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17
Algorithms for QuadDouble Precision Floating Point Arithmetic
 Proceedings of the 15th Symposium on Computer Arithmetic
, 2001
"... A quaddouble number is an unevaluated sum of four IEEE double precision numbers, capable of representing at least 212 bits of significand. We present the algorithms for various arithmetic operations (including the four basic operations and various algebraic and transcendental operations) on quaddo ..."
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Cited by 55 (9 self)
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A quaddouble number is an unevaluated sum of four IEEE double precision numbers, capable of representing at least 212 bits of significand. We present the algorithms for various arithmetic operations (including the four basic operations and various algebraic and transcendental operations) on quaddouble numbers. The performance of the algorithms, implemented in C++, is also presented. 1.
ARPREC: An arbitrary precision computation package
, 2002
"... This paper describes a new software package for performing arithmetic with an arbitrarily high level of numeric precision. It is based on the earlier MPFUN package [2], enhanced with special IEEE floatingpoint numerical techniques and several new functions. This package is written in C++ code for h ..."
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Cited by 54 (18 self)
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This paper describes a new software package for performing arithmetic with an arbitrarily high level of numeric precision. It is based on the earlier MPFUN package [2], enhanced with special IEEE floatingpoint numerical techniques and several new functions. This package is written in C++ code for high performance and broad portability and includes both C++ and Fortran90 translation modules, so that conventional C++ and Fortran90 programs can utilize the package with only very minor changes. This paper includes a survey of some of the interesting applications of this package and its predecessors.
Central Binomial Sums and Multiple Clausen Values (with Connections to Zeta Values
"... We find and prove relationships between Riemann zeta values and central binomial sums. We also investigate alternating binomial sums (also called Apéry sums). The study of nonalternating sums leads to an investigation of different types of sums which we call multiple Clausen values. The study of al ..."
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Cited by 37 (14 self)
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We find and prove relationships between Riemann zeta values and central binomial sums. We also investigate alternating binomial sums (also called Apéry sums). The study of nonalternating sums leads to an investigation of different types of sums which we call multiple Clausen values. The study of alternating sums leads to a tower of experimental results involving polylogarithms in the golden ratio. In the nonalternating case, there is a strong connection to polylogarithms of the sixth root of unity, encountered in the 3loop Feynman diagrams of hepth/9803091 and subsequently in hepph/9910223, hepph/9910224, condmat/9911452 and hepth/0004010.
AUTOMATIC CLASSIFICATION OF RESTRICTED LATTICE WALKS
"... Abstract. We propose an experimental mathematics approach leading to the computerdriven discovery of various structural properties of general counting functions coming from enumeration of walks. 1. ..."
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Cited by 34 (9 self)
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Abstract. We propose an experimental mathematics approach leading to the computerdriven discovery of various structural properties of general counting functions coming from enumeration of walks. 1.
A comparison of three highprecision quadrature schemes
 Experimental Mathematics
, 2004
"... The authors have implemented three numerical quadrature schemes, using the Arbitrary Precision (ARPREC) software package. The objective here is a quadrature facility that can efficiently evaluate to very high precision a large class of integrals typical of those encountered in experimental mathemati ..."
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Cited by 30 (13 self)
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The authors have implemented three numerical quadrature schemes, using the Arbitrary Precision (ARPREC) software package. The objective here is a quadrature facility that can efficiently evaluate to very high precision a large class of integrals typical of those encountered in experimental mathematics, relying on a minimum of a priori information regarding the function to be integrated. Such a facility is useful, for example, to permit the experimental identification of definite integrals based on their numerical values. The performance and accuracy of these three quadrature schemes are compared using a suite of 15 integrals, ranging from continuous, wellbehaved functions on finite intervals to functions with infinite derivatives and blowup singularities at endpoints, as well as several integrals on an infinite interval. In results using 412digit arithmetic, we achieve at least 400digit accuracy, using two of the programs, for all problems except one highly oscillatory function on an infinite interval. Similar results were obtained using 1012digit arithmetic.
