Computation of oscillating integrals

An adaptive quadrature method for the automatic computation of integrals with strongly oscillating integrand is presented. The integration method is based on a truncated Chebyshev series approximation. The algorithm uses a global subinterval division strategy. There is a protection against the influence of round-off errors. A Fortran implementation of the algorithm is given.     

Automatic integration over a sphere

We present an algorithm for automatic integration over an N-dimensional sphere. The quadrature formula is obtained by using a trapezoidal rule after a non-linear transformation, and allows to deal with integrand singularities on the surface or in the centre of the sphere.At the basis of the theoretical development lie the construction and the selection of suitable transformations.The algorithm is cast into an automatic integration program coded as a Standard Fortran sub-routine.     

Derivative corrections for quadrature formulas

In this paper, we develop corrected quadrature formulas by approximating the derivatives of the integrand that appear in the asymptotic error expansion of the quadrature, using only the function values in the original quadrature rule. A higher order convergence is achieved without computing additional function values of the integrand.This author is in part supported by National Science Foundation under grant DMS-9504780 and by NASA-OAI Summer Faculty Fellowship (1995).     

Numerical solution of Volterra integral equations

Summary The numerical method discussed in this paper is based on quadrature formulae. With some assumptions on the coefficients of the quadrature formula and on the integrand, convergence properties of the method for both linear and non-linear equations are established.This article is a part of the author's D. Sc. Thesis.     

A High-Order Piecewise Quartic Spline Rule for Hadamard Integral and Its Application of the Cavity Scattering

We develop a fourth-order piecewise quartic spline rule for Hadamard integral. The quadrature formula of Hadamard integral is obtained by replacing the integrand function with the piecewise quartic spline interpolation function. We establish corresponding error estimates and analyze the numerical stability. The rule can achieve fourth-order convergence at any point in the interval, even when the singular point coincides with the grid point. Since the derivative information of the integrand is not required, the rule can be easily applied to solve many practical problems. Finally, the quadrature formula is applied to solve the electromagnetic scattering from cavities with different wave numbers, which improves the whole accuracy of the solution. Numerical experiments are presented to show the efficiency and accuracy of the theoretical analysis.     

Multilevel quasi-interpolation on a sparse grid with the Gaussian

Motivated by the recent multilevel sparse kernel-based interpolation (MuSIK) algorithm proposed in Georgoulis et al. (SIAM J. Sci. Comput. 35, 815–832, 2013), we introduce the new quasi-multilevel sparse interpolation with kernels (Q-MuSIK) via the combination technique. The Q-MuSIK scheme achieves better convergence and run time when compared with classical quasi-interpolation. Also, the Q-MuSIK algorithm is generally superior to the MuSIK methods in terms of run time in particular in high-dimensional interpolation problems, since there is no need to solve large algebraic systems. We subsequently propose a fast, low complexity, high-dimensional positive-weight quadrature formula based on Q-MuSIKSapproximation of the integrand. We present the results of numerical experimentation for both quasi-interpolation and quadrature in high dimensions.     

Quadpack computation of Feynman loop integrals

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.     

Error bounds of a quadrature formula with multiple nodes for the Fourier-Chebyshev coefficients for analytic functions

Three kinds of effective error bounds of the quadrature formulas with multiple nodes that are generalizations of the well-known Micchelli-Rivlin quadrature formula, when the integrand is a function analytic in the regions bounded by confocal ellipses, are given. A numerical example which illustrates the calculation of these error bounds is included.     

Integration formulas of tenth order and higher

Nowadays, due to the considerable growth of computer capacities, the development of more efficient quadrature formulas may seem unnecessary. However, if the calculation of each integrand value requires much computational time or we have to study the dependence of the integral on a large number of parameters the integrand is determined through, then it is necessary to use more efficient formulas.     

一类广义Gauss型求积公式

基于被积函数在n次第一类和第二类Chebyshev多项式的零点处的差商,该本构造了两种Gauss型求积公式. 这些求积公式包含了某些已知结果作为特例.更重要的是这些新结果与Gauss-Turan求积公式有密切的联系.     

