Quadrature rules with multiple nodes for evaluating integrals with strong singularities

Quadrature rules with multiple nodes

In this paper a brief historical survey of the development of quadrature rules with multiple nodes and the maximal algebraic degree of exactness is given. The natural generalization of such rules are quadrature rules with multiple nodes and the maximal degree of exactness in some functional spaces that are different from the space of algebraic polynomial. For that purpose we present a generaliz...

Weighted quadrature rules with binomial nodes

In this paper, a new class of a weighted quadrature rule is represented as --------------------------------------------  where  is a weight function,  are interpolation nodes,  are the corresponding weight coefficients and denotes the error term. The general form of interpolation nodes are considered as   that  and we obtain the explicit expressions of the coefficients  using the q-...

Quadrature Rules for Fuzzy Integrals

Asymptotic expansions of Gauss-Legendre quadrature rules for integrals with endpoint singularities

Let I[f ] = ∫ 1 −1 f(x) dx, where f ∈ C ∞(−1, 1), and let Gn[f ] = ∑n i=1 wnif(xni) be the n-point Gauss–Legendre quadrature approximation to I[f ]. In this paper, we derive an asymptotic expansion as n → ∞ for the error En[f ] = I[f ]−Gn[f ] when f(x) has general algebraic-logarithmic singularities at one or both endpoints. We assume that f(x) has asymptotic expansions of the forms f(x) ∼ ∞ ∑ ...

Numerical Quadrature and Nonlinear Sequence Transformations; Unified Rules for Efficient Computation of Integrals With Algebraic and Logarithmic Endpoint Singularities

Some nonlinear transformations for accelerating the convergence of infinite sequences due to Levin are reviewed, and new results of practical importance in applications are given. Using these results, the transformations of Levin are modified and used to obtain new numerical integration formulas for weight functions with algebraic and logarithmic endpoint singularities, which are simpler to com...