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.

     

Average quadrature formulas of Gauss type

In the average quadrature formulas the values of a given functionat given points are replaced by its averages over some distinctintervals. If all the intervals are of the same length, thequadrature formulas of interpolatory type and in particular,of Newton-Cotes type were constructed in Omladi (1978). Here,we construct average quadrature formulas of Gauss type for intervalsof the same length. The middle points of the intervals are zerosof polynomials, orthogonal in a technical sense.     

Blending product-type quadrature rules

The idea of blending which was originally used for bivariate approximation is utilized for the numerical integration of the product of two functions. The combination of three product-type quadrature rules results in a rule with a lower error than each of the original rules. Rules of different exactness degrees as well as compounded rules of different step sizes can be taken for such a combination. Two explicit rules are constructed for demonstration; numerical examples confirm the asymptotic rates of convergence of these rules.     

Numerical solution of nonlinear two-dimensional Fredholm integral equations of the second kind using Gauss product quadrature rules

The Gauss product quadrature rules and collocation method are applied to reduce the second-kind nonlinear two-dimensional Fredholm integral equations (FIE) to a nonlinear system of equations. The convergence of the proposed numerical method is proved under certain conditions on the kernel of the integral equation. An iterative method for approximating the solution of the obtained nonlinear system is provided and its convergence is proved. Also, some numerical examples are presented to show the efficiency and accuracy of the proposed method.     

On Gauss quadrature formulas with equal weights

Summary It is well known that the Chebyshev weight function (1–x ²)^–1/2 is the only weight function (up to a linear transformation) for which then point Gauss quadrature formula has equal weights for alln. In this paper we describe all weight functions for which thenm point Gauss quadrature formula has equal weights for alln, wherem is fixed.     

Gauss legendre quadrature formulas over a tetrahedron

In this article we consider the Gauss Legendre Quadrature method for numerical integration over the standard tetrahedron: {(x, y, z)|0 ≤ x, y, z ≤ 1, x + y + z ≤ 1} in the Cartesian three‐dimensional (x, y, z) space. The mathematical transformation from the (x, y, z) space to (ξ, η, ζ) space is described to map the standard tetrahedron in (x, y, z) space to a standard 2‐cube: {(ξ, η, ζ)| ? 1 ≤ ζ, η, ζ ≤ 1} in the (ξ, η, ζ) space. This overcomes the difficulties associated with the derivation of new weight coefficients and sampling points. The effectiveness of the formulas is demonstrated by applying them to the integration of three nonpolynomial, three polynomial functions and to the evaluation of integrals for element stiffness matrices in linear three‐dimensional elasticity. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

On sensitivity of Gauss–Christoffel quadrature

In numerical computations the question how much does a function change under perturbations of its arguments is of central importance. In this work, we investigate sensitivity of Gauss–Christoffel quadrature with respect to small perturbations of the distribution function. In numerical quadrature, a definite integral is approximated by a finite sum of functional values evaluated at given quadrature nodes and multiplied by given weights. Consider a sufficiently smooth integrated function uncorrelated with the perturbation of the distribution function. Then it seems natural that given the same number of function evaluations, the difference between the quadrature approximations is of the same order as the difference between the (original and perturbed) approximated integrals. That is perhaps one of the reasons why, to our knowledge, the sensitivity question has not been formulated and addressed in the literature, though several other sensitivity problems, motivated, in particular, by computation of the quadrature nodes and weights from moments, have been thoroughly studied by many authors. We survey existing particular results and show that even a small perturbation of a distribution function can cause large differences in Gauss–Christoffel quadrature estimates. We then discuss conditions under which the Gauss–Christoffel quadrature is insensitive under perturbation of the distribution function, present illustrative examples, and relate our observations to known conjectures on some sensitivity problems. The work of the first author was supported by the National Science Foundation under Grants CCR-0204084 and CCF-0514213. The work of the other two authors was supported by the Program Information Society under project 1ET400300415 and by the Institutional Research Plan AV0Z100300504. P. Tichy in the years 2003–2006 on leave at the Institute of Mathematics, TU Berlin, Germany.     

