Integrals of the Ising class

Bailey, David H.; Borwein, Jonathan M.; Crandall, Richard E.

doi:10.1088/0305-4470/39/40/001

Cited by 59 publications

(145 citation statements)

References 12 publications

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“…We tried but failed to recognize D 5 in terms of similar constants (the 500-digit numerical value is available if anyone wishes to try). The conjectured identity shown here for E 5 was confirmed to 240-digit accuracy, which is 180 digits beyond the level that could reasonably be ascribed to numerical round-off error; thus we are quite confident in this result even though we do not have a formal proof [6]. In a follow-on study [8], we examined the following generalization of the C n integrals:…”

Section: Ising Integralsmentioning

confidence: 65%

“…By using the new edition of the Inverse Symbolic Calculator, available at http://carma-lx1.newcastle.edu.au:8087, this numerical value can be identified as lim n→∞ C n = 2e −2γ , where γ is Euler's constant. We later were able to prove this fact-this is merely the first term of an asymptotic expansion-and thus showed that the C n integrals are fundamental in this context [6].…”

Section: Ising Integralsmentioning

confidence: 82%

“…For many integrand functions, these schemes exhibit "quadratic" or "exponential" convergence -dividing the integration interval in half (or, equivalently, doubling the number of evaluation points) approximately doubles the number of correct digits in the result [13]. In a recent study, the present authors together with Richard Crandall applied tanhsinh quadrature, implemented using the ARPREC package, to study the following classes of integrals [6]. The D n integrals arise in the Ising theory of mathematical physics, and the C n have tight connections to quantum field theory.…”

Section: Ising Integralsmentioning

confidence: 99%

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High-precision computation: Mathematical physics and dynamics

Bailey

Barrio

Borwein

2012

Applied Mathematics and Computation

105

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At the present time, IEEE 64-bit floating-point 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 high-precision software packages that include high-level 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 n-body atomic systems, studies of the fine structure constant, scattering amplitudes of quarks, gluons and bosons, nonlinear oscillator theory, experimental mathematics, evaluation of orthogonal polynomials, numerical integration of ODEs, computation of periodic orbits, studies of the splitting of separatrices, detection of strange nonchaotic attractors, Ising theory, quantum field theory, and discrete dynamical systems. We conclude that high-precision arithmetic facilities are now an indispensable component of a modern large-scale scientific computing environment.

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Section: Ising Integralsmentioning

confidence: 65%

Section: Ising Integralsmentioning

confidence: 82%

Section: Ising Integralsmentioning

confidence: 99%

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High-precision computation: Mathematical physics and dynamics

Bailey

Barrio

Borwein

2012

Applied Mathematics and Computation

105

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“…In many domains of mathematical physics [14], the computation of multiple integrals like the evaluation of Feynman diagrams [10], or various correlation functions in quantum field theory and statistical mechanics [15,16] is an important problem. The n-particle contribution to the magnetic susceptibility of the Ising model have integral representations.…”

Section: Introductionmentioning

confidence: 99%

Landau singularities and singularities of holonomic integrals of the Ising class

Boukraa¹,

Hassani²,

Maillard³

et al. 2007

J. Phys. A: Math. Theor.

159

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Abstract.We consider families of multiple and simple integrals of the "Ising class" and the linear ordinary differential equations with polynomial coefficients they are solutions of. We compare the full set of singularities given by the roots of the head polynomial of these linear ODE's and the subset of singularities occurring in the integrals, with the singularities obtained from the Landau conditions. For these Ising class integrals, we show that the Landau conditions can be worked out, either to give the singularities of the corresponding linear differential equation or the singularities occurring in the integral. The singular behavior of these integrals is obtained in the self-dual variable w = s/2/(1 + s 2 ), with s = sinh(2K), where K = J/kT is the usual Ising model coupling constant. Switching to the variable s, we show that the singularities of the analytic continuation of series expansions of these integrals actually break the Kramers-Wannier duality. We revisit the singular behavior (J. Phys. A 38 (2005) 9439-9474) of the third contribution to the magnetic susceptibility of Ising model χ (3) at the points 1+3w +4w 2 = 0 and show that χ (3) (s) is not singular at the corresponding points inside the unit circle |s| = 1, while its analytical continuation in the variable s is actually singular at the corresponding points 2 + s + s 2 = 0 oustside the unit circle (|s| > 1).

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“…Indeed, numerous demanding high-precision computations have been done in this way, most often by invoking parallelism at the level of loops in the application rather than within individual high-precision arithmetic operations. For example, up to 512 processors were employed to compute some two-and three-dimensional integrals in [29]. Parallel high-precision computation will be further discussed in Section 9.…”

Section: Techniques and Software For High-precision Arithmeticmentioning

confidence: 99%

High-Precision Arithmetic in Mathematical Physics

Bailey

Borwein

2015

Mathematics

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For many scientific calculations, particularly those involving empirical data, IEEE 32-bit floating-point arithmetic produces results of sufficient accuracy, while for other applications IEEE 64-bit floating-point is more appropriate. But for some very demanding applications, even higher levels of precision are often required. This article discusses the challenge of high-precision computation, in the context of mathematical physics, and highlights what facilities are required to support future computation, in light of emerging developments in computer architecture.

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Integrals of the Ising class

Cited by 59 publications

References 12 publications

High-precision computation: Mathematical physics and dynamics

High-precision computation: Mathematical physics and dynamics

Landau singularities and singularities of holonomic integrals of the Ising class

High-Precision Arithmetic in Mathematical Physics

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