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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 21 (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.
Effective error bounds for EulerMaclaurinbased quadrature schemes
 PROC. 2006 CONF. ON HIGHPERFORMANCE COMPUTING SYSTEMS, IEEE COMPUTER SOCIETY, 2006, AVAILABLE AT HTTP://CRD.LBL.GOV/~DHBAILEY/DHBPAPERS/HPCS06.PDF
, 2005
"... We analyze the behavior of EulerMaclaurinbased integration schemes with the intention of deriving accurate and economic estimations of the error term. ..."
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Cited by 12 (10 self)
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We analyze the behavior of EulerMaclaurinbased integration schemes with the intention of deriving accurate and economic estimations of the error term.
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 11 (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.
COMPUTING A α, log(A) AND RELATED MATRIX FUNCTIONS BY CONTOUR INTEGRALS
, 2007
"... Abstract. New methods are proposed for the numerical evaluation of f(A) or f(A)b, where f(A) is a function such as A 1/2 or log(A) with singularities in (−∞, 0] and A is a matrix with eigenvalues on or near (0, ∞). The methods are based on combining contour integrals evaluated by the periodic trapez ..."
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Cited by 11 (1 self)
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Abstract. New methods are proposed for the numerical evaluation of f(A) or f(A)b, where f(A) is a function such as A 1/2 or log(A) with singularities in (−∞, 0] and A is a matrix with eigenvalues on or near (0, ∞). The methods are based on combining contour integrals evaluated by the periodic trapezoid rule with conformal maps involving Jacobi elliptic functions. The convergence is geometric, so that the computation of f(A)b is typically reduced to one or two dozen linear system solves, which can be carried out in parallel. Key words. Cauchy integral, conformal map, contour integral, matrix function, quadrature, rational approximation, trapezoid rule AMS subject classifications. 65F30, 65D30 1. Introduction. It
Elliptic integral evaluations of Bessel moments and applications
, 2008
"... We record and substantially extend what is known about the closed forms for various Bessel function moments arising in quantum field theory, condensed matter theory and other parts of mathematical physics. In particular, we develop formulae for integrals of products of six or fewer Bessel functions. ..."
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Cited by 8 (6 self)
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We record and substantially extend what is known about the closed forms for various Bessel function moments arising in quantum field theory, condensed matter theory and other parts of mathematical physics. In particular, we develop formulae for integrals of products of six or fewer Bessel functions. In consequence, we are able to discover and prove closed forms for cn,k: = � ∞ (t)dt with integers n = 1,2,3,4 and k ≥ 0, 0 tkKn 0 obtaining new results for the even moments c3,2k and c4,2k. We also derive new closed forms for the odd moments sn,2k+1: = � ∞ 0 t2k+1I0 (t)K n−1 0 (t)dt with n = 3,4 and for tn,2k+1: = � ∞ 0 t2k+1I2 0 (t)Kn−2 0 (t)dt with n = 5, relating the latter to Green functions on hexagonal, diamond and cubic lattices. We conjecture the values of s5,2k+1, make substantial progress on the evaluation of c5,2k+1, s6,2k+1 and t6,2k+1 and report more limited progress regarding c5,2k, c6,2k+1 and c6,2k. In the process, we obtain 8 conjectural evaluations, each of which has been checked to 1200 decimal places. One of these lies deep in 4dimensional quantum field theory and two are probably provable by delicate combinatorics. There remains a hard core of five conjectures whose proofs would be most instructive, to mathematicians and physicists alike.
Handtohand combat with thousanddigit integrals
, 2010
"... In this paper we describe numerical investigations of definite integrals that arise by considering the moments of multistep uniform random walks in the plane, together with a closely related class of integrals involving the elliptic functions K, K ′ , E and E ′. We find that in many cases such inte ..."
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Cited by 6 (3 self)
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In this paper we describe numerical investigations of definite integrals that arise by considering the moments of multistep uniform random walks in the plane, together with a closely related class of integrals involving the elliptic functions K, K ′ , E and E ′. We find that in many cases such integrals can be “experimentally ” evaluated in closed form or that intriguing linear relations exist within a class of similar integrals. Discovering these identities and relations often requires the evaluation of integrals to extreme precision, combined with largescale runs of the “PSLQ ” integer relation algorithm. This paper presents details of the techniques used in these calculations and mentions some of the many difficulties that can arise.
NEW QUADRATURE FORMULAS FROM CONFORMAL MAPS
, 2008
"... Gauss and Clenshaw–Curtis quadrature, like Legendre and Chebyshev spectral methods, make use of grids strongly clustered at boundaries. From the viewpoint of polynomial approximation this seems necessary and indeed in certain respects optimal. Nevertheless such methods may “waste” a factor of π/2 ..."
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Cited by 6 (4 self)
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Gauss and Clenshaw–Curtis quadrature, like Legendre and Chebyshev spectral methods, make use of grids strongly clustered at boundaries. From the viewpoint of polynomial approximation this seems necessary and indeed in certain respects optimal. Nevertheless such methods may “waste” a factor of π/2 with respect to each space dimension. We propose new nonpolynomial quadrature methods that avoid this effect by conformally mapping the usual ellipse of convergence to an infinite strip or another approximately straightsided domain. The new methods are compared with related ideas of Bakhvalov, Kosloff and TalEzer, Rokhlin and Alpert, and others. An advantage of the conformal mapping approach is that it leads to theorems guaranteeing geometric rates of convergence for analytic integrands. For example, one of the formulas presented is proved to converge 50 % faster than Gauss quadrature for functions analytic in an εneighborhood of [−1, 1].
Summary of Sinc Numerical Methods
"... Sinc approximation methods excel for problems whose solutions may have singularities, or infinite domains, or boundary layers. This article summarizes results obtained to date, on Sinc numerical methods of computation. Sinc methods provide procedures for function approximation over bounded or unbou ..."
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Cited by 5 (0 self)
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Sinc approximation methods excel for problems whose solutions may have singularities, or infinite domains, or boundary layers. This article summarizes results obtained to date, on Sinc numerical methods of computation. Sinc methods provide procedures for function approximation over bounded or unbounded regions, encompassing interpolation, approximation of derivatives, approximate definite and indefinite integration, solving initial value ordinary differential equation problems, approximation and inversion of Fourier and Laplace transforms, approximation of Hilbert transforms, and approximation of indefinite convolutions, the approximate solution of partial differential equations, and the approximate solution of integral equations, methods for constructing conformal maps, and methods for analytic continuation. Indeed, Sinc are ubiquitous for approximating every operation of calculus. 1 Introduction and Summary This article attempts to summarize the existing numerical methods based on ...