Higher order nets were introduced by Dick as a generalisation of classical (t,m, s)nets, which are point sets frequently used in quasi-Monte Carlo integration algorithms. Essential tools in finding such point sets of high quality are propagation rules, which make it possible to generate new higher order nets from existing higher order nets and even classical (t,m, s)-nets. Such propagation rule...

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...

This work is concerned with the numerical solution of a nonlinear weakly singular Volterra integral equation. We investigate the application of product integration methods and a detailed analysis of the Trapezoidal method is given. In order to improve the numerical results we consider extrapolation procedures and collocation methods based on graded meshes. Several examples are presented illustr...

In this paper, a numerical method for solving fuzzy Fredholm integral equations of the second kind is introduced. We apply the trapezoidal rule to compute the Riemann integrals.This kind of integral equations convert to a linear system. Then, by solving the linear system , unknowns are determined. Finally, an algorithm is presented to solve the fuzzy integral equation by using the trapezoidal r...

A numerical integration method that has rapid convergence for integrands with known singularities is presented. Based on endpoint corrections to the trapezoidal rule, the quadratures are suited for the discretization of a variety of integral equations encountered in mathematical physics. The quadratures are based on a technique introduced by Rokhlin (Computers Math. Applic. 20, pp. 51-62, 1990)...

Rates of convergence (or divergence) are obtained in the application of Gauss, Lobatto, and Radau integration rules to functions with an algebraic or logarithmic singularity inside, or at an endpoint of, the interval of integration. A typical result is the following: For a generalized Jacobi weight function on [-1,1], the error in applying an «-point rule to f(x) = \x -y\~* isO(n~2 + 2i), if y ...

Last Time: We discussed quadrature rules, which are means of estimating integrals of functions using a weighted sum of function values. Among these, one of the simplest is the trapezoidal rule, which derives its name from the fact that it attempts to approximate the area under the curve using a trapezoid. Although it can be proved the accuracy of this rule increases quadratically with the numbe...

In this study, we propose a new method to enhance the accuracy of Modified Multi-class Classification based on Association Rule (MMCAR) classifier. We introduce a Partial Rule Match Filtering (PRMF) method that allows a minimal match of the items in the rule’s body in order for the rule to be added into a classifier. Experiments on Reuters-21578 data sets are performed in order to evaluate the ...