Elise de Doncker

Algorithm 612: TRIEX: Integration Over a TRIangle Using Nonlinear EXtrapolation

1. USAGE OF THE INTEGRATOR We adopt notation such as T R I E X () for a subroutine or a function, R E S U L T for a variable name, and V E R (, , ,) for a two-dimensional array. Permission to copy without fee all or part of this material is granted provided that the copies are not made or &stnbuted for direct commercial advantage, the ACM copyright notice and the title of the publication and its date appear, and notice is given that copying Is by permission of the Association for Computing Machinery. To copy otherwise, or to repubhsh, requires a fee and/or specific permission.

The Integration of the Multivariate Normal Density Function for the Triangular Method

The model involved in the triangular method is presented, which leads to the need for evaluating a multidimensional integral of the multidimensional normal density function over an irregular region. Work done on the numerical evaluation of this integral is discussed.

Algorithm 45. Automatic computation of improper integrals over a bounded or unbounded planar region

An automatic quadrature algorithm especially designed for double integration of functions with some form of singular behaviour on the boundary of the integration region is described, and its FORTRAN code is presented.The algorithm is based on the use of the product trapezoidal rule, after a non-linear transformation of the integrand in both variables renders a new integrand function whose derivatives vanish on the (transformed) boundary.Numerical results demonstrate the ability of the algorithm to obtain high accuracies in dealing automatically with pathological singularities of non-specific types.ZusammenfassungEs wird ein Algorithmus zur automatischen zweidimensionalen Quadratur beschrieben, der speziell für Integranden mit singulärem Verhalten am Rand des Integrationsgebietes geeignet ist. Eine Implementierung dieses Algorithmus in Standard FORTRAN wird vorgestellt.Der Algorithmus beruht auf einer nichtlinearen Transformation des Integranden in beiden Veränderlichen, die so gewählt, wird, daß der transformierte Integrand und alle seine Ableitungen am Rand des (transformierten) Integrationsgebiets verschwinden. Auf den transformierten Integranden wird die Trapezregel in der Produktform angewendet.An Hand von numerischen Testresultaten wird die Effizienz des vorgestellten Algorithmus bei der automatischen Ermittlung von sehr genauen Näherungswerten für Integrale mit unspezifischen, pathologischen Singularitäten demonstriert.

Algorithm 32 automatic computation of integrals with singular integrand, over a finite or an infinite range

A numerical quadrature algorithm is developed, for integrands which may exhibit some kind of singular behaviour within the finite of infinite integration range.Using the automatical FORTRAN IV integration program, one should provide the abscissae the function is not “smooth” at.The quadrature formula has been obtained by applying the trapezoidal rule after transformation of the integrand.Standing severe tests which were based on the test functions of Casaletto et al. and on Kahaners sample set, the integration scheme turned out to be of a remarkable reliability, efficiency and accuracy.ZusammenfassungEs wird ein Algorithmus zur numerischen Quadratur beschrieben, der für Integranden mit singulärem Verhalten in einem endlichen oder unendlichen Integrationsbereich geeignet ist.Die Quadraturformel wurde durch Transformation des Integranden und anschließende Anwendung der Trapezregel erhalten.Es wird ein FORTRAN IV Programm vorgestellt, das auf der Anwendung dieser Quadraturformel beruht. Beim Aufruf dieses Programms sollte der Benutzer jene Abszissenwerte vorgeben, an denen der Integrand Singularitäten aufweist.Unter Verwendung der Testfunktionen von Casaletto et al. und von Kahaner konnte die Zuverlässigkeit, Effizienz und Genauigkeit dieses Integrationsprogramms nachgewiesen werden.

New Euler-Maclaurin Expansions and Their Application to Quadrature Over the s-Dimensional Simplex

