On the monotonicity of sequences of quadrature formulae

Summary LetI(f)L(f)= _k=0 ^r ₌₀ ^vk–1 a _k f ⁽⁾(X _k ) be a quadrature formula, and let {S _n (f)} _n=1 be successive approximations of the definite integralI(f)= ₀ ¹ f(x)dx obtained by the composition ofL, i.e.,S _n(f)=L( _n ), where .We prove sufficient conditions for monotonicity of the sequence {S _n (f)} _n=1 . As particular cases the monotonicity of well-known Newton-Cotes and Gauss quadratures is shown. Finally, a recovery theorem based on the monotonicity results is presented     

A modified Gaussian quadrature rule for integrals involving poles of any order

By applying the theory of completely symmetric functions we derive a Gaussian quadrature rule which generalizes that due to McNamee. A feature of this generalization is the inclusion of an explicit correction term taking account of the presence of poles (of any order) of the integrand close to the integration-interval. A numerical example is provided to illustrate the formulae.     

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     

Error bounds of certain Gaussian quadrature formulae

We study the kernel of the remainder term of Gauss quadrature rules for analytic functions with respect to one class of Bernstein-Szegö weight functions. The location on the elliptic contours where the modulus of the kernel attains its maximum value is investigated. This leads to effective error bounds of the corresponding Gauss quadratures.     

Local error estimates in quadrature

A theoretical error estimate for quadrature formulas, which depends on four approximations of the integral, is derived. We obtain a bound, often sharper than the trivial one, which requires milder conditions to be satisfied than a similar result previously presented by Laurie. A selection of numerical tests with one-dimensional integrals is reported, to show how the error estimate works in practice.     

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.     

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).     

On chebyshev quadrature and variance of quadrature formulas

The purpose of this note is to give an example which demonstrates that one can achieve much higher algebraic precision with a quadrature rule with small but not minimal variance than with a Chebyshev rule with minimal variance.     

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     

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.     

On the uniform convergence of Gaussian quadrature rules for Cauchy principal value integrals

Summary In a previous paper the authors proposed a modified Gaussian rule _* ^m (wf;t)to compute the integral (wf;t) in the Cauchy principal value sense associated with the weightw, and they proved the convergence in closed sets contained in the integration interval. The main purpose of the present work is to prove uniform convergence of the sequence { _* ^m (wf;t)} on the whole integration interval and to give estimates for the remainder term. The same results are shown for particular subsequences of the Gaussian rules _m (wf;t) for the evaluation of Cauchy principal value integrals. A result on the uniform convergence of the product rules is also discussed and an application to the numerical solution of singular integral equations is made.     

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.     

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.     

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.     

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.     

Best quadrature formula on Sobolev class with Chebyshev weight

Using best interpolation function based on a given function information, we present a best quadrature rule of function on Sobolev class KW^r[-1,1]${\mathit{KW}}^r[-1,1]$ with Chebyshev weight. The given function information means that the values of a function f∈KW^r[-1,1]$f\in {\mathit{KW}}^r[-1,1]$ and its derivatives up to r-1$r-1$ order at a set of nodes x$\mathbf{x}$ are given. Error bounds are obtained, and the method is illustrated by some examples.     

Complex Gaussian quadrature of oscillatory integrals

We construct and analyze Gauss-type quadrature rules with complex- valued nodes and weights to approximate oscillatory integrals with stationary points of high order. The method is based on substituting the original interval of integration by a set of contours in the complex plane, corresponding to the paths of steepest descent. Each of these line integrals shows an exponentially decaying behaviour, suitable for the application of Gaussian rules with non-standard weight functions. The results differ from those in previous research in the sense that the constructed rules are asymptotically optimal, i.e., among all known methods for oscillatory integrals they deliver the highest possible asymptotic order of convergence, relative to the required number of evaluations of the integrand.     

A modified sinc quadrature rule for functions with poles near the arc of integration

Sinc function approach is used to obtain a quadrature rule for estimating integrals of functions with poles near the are of integration. Special treatment is given to integration over the intervals (–, ), (0, ), and (–1, 1). It is shown that the error of the quadrature rule converges to zero at the rateO(exp(–cN)) asN , whereN is the number of nodes used, and wherec is a positive constant which is independent ofN.