Error estimates for Gaussian quadrature formulae

Summary We derive both strict and asymtotic error bounds for the Gauss-Jacobi quadrature formula with respect to a general measure. The estimates involve the maximum modulus of the integrand on a contour in the complex plane. The methods are elementary complex analysis.     

Error estimates for quadrature formulas

We construct two-sided polynomials of collocation type of the same order as a given system of basis functions according to a given ordered system of nodes of arbitrary multiplicity and according to a system of nodes displaced to the right (or to the left) at one position. Numerical estimates are given for the remaining terms of the quadrature formulas.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 70, pp. 21–31, 1990.     

Error estimates for Filon quadrature formulae

In this paper, upper bounds for the error of (generalized) Filon quadrature formulae are stated. Furthermore, the main term of this error is derived, yielding simple modified quadrature rules of higher asymptotical precision.     

Error estimates for interpolatory quadrature formulae

Summary In this paper we study the remainder of interpolatory quadrature formulae. For this purpose we develop a simple but quite general comparison technique for linear functionals. Applied to quadrature formulae it allows to eliminate one of the nodes and to estimate the remainder of the old formula in terms of the new one. By repeated application we may compare with quadrature formulae having only a few nodes left or even no nodes at all. With the help of this method we obtain asymptotically best possible error bounds for the Clenshaw-Curtis quadrature and other Pólya type formulae.Our comparison technique can also be applied to the problem of definiteness, i.e. the question whether the remainderR[f] of a formula of orderm can be represented asc·f ^(m)(). By successive elimination of nodes we obtain a sequence of sufficient criteria for definiteness including all the criteria known to us as special cases.Finally we ask for good and worst quadrature formulae within certain classes. We shall see that amongst all quadrature formulae with positive coefficients and fixed orderm the Gauss type formulae are worst. Interpreted in terms of Peano kernels our theorem yields results on monosplines which may be of interest in themselves.     

Error estimates for generalized compound quadrature formulas

We define quadrature formulas for integrals with weight functionsby applying a given approximation method locally. This allowsthe generalisation of different quadrature formulas, e.g., thecompound Newton-Cotes formulas, Gauss summation formulas, orGregory's formulas, to the case of weighted integrals, as wellas to construct new quadrature formulas, and to derive errorestimates for all these quadrature formulas. The estimates consideredhere are mainly of the form |R[f]|c||f^(r)||, provided the underlyingapproximation method is exact for polynomials of degree <r(R[f] is the quadrature error). Explicit, asymptotically sharperror estimates are obtained for arbitrary integrable weightfunctions. Further, estimates are obtained for the case thatthe quadrature error is of higher order than the approximationerror.     

Error estimates for a class of quadrature formulae

The problem considered is that of estimating the error of a class of quadrature formulae for _–1 ¹ w _r (x)f(x)dx, (w _r (x) being a positive weight-function), where only values off(x) in (–1,1) and off(x) and its derivatives at the end-points of the interval are considered.     

Error estimates of anti-Gaussian quadrature formulae

Anti-Gauss quadrature formulae associated with four classical Chebyshev weight functions are considered. Complex-variable methods are used to obtain expansions of the error in anti-Gaussian quadrature formulae over the interval [−1,1]$[-1,1]$. The kernel of the remainder term in anti-Gaussian quadrature formulae is analyzed. The location on the elliptic contours where the modulus of the kernel attains its maximum value is investigated. This leads to effective L^∞$L^{\infty }$-error bounds of anti-Gauss quadratures. Moreover, the effective L¹$L^1$-error estimates are also derived. The results obtained here are an analogue of some results of Gautschi and Varga (1983) [11], Gautschi et al. (1990) [9] and Hunter (1995) [10] concerning Gaussian quadratures.     

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.     

Error bounds of certain Gaussian quadrature formulae

We study the kernel of the remainder term of Gauss quadrature rules for analytic functions with respect to one class of Bernstein-Szegö weight functions. The location on the elliptic contours where the modulus of the kernel attains its maximum value is investigated. This leads to effective error bounds of the corresponding Gauss quadratures.     

Error estimates for Gaussian quadratures of analytic functions

For analytic functions the remainder term of Gaussian quadrature formula and its Kronrod extension can be represented as a contour integral with a complex kernel. We study these kernels on elliptic contours with foci at the points ±1 and the sum of semi-axes ?>1 for the Chebyshev weight functions of the first, second and third kind, and derive representation of their difference. Using this representation and following Kronrod’s method of obtaining a practical error estimate in numerical integration, we derive new error estimates for Gaussian quadratures.     

Error estimates for Gauss-Legendre quadrature of integrands possessing Dirichlet series expansions

A number of formulae are derived for the estimation of the error in the numerical evaluation of integrals of the form _–1 ¹ f(x)dx wheref possesses a Dirichlet series expansion which contains the interval [–1,1] within its region of convergence. The formulae are based on Gauss-Legendre quadrature.     

Error estimates for some quadrature rules with maximal trigonometric degree of exactness

In this paper, we give error estimates for quadrature rules with maximal trigonometric degree of exactness with respect to an even weight function on ( ? π,π) for integrand analytic in a certain domain of complex plane. Copyright © 2013 John Wiley & Sons, Ltd.     

Error estimates of quadrature formulas for classes of functions with bounded mixed derivative

Translated from Matematicheskie Zametki, Vol. 46, No. 2, pp. 128–134, August, 1989.     

Error estimates for approximate approximations with Gaussian kernels on compact intervals

The aim of this paper is the investigation of the error which results from the method of approximate approximations applied to functions defined on compact intervals, only. This method, which is based on an approximate partition of unity, was introduced by Maz’ya in 1991 and has mainly been used for functions defined on the whole space up to now. For the treatment of differential equations and boundary integral equations, however, an efficient approximation procedure on compact intervals is needed.In the present paper we apply the method of approximate approximations to functions which are defined on compact intervals. In contrast to the whole space case here a truncation error has to be controlled in addition. For the resulting total error pointwise estimates and L₁-estimates are given, where all the constants are determined explicitly.     

Stopping functionals for Gaussian quadrature formulas

Gaussian formulas are among the most often used quadrature formulas in practice. In this survey, an overview is given on stopping functionals for Gaussian formulas which are of the same type as quadrature formulas, i.e., linear combinations of function evaluations. In particular, methods based on extended formulas like the important Gauss–Kronrod and Patterson schemes, and methods which are based on Gaussian nodes, are presented and compared.     

Error bounds for Romberg quadrature

Denote by the error of a Romberg quadrature rule applied to the function f. We determine approximately the constants in the bounds of the types and for all classical Romberg rules. By a comparison with the corresponding constants of the Gaussian rule we give the statement “The Gaussian quadrature rule is better than the Romberg method” a precise meaning. Received September 10, 1997 / Revised version received February 16, 1998     

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.     

Error estimates with explicit constants for Sinc approximation, Sinc quadrature and Sinc indefinite integration

Error estimates with explicit constants are given for approximations of functions, definite integrals and indefinite integrals by means of the Sinc approximation. Although in the literature various error estimates have already been given for these approximations, those estimates were basically for examining the rates of convergence, and several constants were left unevaluated. Giving more explicit estimates, i.e., evaluating these constants, is of great practical importance, since by this means we can reinforce the useful formulas with the concept of “verified numerical computations.” In this paper we reveal the explicit form of all constants in a computable form under the same assumptions of the existing theorems: the function to be approximated is analytic in a suitable region. We also improve some formulas themselves to decrease their computational costs. Numerical examples that confirm the theory are also given.