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Parallel Integer Relation Detection: Techniques and Applications
 Mathematics of Computation
, 2000
"... Let {x1,x2, ···,xn} be a vector of real numbers. An integer relation algorithm is a computational scheme to find the n integers ak, if they exist, such that a1x1 +a2x2 +···+ anxn = 0. In the past few years, integer relation algorithms have been utilized to discover new results in mathematics and phy ..."
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Cited by 43 (32 self)
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Let {x1,x2, ···,xn} be a vector of real numbers. An integer relation algorithm is a computational scheme to find the n integers ak, if they exist, such that a1x1 +a2x2 +···+ anxn = 0. In the past few years, integer relation algorithms have been utilized to discover new results in mathematics and physics. Existing programs for this purpose require very large amounts of computer time, due in part to the requirement for multiprecision arithmetic, yet are poorly suited for parallel processing. This paper presents a new integer relation algorithm designed for parallel computer systems, but as a bonus it also gives superior results on single processor systems. Singleand multilevel implementations of this algorithm are described, together with performance results on a parallel computer system. Several applications of these programs are discussed, including some new results in mathematical number theory, quantum field theory and chaos theory.
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 33 (16 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.
Experimental Mathematics: Recent Developments and Future Outlook
 CECM PREPRINT 99:143] FFL J.M. BORWEIN AND P.B. BORWEIN, &QUOT;CHALLENGES FOR MATHEMATICAL COMPUTING,&QUOT; COMPUTING IN SCIENCE & ENGINEERING, 2001. [CECM PREPRINT 01:160
, 2000
"... ..."
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 22 (9 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.
Highly parallel, highprecision numerical integration

, 2008
"... This paper describes schemes for rapidly computing numerical values of definite integrals to very high accuracy (hundreds to thousands of digits) on highly parallel computer systems. Such schemes are of interest not only in computational physics and computational chemistry, but also in experimental ..."
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Cited by 21 (19 self)
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This paper describes schemes for rapidly computing numerical values of definite integrals to very high accuracy (hundreds to thousands of digits) on highly parallel computer systems. Such schemes are of interest not only in computational physics and computational chemistry, but also in experimental mathematics, where highprecision numerical values of definite integrals can be used to numerically discover new identities. This paper presents performance results for 1D and 2D integral test suites on highly parallel computer systems. Results are also given for certain problems that derive from mathematical physics. One of these results confirms a conjecture to 20,000 digit accuracy. The performance rate for this calculation is 690 Gflop/s on 1024 CPUs of a stateoftheart parallel system. Other results, which range in precision from 120 to 500 digits, and for 1D, 2D, 3D and 4D integrals, derive from Ising theory. The largest of these calculations required 28 hours on 256 CPUs. We believe that these are the first instances of evaluations of nontrivial 3D and 4D integrals to multihundreddigit accuracy.
Highprecision floatingpoint arithmetic in scientific computation
 Computing in Science and Engineering, May–June
, 2005
"... 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: some of these applications require roughly twice ..."
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Cited by 8 (1 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: some of these applications require roughly twice this level; others require four times; while still others require hundreds or more digits to obtain numerically meaningful results. Such calculations have been facilitated by new highprecision software packages that include highlevel language translation modules to minimize the conversion effort. These activities have yielded a number of interesting new scientific results in fields as diverse as quantum theory, climate modeling and experimental mathematics, a few of which are described in this article. Such developments suggest that in the future, the numeric precision used for a scientific computation may be as important to the program design as are the algorithms and data structures.
2001 Parallel integer relation detection: techniques and applications
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Cited by 1 (0 self)
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For guidance on citations see FAQs. c ○ [not recorded] Version: [not recorded] Link(s) to article on publisher’s website:
Central Binomial Sums and Multiple Clausen Values (with Connections to Zeta Values)
, 1999
"... We nd relationships between Riemann zeta values and central binomial sums. We also investigate alternating binomial sums (also called Apery sums). The study of nonalternating sums leads to an investigation of a dierent type of sums which we call multiple Clausen values. The study of alternating sum ..."
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We nd relationships between Riemann zeta values and central binomial sums. We also investigate alternating binomial sums (also called Apery sums). The study of nonalternating sums leads to an investigation of a dierent type of sums which we call multiple Clausen values. The study of alternating sums leads to a tower of results involving polylogarithms in the golden ratio. AMS (1991) subject classication: Primary 40B05, 33E20, Secondary 11M99, 11Y99. Key words: binomial sums, multiple zeta values, logsine integrals, Clausens function, multiple Clausen values, polylogarithms, Apery sums. 1 1 Introduction We began by studying the central binomial sum S(k), given as: S(k) := 1 X n=1 1 n k 2n n (1) for integer k. A classical evaluation is S(4) = 17 36 (4). Using a mixture of integer relation and other computational techniques, we uncover remarkable links to multidimensional polylogarithms of sixth roots of unity and to multidimensional Clausen functions (MCV's) at =...