Computation of the roots of cauchy type principal value integrals

A new method for the computation of the roots of Cauchy type principal value integrals (inside the integration interval) is proposed. This method is based on the extension of the formulae defining such integrals (by analytic continuation) outside the integration interval and, furthermore, on the application of the method of Burniston and Siewert for the computation of the zeros of sectionally analytic functions. A recent analogous method proposed by the author can alternatively be used. Gaussian quadrature rules are used for the numerical computation of the integrals and, hence, of the sought roots. Numerical results are presented for the roots of the Legendre function of the second kind Q₁(x).     

Numerical contour integration for loop integrals

A fully numerical method to calculate loop integrals, a numerical contour-integration method, is proposed. Loop integrals can be interpreted as a contour integral in a complex plane for an integrand with multi-poles in the plane. Stable and efficient numerical integrations an along appropriate contour can be performed for scalar and tensor integrals appearing in loop calculations of the standard model. Examples of 3- and 4-point diagrams in 1-loop integrals and 2- and 3-point diagrams in 2-loop integrals with arbitrary masses are shown.Moreover it is shown that numerical evaluations of the Hypergeometric function, which often appears in the loop integrals, can be performed using the numerical contour-integration method.     

Numerical approximation of electromagnetic signals arising in the evaluation of geological formations

The integration of complicated oscillatory functions arises in computational electromagnetics when evaluating signals produced by the propagation of electromagnetic radiation through the anisotropic layers of a geological formation. The computation of exact integrals involves the evaluation of Sommerfeld integrals. The matrix-pencil method is used for the numerical approximation of such signals. Numerical results show accuracy and robustness of the method for the approximation of these signals, and efficiency in their numerical integration. Sampling frequency is discussed and numerical efficiency is improved by down-sampling.     

The adjustment-stabilization method for constrained systems

For constrained system which has several independent first integrals, we give a new stabilization method which named adjustment-stabilization method. It can stabilize all known constants of motion for a given dynamical system very well instead of the stabilization and post-stabilization methods which only conserves one of all first integrals. Further more, new method can improve numerical accuracy too. We also point out the post-stabilization is just a simplest case of the new method.     

Application of the modified Taylor approximation

In this paper we introduce and describe a new scheme for the numerical integration of smooth functions. The scheme is based on the modified Taylor expansion and is suitable for functions that exhibit near-sinusoidal or repetitive behaviour. We discuss the method and its rate of convergence, then implement it for the approximation of certain integrals. Examples include integrands involving Airy wave, Bessel, Gamma, and elliptic functions. The results, from the data in the tables, demonstrate that the method converges rapidly and approximates the integral as well as some well-known numerical integration methods used with sufficiently small step sizes.     

On the evaluation of Fermi-Dirac integral and its derivatives by IMT and DE quadrature methods

The numerical values of the derivatives of the Fermi-Dirac integral, up to the third order, with respect to its parameters are evaluated by both the IMT and the Double Exponential integration methods. These integrals find extensive use in astrophysical problems. It is found that both the IMT and the DE methods are very efficient, especially for handling the end point integrable singularities. Also, the DE method is slightly superior to the IMT method. Unlike the IMT or Gauss methods, the nodes of DE method can be evaluated in terms of built in elementary functions, which is a big advantage.     

Dynamic elastoplastic analysis of 3-D structures by the domain/boundary element method

The main objective of this paper is to present a general three-dimensional boundary element methodology for solving transient dynamic elastoplastic problems. The elastostatic fundamental solution is used in writing the integral representation and this creates in addition to the surface integrals, volume integrals due to inertia and inelasticity. Thus, an interior discretization in addition to the usual surface discretization is necessary. Isoparametric linear quadrilateral elements are used for the surface discretization and isoparametric linear hexahedra for the interior discretization. Advanced numerical integration techniques for singular and nearly singular integrals are employed. Houbolt's step-by-step numerical time integration algorithm is used to provide the dynamic response. Numerical examples are presented to illustrate the method and demonstrate its accuracy.     