基于切比雪夫节点的一类带显式系数的广义高斯求积公式(英文)

The goal here is to give a simple approach to a quadrature formula based on the divided diffierences of the integrand at the zeros of the nth Chebyshev polynomial of the first kind,and those of the（n-1）st Chebyshev polynomial of the second kind.Explicit expressions for the corresponding coefficients of the quadrature rule are also found after expansions of the divided diffierences,which was proposed in[14].     

An analogue of the four-point Newton-Cotes formula for a function with a boundary-layer component

The construction of the Newton-Cotes formulas is based on approximating the integrand by a Lagrange polynomial. The error of such quadrature formulas can be great for a function with a boundary-layer component. In this paper, an analog of the four-point Newton-Cotes rule is constructed. The construction is based on using a nonpolynomial interpolation that is exact for the boundary layer component. Error estimates of the quadrature rule independent of the boundary layer component gradients are obtained. Numerical experiments are performed.     

Computation of the attractive force of an ellipsoid

A numerical method for computing the attractive force of an ellipsoid is proposed that does not involve separating subdomains with singularities. The sought function is represented as a triple integral such as the inner integral of the kernel can be evaluated analytically with the kernel treated as a weight function. The inner integral is approximated by a quadrature for the product of functions, of which one has an integrable singularity. As a result, the integrand obtained before the second integration has only a weak logarithmic singularity. The subsequent change of variables yields an integrand without singularities. Based on this approach, at each stage of integral evaluation with respect to a single variable, quadrature formulas are derived that do not have singularities at integration nodes and do not take large values at these nodes. For numerical experiments, a rather complicated test function is constructed that is the exact attractive force of an ellipsoid of revolution with an elliptic density distribution.     

Bounds on the error of fejer and clenshaw-curtis type quadrature for analytic functions

We consider the problem of integrating a function f : [-1,1] → R which has an analytic extension to an open disk D^r of radius r and center the origin, such that for any . The goal of this paper is to study the minimal error among all algorithms which evaluate the integrand at the zeros of the n-degree Chebyshev polynomials of first or second kind (Fejer type quadrature formulas) or at the zeros of (n-2)-degree Chebyshev polynomials jointed with the endpoints -1,1 (Clenshaw-Curtis type quadrature formulas), and to compare this error to the minimal error among all algorithms which evaluate the integrands at n points. In the case r > 1, it is easy to prove that Fejer and Clenshaw-Curtis type quadrature are almost optimal. In the case r = 1, we show that Fejer type formulas are not optimal since the error of any algorithm of this type is at least about n^-2. These results hold for both the worst-case and the asymptotic settings.     

Numerical approximation of highly oscillatory integrals on semi-finite intervals by steepest descent method

A numerical steepest descent method, based on the Laguerre quadrature rule, is developed for integration of one-dimensional highly oscillatory functions on [0,?∞?) of a general class. It is shown that if the integrand is analytic, then in the absence of stationary points, the method is rapidly convergent. The method is extended to the case when there are a finite number of stationary points in [0,?∞?). It can be further extended to the case when the integrand is only smooth to some degree (not necessarily analytic). We illustrate the theoretical results using some numerical experiences.     

Error estimates for Gaussian quadrature formulae

Summary We derive both strict and asymtotic error bounds for the Gauss-Jacobi quadrature formula with respect to a general measure. The estimates involve the maximum modulus of the integrand on a contour in the complex plane. The methods are elementary complex analysis.     

A Doubly Adaptive Integration Algorithm Using Stratified Rules

A doubly adaptive integration algorithm chooses between a higher order rule applied on the current subinterval or the subdivision of the interval. We describe one such algorithm using a stratified sequence of integration rules. We present a criterion to select the suitable strategy, depending on the type of integrand, using available information.     

Comparison of numerical integration formulas

In this paper we study the mean square error of numerical integration, when the integrand is a random stationary process. We obtain exact asymptotic errors of classical quadrature formulas and give lower and upper bounds for the least mean square error.     

Quadrature formulae on half-infinite intervals

We develop two classes of quadrature rules for integrals extended over the positive real axis, assuming given algebraic behavior of the integrand at the origin and at infinity. Both rules are expressible in terms of Gauss-Jacobi quadratures. Numerical examples are given comparing these rules among themselves and with recently developed quadrature formulae based on Bernstein-type operators.Work supported, in part, by the National Science Foundation under grant CCR-8704404.