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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.
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 27 (22 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.
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 24 (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 ...
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 24 (8 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.
Doubleexponential fast Gauss transform algorithms for pricing discrete lookback options
, 2005
"... This paper presents fast and accurate algorithms for computing the prices of discretely sampled lookback options. Under the BlackScholes framework, the pricing of a discrete lookback option can be reduced to a series of convolutions of a function with the Gaussian distribution. Using this fact, an ..."
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Cited by 19 (0 self)
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This paper presents fast and accurate algorithms for computing the prices of discretely sampled lookback options. Under the BlackScholes framework, the pricing of a discrete lookback option can be reduced to a series of convolutions of a function with the Gaussian distribution. Using this fact, an efficient algorithm, which computes these convolutions by a combination of the doubleexponential integration formula and the fast Gauss transform, has been proposed recently. We extend this algorithm to lookback options under Merton’s jumpdiffusion model and American lookback options. Numerical experiments show that our method is much faster and more accurate than conventional methods for lookback options under Merton’s model. For American lookback options, our method outperforms conventional methods when required accuracy is relatively high. A lookback option is the right to sell an asset at the end of a time period at the highest price the asset took during the period (lookback put option),
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 17 (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].
THE EXPONENTIALLY CONVERGENT TRAPEZOIDAL RULE
"... Abstract. It is well known that the trapezoidal rule converges geometrically when applied to analytic functions on periodic intervals or the real line. The mathematics and history of this phenomenon are reviewed and it is shown that far from being a curiosity, it is linked with computational methods ..."
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Cited by 17 (3 self)
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Abstract. It is well known that the trapezoidal rule converges geometrically when applied to analytic functions on periodic intervals or the real line. The mathematics and history of this phenomenon are reviewed and it is shown that far from being a curiosity, it is linked with computational methods all across scientific computing, including algorithms related to inverse Laplace transforms, special functions, complex analysis, rational approximation, integral equations, and the computation of functions and eigenvalues of matrices and operators.
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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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.
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.