Accurate numerical method for the solutions of the Schrödinger equation and the radial integrals based on the CIP method

A new accurate numerical method based on the constrained interpolation profile (CIP) method to solve the Schrödinger wave equation for bound and free states in central fields and to calculate radial integrals is presented. The radial wave equation is integrated on an arbitrary grid system by the adaptive stepsize controlled Runge-Kutta method controlling the truncation errors within a prescribed accuracy. For the continuum orbitals in the highly oscillating region, the non-linear radial wave equation in the phase-amplitude representation is used. In the evaluation of the derivatives of the radial wave function, the potential energy is approximated by the CIP method. In addition, the radial integrals encountered in the computation of various atomic process are accomplished with the CIP method using the values and their analytical derivatives at the grids. This numerical procedure can be extended in a straightforward way to solve the Dirac wave equation.     

Evaluation of singular integrals in the symmetric Galerkin boundary element method

The implementation of the symmetric Galerkin boundary element method (SGBEM) involves extensive work on the evaluation of various integrals, ranging from regular integrals to hypersingular integrals. In this paper, the treatments of weak singular integrals in the time domain are reviewed, and analytical evaluations for the spatial double integrals which contain weak singular terms are derived. A special scheme on the allocation of Gaussian integration points for regular double integrals in the SGBEM is developed to improve the efficiency of the Gauss–Legendre rule. The proposed approach is implemented for the two-dimensional elastodynamic problems, and two numerical examples are presented to verify the accuracy of the numerical implementation.     

An effective method for numerical evaluation of general 2D and 3D high order singular boundary integrals

In this paper, a robust method is presented for numerical evaluation of weakly, strongly, hyper- and super-singular boundary integrals, which exist in the Cauchy principal value sense in two- and three-dimensional problems. In this method, the singularities involved in integration kernels are analytically removed by expressing the non-singular parts of the integration kernels as power series in the local distance ρ of the intrinsic coordinate system. For three-dimensional boundary integrals, the radial integration method [1] is applied to transform the surface integral into a line integral over the contour of the surface and to remove various orders of singularities within the radial integrals. Some examples are provided to verify the correctness and robustness of the presented method.     

On the selection of collocation points for the numerical solution of singular integral equations with generalized kernels appearing in elasticity problems

A modification of the collocation method for the numerical solution of Cauchy type singular integral equations with generalized kernels is proposed. In accordance with this modification, although the abscissas and weights used in the numerical integration rule for the approximation of the integrals of the integral equation remain unaltered, yet the collocation points are selected in such a way that the poles of the integrands due not only to the Cauchy principal value part of the kernel, but also to the singularities of the generalized part of the kernel are taken into account. This modification assures the convergence of the method to the correct results since the error terms, usually neglected for the collocation points nearest to the end-points of the integration interval and generally tending to infinity, are now taken into consideration for the selection of the collocation points. The method was applied to the singular integral equations derived for the antiplane and plane elasticity problems of a crack terminating at a bimaterial interface.     

The boundary element-free method for 2D interior and exterior Helmholtz problems

The boundary element-free method (BEFM) is developed in this paper for numerical solutions of 2D interior and exterior Helmholtz problems with mixed boundary conditions of Dirichlet and Neumann types. A unified boundary integral equation is established for both interior and exterior problems. By using the improved interpolating moving least squares method to form meshless shape functions, mixed boundary conditions in the BEFM can be satisfied directly and easily. Detailed computational formulas are derived to compute weakly and strongly singular integrals over linear and higher order integration cells. Three numerical integration procedures are developed for the computation of strongly singular integrals. Numerical examples involving acoustic scattering and radiation problems are presented to show the accuracy and efficiency of the meshless method.     

Numerical solution of the two-dimensional Helmholtz equation with variable coefficients by the radial integration boundary integral and integro-differential equation methods

This paper presents new formulations of the boundary–domain integral equation (BDIE) and the boundary–domain integro-differential equation (BDIDE) methods for the numerical solution of the two-dimensional Helmholtz equation with variable coefficients. When the material parameters are variable (with constant or variable wave number), a parametrix is adopted to reduce the Helmholtz equation to a BDIE or BDIDE. However, when material parameters are constant (with variable wave number), the standard fundamental solution for the Laplace equation is used in the formulation. The radial integration method is then employed to convert the domain integrals arising in both BDIE and BDIDE methods into equivalent boundary integrals. The resulting formulations lead to pure boundary integral and integro-differential equations with no domain integrals. Numerical examples are presented for several simple problems, for which exact solutions are available, to demonstrate the efficiency of the proposed methods.     