HighPrecision Computation and Mathematical Physics
"... At the present time, IEEE 64bit floatingpoint arithmetic is sufficiently accurate for most scientific applications. However, for a rapidly growing body of important scientific computing applications, a higher level of numeric precision is required. Such calculations are facilitated by highpreci ..."
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Cited by 17 (3 self)
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At the present time, IEEE 64bit floatingpoint arithmetic is sufficiently accurate for most scientific applications. However, for a rapidly growing body of important scientific computing applications, a higher level of numeric precision is required. Such calculations are facilitated by highprecision software packages that include highlevel language translation modules to minimize the conversion effort. This paper presents a survey of recent applications of these techniques and provides some analysis of their numerical requirements. These applications include supernova simulations, climate modeling, planetary orbit calculations, Coulomb nbody atomic systems, scattering amplitudes of quarks, gluons and bosons, nonlinear oscillator theory, Ising theory, quantum field theory and experimental mathematics. We conclude that highprecision arithmetic facilities are now an indispensable component of a modern largescale scientific computing environment.
A compendium of BBPtype formulas for mathematical constants
, 2000
"... A 1996 paper by the author, Peter Borwein and Simon Plouffe showed that any mathematical constant given by an infinite series of a certain type has the property that its nth digit in a particular number base could be calculated directly, without needing to compute any of the first n−1 digits, by me ..."
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Cited by 15 (2 self)
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A 1996 paper by the author, Peter Borwein and Simon Plouffe showed that any mathematical constant given by an infinite series of a certain type has the property that its nth digit in a particular number base could be calculated directly, without needing to compute any of the first n−1 digits, by means of a simple algorithm that does not require multipleprecision arithmetic. Several such formulas were presented in that paper, including formulas for the constants π and log 2. Since then, numerous other formulas of this type have been found. This paper presents a compendium of currently known results of this sort, both formal and experimental. Many of these results were found in the process of compiling this collection and have not previously appeared in the literature. Several conjectures suggested by these results are mentioned.
Ten Problems in Experimental Mathematics
, 2006
"... Challenge ” of Nick Trefethen, beautifully described in [12] (see also [13]). Indeed, these ten numeric challenge problems are also listed in [15, pp. 22–26], where they are followed by the ten symbolic/numeric challenge problems that are discussed in this article. Our intent in [15] was to present ..."
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Cited by 11 (5 self)
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Challenge ” of Nick Trefethen, beautifully described in [12] (see also [13]). Indeed, these ten numeric challenge problems are also listed in [15, pp. 22–26], where they are followed by the ten symbolic/numeric challenge problems that are discussed in this article. Our intent in [15] was to present ten problems that are characteristic of the sorts of problems
Supporting extended precision on graphics processors
 Proceedings of the Sixth International Workshop on Data Management on New Hardware (DaMoN 2010), June 7, 2010
, 2010
"... Scientific computing applications often require support for nontraditional data types, for example, numbers with a precision higher than 64bit floats. As graphics processors, or GPUs, have emerged as a powerful accelerator for scientific computing, we design and implement a GPUbased extended pre ..."
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Cited by 10 (1 self)
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Scientific computing applications often require support for nontraditional data types, for example, numbers with a precision higher than 64bit floats. As graphics processors, or GPUs, have emerged as a powerful accelerator for scientific computing, we design and implement a GPUbased extended precision library to enable applications with high precision requirement to run on the GPU. Our library contains arithmetic operators, mathematical functions, and dataparallel primitives, each of which can operate at either multiterm or multidigit precision. The multiterm precision maintains an accuracy of up to 212 bits of signifcand whereas the multidigit precision allows an accuracy of an arbitrary number of bits. Additionally, we have integrated the extended precision algorithms to a GPUbased query processing engine to support efficient query processing with extended precision on GPUs. To demonstrate the usage of our library, we have implemented three applications: parallel summation in climate modeling, Newton’s method used in nonlinear physics, and high precision numerical integration in experimental mathematics. The GPUbased implementation is up to an order of magnitude faster, and achieves the same accuracy as their optimized, quadcore CPUbased counterparts. 1.