This article focuses on the process of approximating a definite integral from values of the integrand when exact mathematical integration is not available. This problem arises in statistics when marginal density functions or expected values of random variables are required. The article describes classical univariate quadrature methods including the trapezoidal rule, Simpson’s rule, Newton-Cotes...

Department of Mathematics, University of Gaziantep, Gaziantep, Turkey e-mail address : [email protected] Abstract For the practical estimation of the error of Gauss quadrature rules Gauss-Kronrod rules are widely used; but, it is well known that for the generalized Hermite weight function, ωα(x) = |x|2α exp(−x2) over [−∞,∞], real positive Gauss-Kronrod rules do not exist. Among the alternati...

We present a fast and accurate algorithm for evaluating singular integral operators on smooth surfaces that are globally parametrized by spherical coordinates. Problems of this type arise, for example, in simulating Stokes flows with particulate suspensions and in multi-particle scattering calculations. For smooth surfaces, spherical harmonic expansions are commonly used for geometry representa...

Fredholm integral equations arise naturally in the context of ordinary and partial differential equations: Two-point boundary value problems can be reformulated as Fredholm integral equations, whose kernels are continuous but have finite jump discontinuities in their derivatives. Two-dimensional elliptic boundary problems can be reformulated as Fredholm integral equations with kernels that have...

The idea of Gauss–Kronrod quadrature, in a germinal form, is traced back to an 1894 paper of R. Skutsch. The idea of inserting n+1 nodes into an n-point Gaussian quadrature rule and choosing them and the weights of the resulting (2n+1)-point quadrature rule in such a manner as to maximize the polynomial degree of exactness is generally attributed to A.S. Kronrod [2], [3]. This is entirely justi...

Several algorithms are given and compared for computing Gauss quadrature rules. It is shown that given the three term recurrence relation for the orthogonal polynomials generated by the weight function, the quadrature rule may be generated by computing the eigenvalues and first component of the orthornormalized eigenvectors of a symmetric tridiagonal matrix. An algorithm is also presented for c...

The error in the trapezoidal rule quadrature formula can be attributed to discretization in the interior and non-periodicity at the boundary. Using a contour integral, we derive a unified bound for the combined error from both sources for analytic integrands. The bound gives the Euler–Maclaurin formula in one limit and the geometric convergence of the trapezoidal rule for periodic analytic func...

We consider the h-p Galerkin boundary element method applied to a weakly singular integral equation of the rst kind on an open curved surface. We focus on the eeective computation of the Galerkin matrix, i.e. we introduce a numerical qudrature rule which is shown (for a model case) to converge exponentially fast with the number of kernel evaluations. Numerical experiments show that the applicat...