The (i-panel offset trapezodial rule for noninteger values of u, is introduced in a one-dimensional context. An asymptotic series describing the error functional is derived. The values of u for which this is an even Euler-Maclaurin expansion are determined, together with the conditions under which it terminates after a finite number of terms. This leads to a new variant of one-dimensional Romberg integration. The theory is then extended to quadrature over the s-dimensional simplex, the basic rules being obtained by an iterated use of one-dimensional rules. The application to Romberg integration is discussed, and it is shown how Romberg integration over the simplex has properties analogous to those for standard one-dimensional Romberg integration and Romberg integration over the hypercube. Using extrapolation, quadrature rules for the i-simplex can be generated, and a set of formulas can be obtained which are the optimum so far discovered in the sense of requiring fewest function values to obtain a specific polynomial degree. 1. Introduction and Summary. The usual definition of the /i-panel offset trapezoidal rule, as given in Lyness and Puri (4), restricts p to being a positive integer. We extend their definition to apply with arbitrary positive p. An asymptotic expan- sion is derived, describing the error functional which corresponds to the rule sum. In this way it is, for example, revealed that the /i-panel mid-point and end-point offset trapezoidal rules satisfy an even Euler-Maclaurin type expansion not only for integer but also for half-integer values of p. This suggests that one may use also the latter rules for one-dimensional Romberg integration. All of the one-dimensional results are presented in Section 2. Section 3 deals with the extension of these results to the s-dimensional unit simplex A. This extension is carried out following the techniques of Lyness and Puri (4). The s-dimensional rule is defined as an iterated rule operator, obtaining the rule sum by a repeated application of one-dimensional /i-panel rules. If the product is formed using only mid-point or end-point offset trapezoidal rules, it is found to satisfy an even Euler-Maclaurin expansion for half-integer p and

Numerical Evaluation of Feynman Integrals by a Direct Computation Method

A purely numerical method, Direct Computation Method is applied to evaluate Feynman integrals. This method is based on the combination of an efcient numerical integration and an efcient extrapolation. In addition, high-precision arithmetic and parallelization technique can be used in this method if required. We present the recent progress in development of this method and show results such as one-loop 5-point and two-loop 3-point integrals.

A parallelization of adaptive task partitioning algorithms

Abstract An adaptive task partitioning scheme for MIMD architectures is investigated. For many serial adaptive procedures this methodology provides a direct translation into reasonably efficient parallel versions. A proto-type of a two-dimensional integration method has been implemented in this manner. This was facilitated by using a set of high-level macros, layered over the Argonne macro package, which provides the primitives in the adaptive partitioning scheme. Because of the portable nature of the Argonne macro package our code should be readily ported to other MIMD machines with shared memory. A version of the macros for other MIMD architectures is also possible.

Quadpack computation of Feynman loop integrals

Abstract The paper addresses a numerical computation of Feynman loop integrals, which are computed by an extrapolation to the limit as a parameter in the integrand tends to zero. An important objective is to achieve an automatic computation which is effective for a wide range of instances. Singular or near singular integrand behavior is handled via an adaptive partitioning of the domain, implemented in an iterated/repeated multivariate integration method. Integrand singularities possibly introduced via infrared (IR) divergence at the boundaries of the integration domain are addressed using a version of the Dqags algorithm from the integration package Quadpack, which uses an adaptive strategy combined with extrapolation. The latter is justified for a large class of problems by the underlying asymptotic expansions of the integration error. For IR divergent problems, an extrapolation scheme is presented based on dimensional regularization.

Large—scale parallel numerical integration

We present and analyze strategies which can be used for the parallel computation of large numbers of integrals which may be of different levels of difficulty. Parallelization on the integral level, which is generally used for large numbers of integrals, is combined with parallelization on the subregion level, which enables handling local integration difficulties within individual problems. This results in a new, hierarchical algorithm which incorporates load balancing on the integral level and on the subregion level. We report test results of the software and show that the hierarchical approach leads to a scalable integration algorithm.

Multivariate integration on hypercubic and mesh networks

Abstract We analyze a class of adaptive algorithms for integration over N-dimensional hyper-rectangular or simplical regions, on distributed systems. An adaptive algorithm attempts to achieve the requested accuracy by refining the subdivision of the integration region, thus allowing for a concentration of subdivisions near singularities. At the subdivision of a region, the error behaves according to a prescribed model, relating the error of the parent region to that of its children. The analysis can also be applied to problems in other areas, as long as the task selection is based on a priority function which behaves according to a suitable model. Using an efficient management of the subregions, we show that an O (p/ log p) speedup can be achieved on a p-processor hypercubic network, such as shuffle exchange, butterfly and hypercube. Furthermore, a speedup of O ( p ) can be achieved on a p × p mesh network. We also show that our algorithms compare favorably with well-known dynamic load balancing strategies.