Gaussian quadrature rules with simple node-weight relations

In this paper, we consider the symmetric Gaussian and L-Gaussian quadrature rules associated with twin periodic recurrence relations with possible variations in the initial coefficient. We show that the weights of the associated Gaussian quadrature rules can be given as rational functions in terms of the corresponding nodes where the numerators and denominators are polynomials of degree at most 4. We also show that the weights of the associated L-Gaussian quadrature rules can be given as rational functions in terms of the corresponding nodes where the numerators and denominators are polynomials of degree at most 5. Special cases of these quadrature rules are given. Finally, an easy to implement procedure for the evaluation of the nodes is described.

Approximations for the generalized temperature integral: a method based on quadrature rules

The generalized temperature integral I(m, x) appears in non-isothermal kinetic analysis when the frequency factor depends on the temperature. A procedure based on Gaussian quadrature to obtain analytical approximations for the integral I(m, x) was proposed. The results showed good agreement between the obtained approximation values and those obtained by numerical integration. Unless other approximations found in literature, the methodology presented in this paper can be easily generalized in order to obtain approximations with the maximum of accurate.

Associated symmetric quadrature rules

We prove a relation between two different types of symmetric quadrature rules, where one of the types is the classical symmetric interpolatory quadrature rules. Some applications of a new quadrature rule which was obtained through this relation are also considered.

Inversely symmetric interpolatory quadrature rules

We consider interpolatory quadrature rules with nodes and weights satisfying symmetric properties in terms of the division operator. Information concerning these quadrature rules is obtained using a transformation that exists between these rules and classical symmetric interpolatory quadrature rules. In particular, we study those interpolatory quadrature rules with two fixed nodes. We obtain specific examples of such quadrature rules.

Szegö polynomials: quadrature rules on the unit circle and on [-1, 1]

We consider some of the relations that exist between real Szegö polynomials and certain para-orthogonal polynomials defined on the unit circle, which are again related to certain orthogonal polynomials on [-1, 1] through the transformation x = (z1/2+z1/2)/2. Using these relations we study the interpolatory quadrature rule based on the zeros of polynomials which are linear combinations of the orthogonal polynomials on [-1, 1]. In the case of any symmetric quadrature rule on [-1, 1], its associated quadrature rule on the unit circle is also given.

A CHEBYSHEV-TYPE QUADRATURE RULE WITH SOME INTERESTING PROPERTIES

We give here an n-point Chebyshev-type rule of algebraic degree of precision n - 1, but having nodes that can be given explicitly. This quadrature rule also turns out to be one with an ''almost'' highest algebraic degree of precision.

ANOTHER QUADRATURE RULE OF HIGHEST ALGEBRAIC DEGREE OF PRECISION

We consider certain quadrature rules of highest algebraic degree of precision that involve strong Stieltjes distributions (i.e., strong distributions on the positive real axis). The behavior of the parameters of these quadrature rules, when the distributions are strong c-inversive Stieltjes distributions, is given. A quadrature rule whose parameters have explicit expressions for their determination is presented. An application of this quadrature rule for the evaluation of a certain type of integrals is also given.

A class of hypergeometric polynomials with zeros on the unit circle: Extremal and orthogonal properties and quadrature formulas

The class of hypergeometric polynomials F12(-m,b;b+b̄;1-z) with respect to the parameter b=λ+iη, where λ>0, are known to have all their zeros simple and exactly on the unit circle |z|=1. In this note we look at some of the associated extremal and orthogonal properties on the unit circle and on the interval (-1,1). We also give the associated Gaussian type quadrature formulas. © 2012 IMACS.

Calculation of line integrals in boundary integral methods

We propose quadrature rules for the approximation of line integrals possessing logarithmic singularities and show their convergence. In some instances a superconvergence rate is demonstrated.

Numerical-asymptotic boundary integral methods in high-frequency acoustic scattering

In this article we describe recent progress on the design, analysis and implementation of hybrid numerical-asymptotic boundary integral methods for boundary value problems for the Helmholtz equation that model time harmonic acoustic wave scattering in domains exterior to impenetrable obstacles. These hybrid methods combine conventional piecewise polynomial approximations with high-frequency asymptotics to build basis functions suitable for representing the oscillatory solutions. They have the potential to solve scattering problems accurately in a computation time that is (almost) independent of frequency and this has been realized for many model problems. The design and analysis of this class of methods requires new results on the analysis and numerical analysis of highly oscillatory boundary integral operators and on the high-frequency asymptotics of scattering problems. The implementation requires the development of appropriate quadrature rules for highly oscillatory integrals. This article contains a historical account of the development of this currently very active field, a detailed account of recent progress and, in addition, a number of original research results on the design, analysis and implementation of these methods.

SOME CONSEQUENCES OF A SYMMETRY IN STRONG DISTRIBUTIONS

Some polynomials and interpolatory quadrature rules associated with strong Stieltjes distributions are considered, especially when the distributions satisfy a Certain symmetric property. (C) 1995 Academic Press, Inc.

Integration of polyharmonic functions

The results in this paper are motivated by two analogies. First, m-harmonic functions in R(n) are extensions of the univariate algebraic polynomials of odd degree 2m-1. Second, Gauss' and Pizzetti's mean value formulae are natural multivariate analogues of the rectangular and Taylor's quadrature formulae, respectively. This point of view suggests that some theorems concerning quadrature rules could be generalized to results about integration of polyharmonic functions. This is done for the Tchakaloff-Obrechkoff quadrature formula and for the Gaussian quadrature with two nodes.

SZEGO and PARA-ORTHOGONAL POLYNOMIALS on THE REAL LINE: ZEROS and CANONICAL SPECTRAL TRANSFORMATIONS

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Kernel polynomials from L-orthogonal polynomials

A positive measure ψ defined on [a,b] such that its moments μn=∫a btndψ(t) exist for n=0,±1,±2,⋯, is called a strong positive measure on [a,b]. If 0≤aquadrature rules associated with these kernel polynomials. Associated eigenvalue problems and numerical evaluation of the nodes and weights of such quadrature rules are also considered. © 2010 IMACS. Published by Elsevier B.V. All rights reserved.