Special Applications of QUADPACK

Piessens, Robert; Doncker-Kapenga, Elise de; Überhuber, Christoph; Kahaner, David K.

doi:10.1007/978-3-642-61786-7_5

Cited by 14 publications

(14 citation statements)

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“…Then we are left with two nonlinear integral equations in the other two variables. While we have not proved formally that a solution to this system exists, we have solved it numerically to high precision, both in Mathematica and using MINPACK [26] and QUADPACK [30]. We obtain ρ 0 ≈ 0.4074218, σ 0 ≈ 0.4177486, τ 0 ≈ 0.7561107, where again we follow the standard convention that the error is less than ±1 in the last digit shown.…”

Section: Configurations Involving Clasp Arcsmentioning

confidence: 97%

Criticality for the Gehring link problem

Cantarella

Kusner

et al. 2006

Geom. Topol.

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In 1974, Gehring posed the problem of minimizing the length of two linked curves separated by unit distance. This constraint can be viewed as a measure of thickness for links, and the ratio of length over thickness as the ropelength. In this paper we refine Gehring's problem to deal with links in a fixed link-homotopy class: we prove ropelength minimizers exist and introduce a theory of ropelength criticality.Our balance criterion is a set of necessary and sufficient conditions for criticality, based on a strengthened, infinite-dimensional version (Theorem 5.4) of the KuhnTucker theorem. We use this to prove that every critical link is C 1 with finite total curvature. The balance criterion also allows us to explicitly describe critical configurations (and presumed minimizers) for many links including the Borromean rings. We also exhibit a surprising critical configuration for two clasped ropes: near their tips the curvature is unbounded and a small gap appears between the two components. These examples reveal the depth and richness hidden in Gehring's problem and our natural extension.

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Section: Configurations Involving Clasp Arcsmentioning

confidence: 97%

Criticality for the Gehring link problem

Cantarella

Kusner

et al. 2006

Geom. Topol.

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show abstract

“…Then we are left with two nonlinear integral equations in the other two variables. While we have not proved formally that a solution to this system exists, we have solved it numerically to high precision, both in Mathematica and using MINPACK [26] and QUADPACK [30]. We obtain where again we follow the standard convention that the error is less than˙1 in the last digit shown.…”

Section: Symmetry and Convexitymentioning

confidence: 99%

Criticality for the Gehring link problem

Cantarella¹,

Fu²,

Kusner³

et al. 2006

Geom. Topol.

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In 1974, Gehring posed the problem of minimizing the length of two linked curves separated by unit distance. This constraint can be viewed as a measure of thickness for links, and the ratio of length over thickness as the ropelength. In this paper we refine Gehring's problem to deal with links in a fixed link-homotopy class: we prove ropelength minimizers exist and introduce a theory of ropelength criticality. Our balance criterion is a set of necessary and sufficient conditions for criticality, based on a strengthened, infinite-dimensional version (Theorem 5.4) of the Kuhn-Tucker theorem. We use this to prove that every critical link is C 1 with finite total curvature. The balance criterion also allows us to explicitly describe critical configurations (and presumed minimizers) for many links including the Borromean rings. We also exhibit a surprising critical configuration for two clasped ropes: near their tips the curvature is unbounded and a small gap appears between the two components. These examples reveal the depth and richness hidden in Gehring's problem and our natural extension. 57M25; 49Q10, 53A04

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“…In such schemes, which we abbreviate as GKP, the first n and 2n + 1 nodes, respectively, can be reused albeit with different weights. Further, in numerical mathematics, the difference between such nested pairs of approximations of the integrand, I n and I 2n+1 , serves as criterion when to stop a numerical quadrature according to 10,19…”

Section: Gauss-legendre Quadrature (Gau) and The Kronrod-patterson Exmentioning

confidence: 99%

“…The most difficult method in this respect is GKP. For selected numbers of nodes, values for the abscissa and weights can be found in QUADPACK, 19 the original paper by Patterson,15 or the patterson_rule program by Burkardt. 31 In addition, software described in ref.…”

Section: Some Practical Aspects Of the Numerical Quadrature Schemes Usedmentioning

confidence: 99%

“…10 The meaning of small, however, is in the eye of the beholder. E.g., the arguably "simplest" and only nonadaptive subroutine (D)QNG in the QUADPACK library for numerical integration 19 use a 10-21 Gauss-Kronrod rule, 20 requiring the evaluation of the integrand at 21 points. Ideally, we would like to use far fewer than 21 λ-states in free energy simulations since, as shown in the companion paper, 2 21 λ-states usually suffice even when using the trapezoidal rule.…”

Section: Introductory Remarksmentioning

confidence: 99%

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Efficiency of alchemical free energy simulations. II. Improvements for thermodynamic integration

2010

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We attempt to optimize the efficiency of thermodynamic integration, as defined by the minimal number of unphysical intermediate states required for the computation of accurate and precise free energy differences. The suitability of various numerical quadrature methods is tested. In particular, we compare the trapezoidal rule, Simpson's rule, Gauss-Legendre, Gauss-Kronrod-Patterson, and Clenshaw-Curtis integration, as well as integration based on a cubic spline approximation of the integrand. We find that Simpson's rule and spline integration are already significantly more efficient that the trapezoidal rule, i.e., correct free energy differences can be obtained using fewer λ-states. We demonstrate that Simpson's rule can be used advantageously with nonequidistant values of the abscissa, which increases the flexibility of the method. Efficiency is enhanced even further if higher order methods, such as Gauss-Legendre, Gauss-Kronrod-Patterson, or Clenshaw-Curtis integration, are used; no more than seven λ-states, which in the case of Clenshaw-Curtis integration include the physical end states, were required for accurate results in all test problems studied. Thus, the performance of thermodynamic integration can equal that of Bennett's acceptance ratio method. We also show, however, that the high efficiency found here relies on the particular functional form of the soft-core potential used; overall, thermodynamic integration is more susceptible to the details of the hybrid Hamiltonian used than Bennett's acceptance ratio method. Therefore, we recommend Bennett's acceptance ratio method as the most robust method to compute alchemical free energy differences; nevertheless, scenarios when thermodynamic integration may be preferable are discussed.

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Special Applications of QUADPACK

Cited by 14 publications

References 0 publications

Criticality for the Gehring link problem

Criticality for the Gehring link problem

Criticality for the Gehring link problem

Efficiency of alchemical free energy simulations. II. Improvements for thermodynamic integration

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