An IMT-type quadrature formula with the same asymptotic performance as the DE formula

We propose an IMT-type quadrature formula which achieves the same asymptotic error estimate as the DE formula. The point of the idea is to optimize the parameters of the IMT-type transformation depending on the number of sampling points. We also show the performance of our IMT-type quadrature formula by numerical examples.     

Gaussian interval quadrature formula

Summary. We prove the existence of a Gaussian quadrature formula for Tchebycheff systems, based on integrals over non-overlapping subintervals of arbitrary fixed lengths and the uniqueness of this formula in the case the subintervals have equal lengths. Received July 6, 1999 / Published online August 24, 2000     

Gauss-Radau and Gauss-Lobatto interval quadrature rules for Jacobi weight function

In this paper we prove the existence and uniqueness of the Gauss-Lobatto and Gauss-Radau interval quadrature formulae for the Jacobi weight function. An algorithm for numerical construction is also investigated and some suitable solutions are proposed. For the special case of the Chebyshev weight of the first kind and a special set of lengths we give an analytic solution. The authors were supported in parts by the Swiss National Science Foundation (SCOPES Joint Research Project No. IB7320–111079 ``New Methods for Quadrature') and the Serbian Ministry of Science and Environmental Protection. Serbian Ministry of Science and Environmental Protection.     

Modified optimal quadrature extensions

Summary Optimal extensions of quadrature rules are of importance in the construction of automatic integrators but many sequences fail to exist in usable form. The paper considers some techniques for overcoming the problem of inextensibility with a minimal effect on integrating efficiency. A modification to the extension procedure proposed recently by Begumisa and Robinson is shown to be just a special case of the standard theory for quadrature extension. Some illustrative examples are included.     

Error bound for an osculatory quadrature formula

Sharper estimates have been found for Donaldson's osculatory quadrature formula.     

Weight functions admitting repeated positive Kronrod quadrature

In this note it is shown that for weight functions of the formw(t)=(1 –t ²)^1/2/s _m (t), wheres _m is a polynomial of degreem which is positive on [–1, +1], successive Kronrod extension of a certain class ofN-point interpolation quadrature formulas, including theN-point Gauss-formula, is always possible and that each Kronrod extension has the positivity and interlacing property.     

Efficient quadrature for highly oscillatory integrals involving critical points

This paper based on the Levin collocation method and Levin-type method together with composite two-point Gauss–Legendre quadrature presents efficient quadrature for integral transformations of highly oscillatory functions with critical points. The effectiveness and accuracy of the quadrature are tested.     

Optimal quadrature formula of Markov's and Locher's type with weight function

Summary A quadrature formula of Markov's type with a weight functionx (1–x), which has properties of formulas exact for polynomials of a given degree and properties of optimal formulas on some sets of functions, is given. The particular case of formula (where ==p=q=0) is the formula of Locher [1, 2].     

A discussion of a new error estimate for adaptive quadrature

Previously D. P. Laurie has introduced a new and sharper error estimate for adaptive quadrature routines with the attractive property that the error is guaranteed to be in a small interval if some constraints are satisfied. In this paper we discuss how to test whether or not the constraints are satisfied, and we report a selection of results from our tests with one dimensional integrals to see how the error estimate works in practice. It turns out that we get a more economic routine using this error estimate, but the loss in reliability, even with the new tests, can be catastrophic.This work was supported by the Norwegian Research Council for Sciences and Huminaties.     

Numerical integration over a disc. A new Gaussian quadrature formula

Summary. We construct a quadrature formula for integration on the unit disc which is based on line integrals over distinct chords in the disc and integrates exactly all polynomials in two variables of total degree . Received August 8, 1996 / Revised version received July 2, 1997     

Weighted quadrature rules for finite element methods

We discuss the numerical integration of polynomials times non-polynomial weighting functions in two dimensions arising from multiscale finite element computations. The proposed quadrature rules are significantly more accurate than standard quadratures and are better suited to existing finite element codes than formulas computed by symbolic integration. We validate this approach by introducing the new quadrature formulas into a multiscale finite element method for the two-dimensional reaction–diffusion equation.     

A generalization of general two-point formula with applications in numerical integration

We derive a general two-point integral quadrature formula using the concept of harmonic polynomials. An improved version of Guessab and Schmeisser’s result is given with new integral inequalities involving functions whose derivatives belong to various classes of functions (_Lp$L_p$ spaces, convex, concave, bounded functions). Furthermore, several special cases of polynomials are considered, and the generalization of well-known two-point quadrature formulae, such as trapezoid, perturbed trapezoid, two-point Newton–Cotes formula, two-point Maclaurin formula, midpoint, are obtained.     

Gregory type quadrature based on quadratic nodal spline interpolation

Summary. Using a method based on quadratic nodal spline interpolation, we define a quadrature rule with respect to arbitrary nodes, and which in the case of uniformly spaced nodes corresponds to the Gregory rule of order two, i.e. the Lacroix rule, which is an important example of a trapezoidal rule with endpoint corrections. The resulting weights are explicitly calculated, and Peano kernel techniques are then employed to establish error bounds in which the associated error constants are shown to grow at most linearly with respect to the mesh ratio parameter. Specializing these error estimates to the case of uniform nodes, we deduce non-optimal order error constants for the Lacroix rule, which are significantly smaller than those calculated by cruder methods in previous work, and which are shown here to compare favourably with the corresponding error constants for the Simpson rule. Received July 27, 1998/ Revised version received February 22, 1999 / Published online January 27, 2000     

Vectorized adaptive quadrature in MATLAB

Adaptive quadrature codes process a collection of subintervals one at a time. We show how to process them all simultaneously and so exploit vectorization and the use of fast built-in functions and array operations that are so important to efficient computation in MATLAB. Using algebraic transformations we have made it just as easy for users to solve problems on infinite intervals and with moderate end point singularities as problems with finite intervals and smooth integrands. Piecewise-smooth integrands are handled effectively with breakpoints.     

The error norm of Gauss-Lobatto quadrature formulae for weight functions of Bernstein-Szegö type

Summary In certain spaces of analytic functions the error term of the Gauss-Lobatto quadrature formula relative to a (nonnegative) weight function is a continuous linear functional. Here we compute the norm of the error functional for the Bernstein-Szegö weight functions consisting of any of the four Chebyshev weights divided by an arbitrary quadratic polynomial that remains positive on [–1, 1]. The norm can subsequently be used to derive bounds for the error functional. The efficiency of these bounds is illustrated with some numerical examples.Work supported in part by a grant from the Research Council of the Graduate School, University of Missouri-Columbia.     

Some quadrature formulae with nonstandard weights

In this paper the authors study “truncated” quadrature rules based on the zeros of Generalized Laguerre polynomials. Then, they prove the stability and the convergence of the introduced integration rules. Some numerical tests confirm the theoretical results.     

Optimal stochastic quadrature formulas for convex functions

We study optimal stochastic (or Monte Carlo) quadrature formulas for convex functions. While nonadaptive Monte Carlo methods are not better than deterministic methods, we prove that adaptive Monte Carlo methods are much better.Supported by a Heisenberg scholarship of the DFG.