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.     

The Beesack–Darst–Pollard inequalities and approximations of the Riemann–Stieltjes integral

Utilising the Beesack version of the Darst–Pollard inequality, some error bounds for approximating the Riemann–Stieltjes integral are given. Some applications related to the trapezoid and mid-point quadrature rules are provided.     

The practical Gauss type rules for Hadamard finite-part integrals using Puiseux expansions

A general framework is constructed for efficiently and stably evaluating the Hadamard finite-part integrals by composite quadrature rules. Firstly, the integrands are assumed to have the Puiseux expansions at the endpoints with arbitrary algebraic and logarithmic singularities. Secondly, the Euler-Maclaurin expansion of a general composite quadrature rule is obtained directly by using the asymptotic expansions of the partial sums of the Hurwitz zeta function and the generalized Stieltjes constant, which shows that the standard numerical integration formula is not convergent for computing the Hadamard finite-part integrals. Thirdly, the standard quadrature formula is recast in two steps. In step one, the singular part of the integrand is integrated analytically and in step two, the regular integral of the remaining part is evaluated using the standard composite quadrature rule. In this stage, a threshold is introduced such that the function evaluations in the vicinity of the singularity are intentionally excluded, where the threshold is determined by analyzing the roundoff errors caused by the singular nature of the integrand. Fourthly, two practical algorithms are designed for evaluating the Hadamard finite-part integrals by applying the Gauss-Legendre and Gauss-Kronrod rules to the proposed framework. Practical error indicator and implementation involved in the Gauss-Legendre rule are addressed. Finally, some typical examples are provided to show that the algorithms can be used to effectively evaluate the Hadamard finite-part integrals over finite or infinite intervals.     

Weighted Convergence of Lagrange Interpolation Based on Gauss-Kronrod Nodes

The Gauss-Kronrod quadrature scheme, which is based on the zeros of Legendrepolynomials and Stieltjes polynomials, is a standard rule for automaticnumerical integration in mathematical software libraries. For a long time,very little was known about the underlying Lagrange interpolationprocesses. Recently, the authors proved new bounds and asymptoticproperties for the Stieltjes polynomials and, subsequently, appliedthese results to investigate the associated interpolation processes. Thepurpose of this paper is to survey the quality of these interpolationprocesses, with additional results that extend and complete the existingones. The principal new results in this paper are necessary and sufficientconditions for weighted convergence. In particular, we show that theLagrange interpolation polynomials associated with the above interpolationprocesses have the same speed of convergence as the polynomials of bestapproximation in certain weighted Besov spaces.     

On Birkhoff-Young quadrature of analytical functions

A simple method is given for constructing quadrature rules for the numerical integration of an analytic function over a line segment in the complex plane. The Birkhoff-Young 5-point, degree 5 rule is obtained as a special case. An error analysis is used to show how rules preferable to the Birkhoff-Young rule are easily developed.     

Product-integration rules and their convergence

An algorithm, based on the use of orthogonal polynomials, for product-integration is outlined. A general discussion on the convergence of such quadrature rules for finite intervals is then given. The paper concludes with five examples for each of which sufficient conditions for convergence of the quadrature rule are given.     

Structure of Stieltjes classes of moment-equivalent probability laws

A Stieltjes class is a one-parameter family of moment-equivalent distribution functions constructed by modulation of a given indeterminate distribution function F, called the center of the class. Members of a Stieltjes class are mutually absolutely continuous, and conversely, any pair of moment-equivalent and mutually absolutely continuous distribution functions can be joined by a Stieltjes class. The center of a Stieltjes class is an equally weighted mixture of its extreme members, and this places restrictions on which distributions can belong to a Stieltjes class with a given center. The lognormal law provides interesting illustrations of the general ideas. In particular, it is possible for two moment equivalent infinitely divisible distributions to be joined by a Stieltjes class, and random scaling can be used to construct new Stieltjes classes from a given Stieltjes class.     

Orthogonal Laurent polynomials corresponding to certain strong Stieltjes distributions with applications to numerical quadratures

In this paper we shall be mainly concerned with sequences of orthogonal Laurent polynomials associated with a class of strong Stieltjes distributions introduced by A.S. Ranga. Algebraic properties of certain quadratures formulae exactly integrating Laurent polynomials along with an application to estimate weighted integrals on with nearby singularities are given. Finally, numerical examples involving interpolatory rules whose nodes are zeros of orthogonal Laurent polynomials are also presented.

     

Quadrature rules based on partial fraction expansions

Quadrature rules are typically derived by requiring that all polynomials of a certain degree be integrated exactly. The nonstandard issue discussed here is the requirement that, in addition to the polynomials, the rule also integrates a set of prescribed rational functions exactly. Recurrence formulas for computing such quadrature rules are derived. In addition, Fejér's first rule, which is based on polynomial interpolation at Chebyshev nodes, is extended to integrate also rational functions with pre-assigned poles exactly. Numerical results, showing a favorable comparison with similar rules that have been proposed in the literature, are presented. An error analysis of a representative test problem is given. This revised version was published online in June 2006 with corrections to the Cover Date.     

