Guidelines for the Use of QUADPACK

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

doi:10.1007/978-3-642-61786-7_4

Cited by 9 publications

(12 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…First note that the limit T → 0 with simultaneously R → 0 is unproblematic since B ≥ 2A > 0 and C ≥ A > 0 in this case. The integrands then are bounded and positive definite, as can be seen by looking at the starting expressions in (21) and (23). Further, a short calculation shows that the limits T → ± √ R with R > 0 are finite, so the integrand is also non-singular in this limit.…”

Section: Integral Representations Of Triangle and Box Functionsmentioning

confidence: 91%

“…Since the integrand of the triangle function (25) is only 1-dimensional, we can rely in this case on the efficient and highly robust QAGS routine of QUADPACK [21], a widely-used package for the numerical computation of definite 1-dimensional integrals. The QAGS algorithm applies an integration rule adaptively in subintervals until the error estimate is sufficiently small.…”

Section: Numerical Evaluationmentioning

confidence: 99%

See 1 more Smart Citation

A numerical evaluation of the scalar hexagon integral in the physical region

2003

View full text Add to dashboard Cite

We derive an analytic expression for the scalar one-loop pentagon and hexagon functions which is convenient for subsequent numerical integration. These functions are of relevance in the computation of next-to-leading order radiative corrections to multi-particle cross sections. The hexagon integral is represented in terms of n-dimensional triangle functions and (n+2)-dimensional box functions. If infrared poles are present this representation naturally splits into a finite and a pole part. For a fast numerical integration of the finite part we propose simple one-and two-dimensional integral representations. We set up an iterative numerical integration method to calculate these integrals directly in an efficient way. The method is illustrated by explicit results for pentagon and hexagon functions with some generic physical kinematics.

show abstract

Section: Integral Representations Of Triangle and Box Functionsmentioning

confidence: 91%

Section: Numerical Evaluationmentioning

confidence: 99%

A numerical evaluation of the scalar hexagon integral in the physical region

2003

View full text Add to dashboard Cite

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: Configurations Involving Clasp Arcsmentioning

confidence: 99%

Criticality for the Gehring link problem

Cantarella

Kusner

et al. 2006

Geom. Topol.

View full text Add to dashboard Cite

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.

show abstract

“…We will refer to these methods as SR, and QAWO, respectively. For the quadrature rule we used a numerical subroutine found in the book QUADPACK [5] which is designed specifically for integrands which are in the form of a function multiplied by an oscillatory function. We present the results of the comparison in the table below.…”

Section: Comparison With Other Methodsmentioning

confidence: 99%

“…Namely, we first express the integral in terms of complete elliptic integrals of the first and second kinds, and then use an algorithm found in Abramowitz and Stegan [1] to compute these elliptic integrals. We compare this technique to two other popular methods: Simpson's Rule, and a quadrature rule subroutine found in the book QUADPACK [5] designed specifically to handle integrals such as (1). Our method proved significantly superior to these other two.…”

Section: Introductionmentioning

confidence: 99%

Numerical evaluation of a symmetric potential function

Carmack¹

1998

Math. Comp.

View full text Add to dashboard Cite

Abstract. We discuss the numerical evaluation of a symmetric potential function which arises naturally in applications. We present a method designed to accurately and efficiently compute this integral, and compare the performance of this method with two other popular techniques. This method requires considerably fewer function evaluations than all other techniques we tested, and is applicable to any integral which can be expressed in terms of complete elliptic integrals.

show abstract

Guidelines for the Use of QUADPACK

Cited by 9 publications

References 0 publications

A numerical evaluation of the scalar hexagon integral in the physical region

A numerical evaluation of the scalar hexagon integral in the physical region

Criticality for the Gehring link problem

Numerical evaluation of a symmetric potential function

Contact Info

Product

Resources

About