it will abolish the nonrobustness issues that has so far defied satisfactory solution. In this paper we describe two computational tools which can be a basis for exact geometric computing: ffl bigFloat: a multiprecision floatingpoint number system with automatic errorhandling. ffl bigExpression: an expressions package based on a precisiondriven mechanism. This package is built on top of bigFloat. We discuss the rationale for the design of these packages. Experimental results are reported. The major contributions of our work are: ffl We demonstrated for the first time that, because of the existence of root...
ErrorBounding in LevelIndex Computer Arithmetic
 in Numerical Methods and Error
, 1966
"... . This paper proposes the use of levelindex (LI) and symmetric levelindex (SLI) computer arithmetic for practical computation with error bounds. Comparisons are made with floatingpoint and several advantages are identified. 1 Introduction Any approach to the general problem of assessing the tot ..."
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Cited by 1 (1 self)
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. This paper proposes the use of levelindex (LI) and symmetric levelindex (SLI) computer arithmetic for practical computation with error bounds. Comparisons are made with floatingpoint and several advantages are identified. 1 Introduction Any approach to the general problem of assessing the total error in the output of computer programs depends on a detailed understanding of the computer arithmetic. The finite precision of the arithmetic gives rise to rounding errors that can be an important component of the total error. Accordingly, much effort has gone into refining the algorithms and circuitry that carry out floatingpoint arithmetic. One goal of this effort has been to minimize rounding errors. Another was to ensure that exceptional conditions, such as underflow and overflow, are detected and reported because their occurrence can completely invalidate the results of a computation. The present state of floatingpoint hardware design [5] is close to optimal, and so the question a...
Numerical Evaluation Of Special Functions
, 1994
"... . This document is an excerpt from the current hypertext version of an article that appeared in Walter Gautschi (ed.), Mathematics of Computation 19431993: A HalfCentury of Computational Mathematics, Proceedings of Symposia in Applied Mathematics 48, American Mathematical Society, Providence, RI ..."
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. This document is an excerpt from the current hypertext version of an article that appeared in Walter Gautschi (ed.), Mathematics of Computation 19431993: A HalfCentury of Computational Mathematics, Proceedings of Symposia in Applied Mathematics 48, American Mathematical Society, Providence, RI 02940, 1994. The symposium was held at the University of British Columbia August 913, 1993, in honor of the fiftieth anniversary of the journal Mathematics of Computation. The original abstract follows. Higher transcendental functions continue to play varied and important roles in investigations by engineers, mathematicians, scientists and statisticians. The purpose of this paper is to assist in locating useful approximations and software for the numerical generation of these functions, and to offer some suggestions for future developments in this field. 4.7. Zeta Function. 4.7.1. Real Arguments. Algorithms: [CHT71], [Luk69b], [PB72]. Software Packages: [Mar65, Algol]. Intermediate Libr...
Basic Linear Algebra Operations In Sli Arithmetic
"... . Symmetric levelindex arithmetic was introduced to overcome recognized limitations of floatingpoint systems, most notably overflow and underflow. The original recursive algorithms for arithmetic operations could be parallelized to some extent, particularly when applied to extended sums or pro ..."
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. Symmetric levelindex arithmetic was introduced to overcome recognized limitations of floatingpoint systems, most notably overflow and underflow. The original recursive algorithms for arithmetic operations could be parallelized to some extent, particularly when applied to extended sums or products, and a SIMD software implementation of some of these algorithms is described. The main purpose of this paper is to present parallel SLI algorithms for arithmetic and basic linear algebra operations. 1. Introduction This paper reports on a continuing project to develop, implement and apply parallel algorithms for SLI (symmetric levelindex) arithmetic. The algorithms are being developed with a view toward a possible future implementation in hardware but at this stage they are being coded for a particular SIMD (single instruction, multiple data) computer system, a DEC MasPar MP1 1 . The algorithms cover individual arithmetic operations and extensions to the BLAs (basic linear alge...
Basic Linear Algebra Operations in SLI Arithmetic
"... . Symmetric levelindex arithmetic was introduced to overcome recognized limitations of floatingpoint systems, most notably overflow and underflow. The original recursive algorithms for arithmetic operations could be parallelized to some extent, particularly when applied to extended sums or pro ..."
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. Symmetric levelindex arithmetic was introduced to overcome recognized limitations of floatingpoint systems, most notably overflow and underflow. The original recursive algorithms for arithmetic operations could be parallelized to some extent, particularly when applied to extended sums or products, and a SIMD software implementation of some of these algorithms is described. The main purpose of this paper is to present parallel SLI algorithms for arithmetic and basic linear algebra operations. 1. Introduction This paper reports on a continuing project to develop, implement and apply parallel algorithms for SLI (symmetric levelindex) arithmetic. The algorithms are being developed with a view toward a possible future implementation in hardware but at this stage they are being coded for a particular SIMD (single instruction, multiple data) computer system, a DEC MasPar MP1 1 . The algorithms cover individual arithmetic operations and extensions to the BLAs (basic linear alge...
Dirty Pages of Logarithm Tables, Lifetime Of The Universe, and (Subjective) Probabilities on Finite and Infinite Intervals
 Reliable Computing
, 2002
"... In many engineering problems, we want a physical characteristic y to lie within given range Y; e.g., for all possible values of the load x from 0 to x0 , the resulting stress y of a mechanical structure should not exceed a given value y0 . If no such design is possible, then, from the purely mat ..."
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In many engineering problems, we want a physical characteristic y to lie within given range Y; e.g., for all possible values of the load x from 0 to x0 , the resulting stress y of a mechanical structure should not exceed a given value y0 . If no such design is possible, then, from the purely mathematical viewpoint, all possible designs are equally bad. Intuitively, however, a design for which y y0 for all values x 2 [0; 0:99 \Delta x0 ] is "more probable" to work well than a design for which y y0 only for the values x 2 [0; 0:5 \Delta x0 ]. In this paper, we describe an interval computationsrelated formalization for this subjective notion of probability. We show that this description is in good accordance with the empirical distribution of numerical data and with the problems related to estimating the lifetime of the Universe.