Suboptimal Kronrod extension formulae for numerical quadrature

Summary We consider cases where the Stieltjes polynomial associated with a Gaussian quadrature formula has complex zeros. In such cases a Kronrod extension of the Gaussian rule does not exist. A method is described for modifying the Stieltjes polynomial so that the resulting polynomial has no complex zeros. The modification is performed in such a way that the Kronrod-type extension rule resulting from the addition of then+1 zeros of the modified Stieltjes polynomial to the original knots of the Gaussian rule has only slightly lower degree of precision than normally achieved when the Kronrod extension rule exists. As examples of the use of the method, we present some new formulae extending the classical Gauss-Hermite quadrature rules. We comment on the limited success of the method in extending Gauss-Laguerre rules and show that several modified extensions of the Gauss Gegenbauer formulae exist in cases where the standard Kronrod extension does not.     

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.     

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.     

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.     

Error estimates for Filon quadrature formulae

In this paper, upper bounds for the error of (generalized) Filon quadrature formulae are stated. Furthermore, the main term of this error is derived, yielding simple modified quadrature rules of higher asymptotical precision.     

Theoretical and practical efficiency measures for symmetric interpolatory quadrature formulas

We study two criteria to evaluate quadrature formulas when used in automatic quadrature programs. The former consists of the computation of a quantity depending on both the truncation error behavior and the geometric properties of the nodes of the rule. This measure allows estimating the asymptotical computational cost in various abstract models of automatic quadrature. The latter is a testing technique which can be used to measure the efficiency of the formulas under consideration in a real environment. The relationships between the two criteria are investigated and the two approaches seem in good agreement.Work supported by CNR, Grant No. 93.00570.CT01.     

An evaluation of Clenshaw–Curtis quadrature rule for integration w.r.t. singular measures

This work is devoted to the study of quadrature rules for integration with respect to (w.r.t.) general probability measures with known moments. Automatic calculation of the Clenshaw–Curtis rules is considered and analyzed. It is shown that it is possible to construct these rules in a stable manner for quadrature w.r.t. balanced measures. In order to make a comparison Gauss rules and their stable implementation for integration w.r.t. balanced measures are recalled. Convergence rates are tested in the case of binomial measures.     

Error estimates for Gaussian quadratures of analytic functions

For analytic functions the remainder term of Gaussian quadrature formula and its Kronrod extension can be represented as a contour integral with a complex kernel. We study these kernels on elliptic contours with foci at the points ±1 and the sum of semi-axes ?>1 for the Chebyshev weight functions of the first, second and third kind, and derive representation of their difference. Using this representation and following Kronrod’s method of obtaining a practical error estimate in numerical integration, we derive new error estimates for Gaussian quadratures.     

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.     

A nodal spline interpolant for the Gregory rule of even order

Summary The Gregory rule is a well-known example in numerical quadrature of a trapezoidal rule with endpoint corrections of a given order. In the literature, the methods of constructing the Gregory rule have, in contrast to Newton-Cotes quadrature,not been based on the integration of an interpolant. In this paper, after first characterizing an even-order Gregory interpolant by means of a generalized Lagrange interpolation operator, we proceed to explicitly construct such an interpolant by employing results from nodal spline interpolation, as established in recent work by the author and C.H. Rohwer. Nonoptimal order error estimates for the Gregory rule of even order are then easily obtained.     

On a conjecture of D. B. Hunter

We consider errors of positive quadrature formulas applied to Chebyshev polynomials. These errors play an important role in the error analysis for many function classes. Hunter conjectured that the supremum of all errors in Gaussian quadrature of Chebyshev polynomials equals the norm of the quadrature formula. We give examples, for which Hunter's conjecture does not hold. However, we prove that the conjecture is valid for all positive quadratures if the supremum is replaced by the limit superior. Considering a fixed positive quadrature formula and the sequence of all Chebyshev polynomials, we show that large errors are rare.     

LANCZOS,GAUSS AND THE PROBLEM OF MOMENTS

Abstract

An algorithm due to Lanczos for solving the problem of weighted moments is discussed including the treatment of double nodes and related to the early work of Gauss on super-interpolatory numerical quadrature.     

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).     

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.     

Computation of the Riesz-Herglotz transform and its application to quadrature formulas over the unit circle

Summary A method is proposed for the computation of the Riesz-Herglotz transform. Numerical experiments show the effectiveness of this method. We study its application to the computation of integrals over the unit circle in the complex plane of analytic functions. This approach leads us to the integration by Taylor polynomials. On the other hand, with the goal of minimizing the quadrature error bound for analytic functions, in the set of quadrature formulas of Hermite interpolatory type, we found that this minimum is attained by the quadrature formula based on the integration of the Taylor polynomial. These two different approaches suggest the effectiveness of this formula. Numerical experiments comparing with other quadrature methods with the same domain of validity, or even greater such as Szeg? formulas, (traditionally considered as the counterpart of the Gauss formulas for integrals on the unit circle) confirm the superiority of the numerical estimations. This work was supported by the ministry of education and culture of Spain under contract PB96-1029.     

On the quadrature error in operational quadrature methods for convolutions

Summary We analyze the quadrature error associated with operational quadrature methods for convolution equations. The assumptions are that the convolution kernel is inL ¹ and that its Laplace transform is analytic and bounded in an obtuse sector of the complex plane. Under these circumstances the Laplace transform has a slow variation property which admits a Fourier analysis of the quadrature error. We provide generalL ^p error estimates assuming suitable smoothness conditions on the function under convolution.     

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.     

Two new formulas for the numerical evaluation of the Hilbert Transform

We develop two algorithms for the numerical evaluation of the semi-infinite Hilbert Transform of functions with a given algebraic behaviour at the origin and at infinity. The first algorithm is connected with a Gauss-Jacobi type quadrature formula for unbounded intervals; the second is based on a rational Bernstein-type operator. Error estimates for different classes of functions are shown. Finally numerical examples are given, comparing the rules among themselves.     

The superconvergence of the Newton-Cotes rule for Cauchy principal value integrals

We consider the general (composite) Newton-Cotes method for the computation of Cauchy principal value integrals and focus on its pointwise superconvergence phenomenon, which means that the rate of convergence of the Newton-Cotes quadrature rule is higher than what is globally possible when the singular point coincides with some a priori known point. The necessary and sufficient conditions satisfied by the superconvergence point are given. Moreover, the superconvergence estimate is obtained and the properties of the superconvergence points are investigated. Finally, some numerical examples are provided to validate the theoretical results.     

Product integration of logarithmic singular integrands based on cubic splines

This paper is concerned with the practical evaluation of the product integral ∫¹_{? 1}f(x)k(x)dx for the case when k(x) = In|x - λ|, λ? (?1, +1) and f is bounded in [?1, +1]. The approximation is a quadrature rule

where the weights {w_n,n,i} are chosen to be exact when f is given by a linear combination of a chosen set of functions {φ_n,j}. In this paper the functions {φ_n,j} are chosen to be cubic B-splines. An error bound for product quadrature rules based on cubic splines is provided. Examples that test the performance of the product quadrature rules for different choices of the function are given. A comparison is made with product quadrature rules based on first kind Chebyshev polynomials.