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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.
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 20 (11 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 Numerical Integration: Progress and Challenges
, 2008
"... Abstract. One of the most fruitful advances in the field of experimental mathematics has been the development of practical methods for very highprecision numerical integration, a quest initiated by Keith Geddes and other researchers in the 1980s and 1990s. These techniques, when coupled with equall ..."
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Cited by 4 (1 self)
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Abstract. One of the most fruitful advances in the field of experimental mathematics has been the development of practical methods for very highprecision numerical integration, a quest initiated by Keith Geddes and other researchers in the 1980s and 1990s. These techniques, when coupled with equally powerful integer relation detection methods, have resulted in the analytic evaluation of many integrals that previously were beyond the realm of symbolic techniques. This paper presents a survey of the current stateoftheart in this area (including results by the present authors and others), mentions some new results, and then sketches what challenges lie ahead. 1
A dynamical strategy for approximation methods
"... The numerical result provided by an approximation method is affected by a global error, which consists of both a truncation error and a roundoff error. Let us consider the converging sequence generated by successively dividing by two the step size used. If computations are performed until, in the c ..."
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Cited by 2 (1 self)
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The numerical result provided by an approximation method is affected by a global error, which consists of both a truncation error and a roundoff error. Let us consider the converging sequence generated by successively dividing by two the step size used. If computations are performed until, in the convergence zone, the difference between two successive approximations is only due to roundoff errors, then the global error on the result obtained is minimal. Furthermore its significant bits which are not affected by roundoff errors are in common with the exact result, up to one. To cite
DOI 10.1007/s1178601101034 Mathematics in Computer Science Stochastic Arithmetic in Multiprecision
"... Abstract Floatingpoint arithmetic precision is limited in length the IEEE single (respectively double) precision format is 32bit (respectively 64bit) long. Extended precision formats can be up to 128bit long. However some problems require a longer floatingpoint format, because of roundoff erro ..."
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Abstract Floatingpoint arithmetic precision is limited in length the IEEE single (respectively double) precision format is 32bit (respectively 64bit) long. Extended precision formats can be up to 128bit long. However some problems require a longer floatingpoint format, because of roundoff errors. Such problems are usually solved in arbitrary precision, but roundoff errors still occur and must be controlled. Interval arithmetic has been implemented in arbitrary precision, for instance in the MPFI library. Interval arithmetic provides guaranteed results, but it is not well suited for the validation of huge applications. The CADNA library estimates roundoff error propagation using stochastic arithmetic. CADNA has enabled the numerical validation of reallife applications, but it can be used in single precision or in double precision only. In this paper, we present a library called SAM (Stochastic Arithmetic in Multiprecision). It is a multiprecision extension of the classic CADNA library. In SAM (as in CADNA), the arithmetic and relational operators are overloaded in order to be able to deal with stochastic numbers. As a consequence, the use of SAM in a scientific code needs only few modifications. This new library SAM makes it possible to dynamically control the numerical methods used and more particularly to determine the optimal number of iterations in an iterative process. We present some applications of SAM in the numerical validation of chaotic