An application of Szegő polynomials to the computation of certain weighted integrals on the real line

In this paper, a new approach in the estimation of weighted integrals of periodic functions on unbounded intervals of the real line is presented by considering an associated weight function on the unit circle and making use of both Szegő and interpolatory type quadrature formulas. Upper bounds for the estimation of the error are considered along with some examples and applications related to the Rogers-Szegő polynomials, the evaluation of the Weierstrass operator, the Poisson kernel and certain strong Stieltjes weight functions. Several numerical experiments are finally carried out.     

Properties of product-type quadrature formulas

In this note some properties of the coefficient matrix associated with a product-interpolatory quadrature formula are determined and certain consequences of exactness of a product-type quadrature rule are deduced. For example, it is shown that the coefficient matrix has maximal rank and it is positive definite when the rule is symmetric. Conditions are stated under which a product-interpolatory rule reduces to a regular quadrature rule. A characterization of the error committed in applying a regular rule to a product of two functions is given.A portion of this research was carried out while the author was a Summer Faculty Research Participant at the Oak Ridge National Laboratory.     

On extrapolation of Jagerman and Stetter rules

In this paper quadrature rules introduced by Jagerman [1] and Stetter [2] are considered and asymptotic expansions for the error given. This allows to make use of the Romberg extrapolation process. Such rules can be viewed as generalizations of the well-known mid-point rule. Thus, numerical examples comparing these rules are finally presented.     

Higher-order Newton–Cotes rules with end corrections

We present higher-order quadrature rules with end corrections for general Newton–Cotes quadrature rules. The construction is based on the Euler–Maclaurin formula for the trapezoidal rule. We present examples with 6 well-known Newton–Cotes quadrature rules. We analyze modified end corrected quadrature rules, which consist on a simple modification of the Newton–Cotes quadratures with end corrections. Numerical tests and stability estimates show the superiority of the corrected rules based on the trapezoidal and the midpoint rules.     

Modified quadrature formulas for functions with nearby poles

This paper is concerned with the numerical integration of functions with poles near the interval of integration. A method is given for modifying known quadrature rules, to obtain rules which are exact for certain classes of rational functions.     

Anti-Szego quadrature rules

Szego quadrature rules are discretization methods for approximating integrals of the form . This paper presents a new class of discretization methods, which we refer to as anti-Szego quadrature rules. Anti-Szego rules can be used to estimate the error in Szego quadrature rules: under suitable conditions, pairs of associated Szego and anti-Szego quadrature rules provide upper and lower bounds for the value of the given integral. The construction of anti-Szego quadrature rules is almost identical to that of Szego quadrature rules in that pairs of associated Szego and anti-Szego rules differ only in the choice of a parameter of unit modulus. Several examples of Szego and anti-Szego quadrature rule pairs are presented.

     

Convergence of Gaussian quadrature formulas on infinite intervals

More general and stronger estimations of bounds for the fundamental functions of Hermite interpolation of high order on an arbitrary system of nodes on infinite intervals are given. Based on this result, convergence of Gaussian quadrature formulas for Riemann–Stieltjes integrable functions on an arbitrary system of nodes on infinite intervals is discussed.     

Error estimates for some composite corrected quadrature rules

The asymptotic behaviour of the error for a general quadrature rule is established and it is applied to some composite corrected quadrature rules.     

An Introduction to Lattice Rules and their Generator Matrices

For the one-dimensional quadrature of a naturally periodic functionover its period, the trapezoidal rule is an excellent choice,its efficiency being predicted theoretically and confirmed inpractice. However, for s-dimensional quadrature over a hypercube,the s-dimensional product trapezoidal rule is not generallycost effective even for naturally periodic functions. The searchfor more effective rules has led first to number theoretic rulesand then more recently to lattice rules. This survey outlinesthe motivation for and present results of this theory. It isparticularly designed to introduce the reader to lattice rules.     

Error estimate and extrapolation of a quadrature formula derived from a quartic spline quasi-interpolant

In this paper we analyze a quadrature rule based on integrating a C ³ quartic spline quasi-interpolant on a bounded interval which has been introduced in Sablonnière (Rend. Semin. Mat. Univ. Pol. Torino 63(3):107–118, 2005). By studying the sign structure of its associated Peano kernel we derive an explicit formula of the quadrature error with an approximation order O(h ⁶). A comparison of this rule with the composite Boole’s and the three-point Gauss-Legendre rules is given. We also compare the Nyström methods associated with the above quadrature formulae for solving the linear Fredholm integral equation of the second kind. Then, by combining the proposed rule with composite Boole’s rule, we construct a new quadrature rule of order O(h ⁷). All the obtained results are illustrated by several numerical tests.     

Mixtures of power series distributions: identifiability via uniqueness in problems of moments

We treat the identifiability problem for mixtures involving power series distributions. Applying an idea of Sapatinas (Ann Inst Stat Math 47:447–459, 1995) we prove and elaborate that a mixture distribution is identifiable if a certain Stieltjes problem of moments has a unique solution while a non-uniqueness leads to a non- identifiable mixture. We describe explicitly models of identifiable mixtures and models of non-identifiable mixtures. Illustrative examples and comments on related questions are also given.