A note on generalized averaged Gaussian formulas

We have recently proposed a very simple numerical method for constructing the averaged Gaussian quadrature formulas. These formulas exist in many more cases than the real positive Gauss–Kronrod formulas. In this note we try to answer whether the averaged Gaussian formulas are an adequate alternative to the corresponding Gauss–Kronrod quadrature formulas, to estimate the remainder term of a Gaussian rule.     

Rational approximation to formal power series

A general method for obtaining rational approximations to formal power series is defined and studied. This method is based on approximate quadrature formulas. Newton-Cotes and Gauss quadrature methods are used. It is shown that Padé approximants and the ε-algorithm are related to Gaussian formulas while linear summation processes are related to Newton-Cotes formulas. An example is exhibited which shows that Padé approximation is not always optimal. An application to e^−t is studied and a method for Laplace transform inversion is proposed.     

Quadrature and orthogonal rational functions

Classical interpolatory or Gaussian quadrature formulas are exact on sets of polynomials. The Szegő quadrature formulas are the analogs for quadrature on the complex unit circle. Here the formulas are exact on sets of Laurent polynomials. In this paper we consider generalizations of these ideas, where the (Laurent) polynomials are replaced by rational functions that have prescribed poles. These quadrature formulas are closely related to certain multipoint rational approximants of Cauchy or Riesz–Herglotz transforms of a (positive or general complex) measure. We consider the construction and properties of these approximants and the corresponding quadrature formulas as well as the convergence and rate of convergence.     

Numerical Quadratures for Hadamard Hypersingular Integrals

In this paper,we develop Gaussian quadrature formulas for the Hadamard fi- nite part integrals.In our formulas,the classical orthogonal polynomials such as Legendre and Chebyshev polynomials are used to approximate the density function f(x)so that the Gaussian quadrature formulas have degree n-1.The error estimates of the formulas are obtained.It is found from the numerical examples that the convergence rate and the accu- racy of the approximation results are satisfactory.Moreover,the rate and the accuracy can be improved by choosing appropriate weight functions.     

一类广义Gauss型求积公式

基于被积函数在n次第一类和第二类Chebyshev多项式的零点处的差商,该本构造了两种Gauss型求积公式. 这些求积公式包含了某些已知结果作为特例.更重要的是这些新结果与Gauss-Turan求积公式有密切的联系.     

New quadrature formulas for the numerical inversion of the Laplace transform

New quadrature formulas for the evaluation of the Bromwich integral, arising in the inversion of the Laplace transform are discussed. They are obtained by optimal addition of abscissas to Gaussian quadrature formulas. A table of abscissas and weights is given.     

Gaussian Versus Optimal Integration of Analytic Functions

We consider error estimates for optimal and Gaussian quadrature formulas if the integrand is analytic and bounded in a certain complex region. First, a simple technique for the derivation of lower bounds for the optimal error constants is presented. This method is applied to Szeg?-type weight functions and ellipses as regions of analyticity. In this situation, the error constants for the Gaussian formulas are close to the obtained lower bounds, which proves the quality of the Gaussian formulas and also of the lower bounds. In the sequel, different regions of analyticity are investigated. It turns out that almost exclusively for ellipses, the Gaussian formulas are near-optimal. For classes of simply connected regions of analyticity, which are additionally symmetric to the real axis, the asymptotic of the worst ratio between the error constants of the Gaussian formulas and the optimal error constants is calculated. As a by-product, we prove explicit lower bounds for the Christoffel-function for the constant weight function and arguments outside the interval of integration. September 7, 1995. Date revised: October 25, 1996.     

Two methods for direct numerical integration of the Prandtl equation and comparative analysis between them

Two methods based on quadrature formulas are proposed for the direct numerical integration of Prandtl’s singular integrodifferential equation. In the first method, Prandtl’s equation is solved directly by applying the method of mechanical quadrature and the circulation along an airfoil section is determined. In the second method, Prandtl’s equation is rewritten for the circulation derivative, which is determined by applying mechanical quadratures, and the circulation is then reconstructed using the same quadrature formulas. Both methods are analyzed numerically and are shown to converge. Their convergence rates are nearly identical, while the second method requires much more CPU time than the first one.     

Quadrature formulas for Fourier coefficients

We consider quadrature formulas of high degree of precision for the computation of the Fourier coefficients in expansions of functions with respect to a system of orthogonal polynomials. In particular, we show the uniqueness of a multiple node formula for the Fourier-Tchebycheff coefficients given by Micchelli and Sharma and construct new Gaussian formulas for the Fourier coefficients of a function, based on the values of the function and its derivatives.     