Automatized analytic continuation of Mellin-Barnes integrals

This paper describe a package written in MATHEMATICA that automatizes typical operations performed during evaluation of Feynman graphs with Mellin-Barnes (MB) techniques. The main procedure allows to analytically continue a MB integral in a given parameter without any intervention from the user and thus to resolve the singularity structure in this parameter. The package can also perform numerical integrations at specified kinematic points, as long as the integrands have satisfactory convergence properties. It is demonstrated that, at least in the case of massive graphs in the physical region, the convergence may turn out to be poor, making naïve numerical integration of MB integrals unusable. Possible solutions to this problem are presented, but full automatization in such cases may not be achievable.

Program summary

Title of program: MBProgram summary URL:http://cpc.cs.qub.ac.uk/summaries/ADYG_v1_0Catalogue identifier: ADYG_v1_0Program obtainable from: CPC Program Library, Queen's University of Belfast, N. IrelandComputers: AllOperating systems: AllProgramming language used:MATHEMATICA, Fortran 77 for numerical evaluationMemory required to execute with typical data: Sufficient for a typical installation of MATHEMATICA.No. of lines in distributed program, including test data, etc.: 12 013No. of bytes in distributed program, including test data, etc.: 231 899Distribution format: tar.gzLibraries used:CUBA [T. Hahn, Comput. Phys. Commun. 168 (2005) 78] for numerical evaluation of multidimensional integrals and CERNlib [CERN Program Library, obtainable from: http://cernlib.web.cern.ch/cernlib/] for the implementation of Γ and ψ functions in Fortran.Nature of physical problem: Analytic continuation of Mellin-Barnes integrals in a parameter and subsequent numerical evaluation. This is necessary for evaluation of Feynman integrals from Mellin-Barnes representations.Method of solution: Recursive accumulation of residue terms occurring when singularities cross integration contours. Numerical integration of multidimensional integrals with the help of the CUBA library.Restrictions on the complexity of the problem: Limited by the size of the available storage space.Typical running time: Depending on the problem. Usually seconds for moderate dimensionality integrals.     

Computation of a generalized Nordsieck integral

In this work we introduce two analytical representations of a generalized Nordsieck integral. These integrals arise in the calculations of scattering properties of systems of Coulomb-charged particles using the natural base, which includes general solutions of the two-body Coulomb equation. We study the numerical convergence of these representations against the direct Fortran numerical integration. We test the performance of the different strategies as a function of the momentum transfer, which is typically a relevant variable in collision processes. We also discuss the advantages and disadvantages of the different approaches.     

Symmetries and exponential error reduction in Yang-Mills theories on the lattice

The partition function of a quantum field theory with an exact symmetry can be decomposed into a sum of functional integrals each giving the contribution from states with definite symmetry properties. The composition rules of the corresponding transfer matrix elements can be exploited to devise a multi-level Monte Carlo integration scheme for computing correlation functions whose numerical cost, at a fixed precision and at asymptotically large times, increases power-like with the time extent of the lattice. As a result the numerical effort is exponentially reduced with respect to the standard Monte Carlo procedure. We test this strategy in the SU(3) Yang-Mills theory by evaluating the relative contribution to the partition function of the parity odd states.     

A note on integration over finite elements

In the application of the finite element method for structural engineering problems, the evaluation of element stiffness matrices involves the computation of an integral over each element. This note shows that these integrals can be evaluated by a mapping, from the integral over a single element, with the same material properties, in all cases where the sides of the elements are linear in the coordinate system chosen. This brings about a remarkable saving in computation time where numerical integration is used.     

Integration points for triangles and tetrahedrons obtained from the gaussian quadrature points for a line

A new set of numerical integration points for triangles and tetrahedrons, derived from basic Gaussian quadrature points for line integrals, are presented.     

Dispersion relations in the nuclear optical model

We have developed a code to calculate integrals arising from the dispersion relation between the real and the imaginary parts of the nuclear optical model potential (OMP). Both, analytical solution for the most common dispersive OMP, and the general numerical solution, are included. In the numerical integration, fast convergence is achieved by means of the Gauss-Legendre integration method, which offers accuracy, easiness of implementation and generality for dispersive optical model calculations. The numerical method is validated versus analytical solution. The use of this package in the OMP parameter search codes allows for an efficient and accurate dispersive analysis.     

Computation of Fourier transform integrals using Chebyshev series expansions

In this paper, we extend the Clenshaw-Curtis integration method for the computation of Fourier transform integrals. In particular, we examine the numerical stability of a recurrence relation occurring in this method.