A Survey of Gauss-Christoffel Quadrature Formulae

  title={A Survey of Gauss-Christoffel Quadrature Formulae},
  author={Walter Gautschi},
We present a historical survey of Gauss-Christoffel quadrature formulae, beginning with Gauss’ discovery of his well-known method of approximate integration and the early contributions of Jacobi and Christoffel, but emphasizing the more recent advances made after the emergence of powerful digital computing machinery. One group of inquiry concerns the development of the quadrature formula itself, e.g. the inclusion of preassigned nodes and the admission of multiple nodes, as well as other… 

Accurate Computation of Weights in Classical Gauss-Christoffel Quadrature Rules

A simple but efficient and general method for improving the accuracy of the computation of the quadrature weights is proposed, which ensures a high level of accuracy for these weights even though the nodes may carry a significant large error.

On sensitivity of Gauss–Christoffel quadrature

It is shown that even a small perturbation of a distribution function can cause large differences in Gauss–Christoffel quadrature estimates.

Is Gauss Quadrature Better than Clenshaw-Curtis?

Comparisons of the convergence behavior of Gauss quadrature with that of its younger brother, Clenshaw-Curtis are compared, and experiments show that the supposed factor-of-2 advantage of Gaussian quadratures is rarely realized.

Stieltjes Polynomials and the Error of Gauss-Kronrod Quadrature Formulas

The Gauss-Kronrod quadrature scheme, which is based on the zeros of Legendre polynomials and Stieltjes polynomials, is the standard method for automatic numerical integration in mathematical software

Anti-Gaussian quadrature formulae based on the zeros of Stieltjes polynomials

It is well known that the Gauss–Kronrod quadrature formula does not always exist with real and distinct nodes and positive weights. In 1996, in an attempt to find an alternative to the Gauss–Kronrod

Quadrature Rules with Multiple Nodes

In this paper a brief historical survey of the development of quadrature rules with multiple nodes and the maximal algebraic degree of exactness is given. The natural generalization of such rules are

Quadrature formulae of non-standard type

We discuss quadrature formulae of highest algebraic degree of precision for integration of functions of one or many variables which are based on non-standard data, i.e., in which the information used

The Australian Journal of Mathematical Analysis and Applications

This paper presents a fast algorithm for computing Gaussian type quadrature formulae which are exact for splinefunctions and which contain boundary terms involving derivatives at both end points and shows that the latter are computed, via eigenvalues and eigenvectors of real symmetric tridiagonal matrices.

Gauss-Kronrod integration rules for Cauchy principal value integrals

Kronrod extensions to two classes of Gauss and Lobatto integration rules for the evaluation of Cauchy principal value integrals are derived. Since in one frequently occurring case, the Kronrod

Gene H. Golub and Gérard Meurant: Matrices, Moments and Quadrature with Applications

In modern computational mathematics, sciences and engineering, Krylov subspace methods and matching moments model reduction can be viewed as nothing but a translation of the classical concepts mentioned above to the language of large-scale matrix computation.



Asymptotic error estimates for the Gauss quadrature formula

1. Introduction. Classical error estimates for the Gauss quadrature formula using derivatives can be used, but they are not of great practical value since the derivatives are not usually available.

Error of the Newton-Cotes and Gauss-Legendre quadrature formulas

Abstract. It was shown by P. J. Davis that the Newton-Cotes quadrature formula is convergent if the integrand is an analytic function that is regular in a sufficiently large region of the complex

Construction of Gauss-Christoffel quadrature formulas

Each of these rules will be called a Gauss-Christoffel quadrature formula if it has maximum degree of exactness, i.e. if (1.1) is an exact equality whenever f is a polynomial of degree 2n 1. It is a

Error-Bounds for the Evaluation of Integrals by the Euler-Maclaurin Formula and by Gauss-Type Formulae

where f(x) is even, can be evaluated with surprising accuracy by means of the trapezoidal-sum formula. It is natural to anticipate that a more general result can be obtained when the integrand is not

Asymptotic Gauss Quadrature Errors as Fourier Coefficients of the Integrand

  • M. Chawla
  • Mathematics
    Journal of the Australian Mathematical Society
  • 1971
The purpose of this paper is to derive asymptotic relations giving the error of a Gauss type quadrature, applied to analytic functions, in terms of certain coefficients in the orthogonal expansion of

Quadrature Formulas for Infinite Integrals

have become increasingly important. The only quadrature generally available for the case b = - a = oo is the Hermite-Gauss formula although the LaguerreGauss formula can also be used if f(x) is an

A Unified Approach to Quadrature Rules with Asymptotic Estimates of Their Remainders

The starting point of this paper is a theorem, based on the theory of functions of a complex variable, which essentially gives a representation of the remainder in a quadrature rule as a contour

On a new type of quadrature formulas

SummaryIn the paper quadrature formulas of the form $$\int\limits_a^b f (x)dx = \sum\nolimits_{k = 1}^n {w_k I(a_k ,b_k ;f) + } E(f)$$ are considered. HereI(ak,bk;f) is the average of the functionf

Gauss interpolation formulas and totally positive kernels

This paper simplifies and generalizes an earlier result of the author's on Gauss interpolation formulas for the one-dimensional heat equation. Such formulas approximate a function at a point (x*, t*)