AN EXTREMAL APPROACH TO BIRKHOFF QUADRATURE FORMULAS

1. Introduction and Main ResultsIn tfor paPer we shaJl use the ddstions and notations of [3l. Let E = (e'k)7t' kt. be anincidence matrir with entries consisting of zeros and ones and satisfying lEl:= Z.,* ei* = n + 1(here we allow a zero row ). Furthermore, in wha follOws we assume that(A) E satisfies the P6lya condition(B) all sequences of E in the interior rows, 0 < i < m + 1, are even.Let Sm denote the set of poiats X = (xo, z1 l "') xm, x.+1) fOr whichand Sm its clOusure. If some O…     

Numerical calculation of singular integrals appearing in three-dimensional potential flow problems

Two approximate methods for calculating singular integrals appearing in the numerical solution of three-dimensional potential flow problems are presented. The first method is a self-adaptive, fully numerical method based on special copy formulas of Gaussian quadrature rules. The singularity is treated by refining the partitions of the copy formula in the vicinity of the singular point. The second method is a semianalytic method based on asymptotic considerations. Under the small curvature hypothesis, asymptotic expansions are derived for the integrals that are involved in the calculation of the scalar potential, the velocity as well as the deformation field induced from curved quadrilateral surface elements. Compared to other methods, the proposed integration schemes, when applied to practical flow field calculations, require less computational effort.     

Numerische Quadratur bei Projektionsverfahren

Summary In this paper we investigate the influence of the numerical quadrature in projection methods. In particular we derive conditions for the order of the quadrature formulas in finite element methods under which the order of convergence is not perturbed. It seems that this question has been discussed only for the Ritz method. There is an essential difference between this method on one side and the Galerkin and least squares methods on the other side. The methods using numerical integration are only in the latter case still projection methods. The resulting conditions for the quadrature formulas are often much weaker than those for the Ritz method. Numerical examples using cubic splines and polynomials show that the conditions derived are realistic. These examples also allow the comparison of some projection methods.

     

Some error expansions for Gaussian quadrature

Complex-variable methods are used to obtain some expansions in the error in Gaussian quadrature formulae over the interval [– 1, 1]. Much of the work is based on an approach due to Stenger, and both circular and elliptical contours are used. Stenger's theorem on monotonicity of convergence of Gaussian quadrature formulae is generalized, and a number of error bounds are 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.     

General Gaussian Quadrature Formulas on Chebyshev Nodes

本文给出了基于（第一类及第二类）Ｃｈｅｂｙｓｈｅｖ节点的广义Ｇａｕｓ求积公式的Ｃｏｔｅｓ数的明显公式及其渐进性态．     

Numerical evaluation of cauchy principal values of integrals

Gaussian quadrature rules for the evaluation of Cauchy principal values of integrals are considered, their relation with Gauss-Legendre formulas is studied, and they are compared with other rules.     

Exit criteria and monotonicity in compound quadrature of Gaussian type

Summary In this short note, for compound quadrature rules of Gaussian type, we prove stopping rules and monotonicity results based on Peano-kernel methods.     

GENERALIZED GAUSSIAN QUADRATURE FORMULASWITH CHEBYSHEV NODES

1.IntroductionThispaperdealswiththegeneralizedGaussianquadratureformulasforChebyshevnodes(of.[2]).Throughoutthepaperweassumethatmandnarepositiveintegers.Asusually,Tn(x)andUn(x)denotethen--thChebyshevpolynomialsofthefirstkindandthesecondkind,respectively.AmonggeneralizedGaussianquadratureformulasoneofthemostimportantcasesistheweightwin(x):~(1~x')[(m ')/']~(" ')/',(1.1)where[rldenotesthelargestinteger5r.In[5]wepointedoutthatifwetakeasnodesofaquadratureformulathezerosof(1--x')Un--100(herewere…     

Chebyshev series method for computing weighted quadrature formulas

In this paper we study convergence and computation of interpolatory quadrature formulas with respect to a wide variety of weight functions. The main goal is to evaluate accurately a definite integral, whose mass is highly concentrated near some points. The numerical implementation of this approach is based on the calculation of Chebyshev series and some integration formulas which are exact for polynomials. In terms of accuracy, the proposed method can be compared with rational Gauss quadrature formula.     

Composition methods in the presence of small parameters

We derive numerical methods for arbitrary small perturbations of exactly solvable differential equations. The methods, based in one instance on Gaussian quadrature, are symplectic if the system is Hamiltonian and are asymptotically more accurate than previously known methods.