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Numerical Evaluation of Special Functions
 In W. Gautschi (Ed.), AMS Proceedings of Symposia in Applied Mathematics 48
, 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, ..."
Abstract

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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. 5.9. Mathieu, Lam'e, and Spheroidal Wave Functions. 5.9.1. Characteristic Values of Mathieu's Equation. Software Packages:...
Augmented precision square roots, 2D norms, and discussion on correctly rounding √ x 2 + y 2
"... Abstract—Define an “augmented precision ” algorithm as an algorithm that returns, in precisionp floatingpoint arithmetic, its result as the unevaluated sum of two floatingpoint numbers, with a relative error of the order of 2 −2p. Assuming an FMA instruction is available, we perform a tight error ..."
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Abstract—Define an “augmented precision ” algorithm as an algorithm that returns, in precisionp floatingpoint arithmetic, its result as the unevaluated sum of two floatingpoint numbers, with a relative error of the order of 2 −2p. Assuming an FMA instruction is available, we perform a tight error analysis of an augmented precision algorithm for the square root, and introduce two slightly different augmented precision algorithms for the 2Dnorm p x2 + y2. Then we give tight lower bounds on the minimum distance (in ulps) between p x2 + y2 and a midpoint when p x2 + y2 is not itself a midpoint. This allows us to determine cases when our algorithms make it possible to return correctlyrounded 2Dnorms. Index Terms—Floatingpoint arithmetic; compensated algorithms; squareroot; Correct rounding; 2Dnorms; accurate computations. I.
Computer Arithmetic (ARITH20), Tübingen: Allemagne (2011)" DOI: 10.1109/ARITH.2011.13 Augmented precision square roots, 2D norms, and discussion on correctly rounding √ x 2 + y 2
, 2011
"... Abstract—Define an “augmented precision ” algorithm as an algorithm that returns, in precisionp floatingpoint arithmetic, its result as the unevaluated sum of two floatingpoint numbers, with a relative error of the order of 2 −2p. Assuming an FMA instruction is available, we perform a tight error ..."
Abstract
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Abstract—Define an “augmented precision ” algorithm as an algorithm that returns, in precisionp floatingpoint arithmetic, its result as the unevaluated sum of two floatingpoint numbers, with a relative error of the order of 2 −2p. Assuming an FMA instruction is available, we perform a tight error analysis of an augmented precision algorithm for the square root, and introduce two slightly different augmented precision algorithms for the 2Dnorm p x2 + y2. Then we give tight lower bounds on the minimum distance (in ulps) between p x2 + y2 and a midpoint when p x2 + y2 is not itself a midpoint. This allows us to determine cases when our algorithms make it possible to return correctlyrounded 2Dnorms. Index Terms—Floatingpoint arithmetic; compensated algorithms; squareroot; Correct rounding; 2Dnorms; accurate computations. I.