一种Gauss型求积公式的收敛性

构造一种有理插值型求积公式(RIQFs),并证明其收敛性.该方法是Gauss求积公式在有理函数空间(Γ)2n中的推广.     

Calculation of Gauss-Kronrod quadrature rules

The Jacobi matrix of the -point Gauss-Kronrod quadrature rule for a given measure is calculated efficiently by a five-term recurrence relation. The algorithm uses only rational operations and is therefore also useful for obtaining the Jacobi-Kronrod matrix analytically. The nodes and weights can then be computed directly by standard software for Gaussian quadrature formulas.

     

Error analysis of quadrature rules

Approaches to the determination of the error in numerical quadrature rules are discussed and compared.     

Onp-generator fully symmetric quadrature rules

Summary We consider fully symmetric quadrature rules for fully symmetricn-dimensional integration regions. When the region is a product region it is well known that product Gaussian rules exist. These obtain an approximation of polynomial degree 4p+1 based on (2p+1) ⁿ function values arranged on a rectangular grid. We term rules using such a grid,p-generator rules. In this paper we determine the necessary conditions on the region of integration forp-generator rules of degree 4p+1 to exist. Regions with this property are termed PropertyQ regions and besides product spaces, this class includes the hypersphere and other related regions.Work performed under the auspices of the U.S. Energy Research and Development Administration     

On quasi degree quadrature rules

Summary In this paper we examine quadrature rules for the integral which are exact for all with +d. We specify three distinct families of solutions which have properties not unlike the standard Gauss and Radau quadrature rules. For each integerd the abscissas of the quadrature rules lie within the closed integration interval and are expressed in terms of the zeros of a polynomialq _d(y). These polynomialsq _d(y), (d=0, 1, ...), which are not orthogonal, satisfy a three term recurrence relation of the type Q_d+1(y)=(y+_d+1)q_d(y)–_d+1yq_d–1(y) and have zeros with the standard interlacing property.This work was supported by the Applied Mathematical Sciences Research Program (KC-04-02) of the Office of Energy Research of the U.S. Department of Energy under Contract W-31-109-Eng-38     

Computation of Gauss-Kronrod quadrature rules

Recently Laurie presented a new algorithm for the computation of -point Gauss-Kronrod quadrature rules with real nodes and positive weights. This algorithm first determines a symmetric tridiagonal matrix of order from certain mixed moments, and then computes a partial spectral factorization. We describe a new algorithm that does not require the entries of the tridiagonal matrix to be determined, and thereby avoids computations that can be sensitive to perturbations. Our algorithm uses the consolidation phase of a divide-and-conquer algorithm for the symmetric tridiagonal eigenproblem. We also discuss how the algorithm can be applied to compute Kronrod extensions of Gauss-Radau and Gauss-Lobatto quadrature rules. Throughout the paper we emphasize how the structure of the algorithm makes efficient implementation on parallel computers possible. Numerical examples illustrate the performance of the algorithm.

     

A historical note on Gauss–Kronrod quadrature

Summary The idea of Gauss–Kronrod quadrature, in a germinal form, is traced back to an 1894 paper of R. Skutsch.     

On the Gauss type quadrature with parametric weight function

Acta Mathematica Hungarica -     

Szegő–Lobatto quadrature rules

Gauss-type quadrature rules with one or two prescribed nodes are well known and are commonly referred to as Gauss–Radau and Gauss–Lobatto quadrature rules, respectively. Efficient algorithms are available for their computation. Szeg? quadrature rules are analogs of Gauss quadrature rules for the integration of periodic functions; they integrate exactly trigonometric polynomials of as high degree as possible. Szeg? quadrature rules have a free parameter, which can be used to prescribe one node. This paper discusses an analog of Gauss–Lobatto rules, i.e., Szeg? quadrature rules with two prescribed nodes. We refer to these rules as Szeg?–Lobatto rules. Their properties as well as numerical methods for their computation are discussed.     

Gauss quadrature approximations to hypergeometric and confluent hypergeometric functions

Integral representations of hypergeometric and confluent hypergeometric functions with real parameters and complex arguments are used to approximate these functions by Gaussian quadrature. An analysis is given of the errors involved and of estimates of the number of Gauss points required to achieve any given accuracy. Numerical examples illustrate the theory.