Gaussian quadrature formulae on the unit circle

Let μ be a probability measure on [0,2π]. In this paper we shall be concerned with the estimation of integrals of the form

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For this purpose we will construct quadrature formulae which are exact in a certain linear subspace of Laurent polynomials. The zeros of Szegö polynomials are chosen as nodes of the corresponding quadratures. We will study this quadrature formula in terms of error expressions and convergence, as well as, its relation with certain two-point Padé approximants for the Herglotz–Riesz transform of μ. Furthermore, a comparison with the so-called Szegö quadrature formulae is presented through some illustrative numerical examples.     

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 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.     

How and how not to check Gaussian quadrature formulae

We discuss characteristic difficulties inherent in the validation of Gaussian quadrature formulae, in particular the futility of moment-related tests. We propose instead more effective tests based on the recursion coefficients of the appropriate orthogonal polynomials and on the sum of the quadrature nodes.Work sponsored in part by the National Science Foundation under grant MCS-7927158A1.     

The error norm of Gaussian quadrature formulae for weight functions of Bernstein-Szegö type

Summary We consider the Gaussian quadrature formulae for the Bernstein-Szegö weight functions consisting of any one of the four Chebyshev weights divided by an arbitrary quadratic polynomial that remains positive on [–1, 1]. Using the method in Akrivis (1985), we compute the norm of the error functional of these quadrature formulae. The quality of the bounds for the error functional, that can be obtained in this way, is demonstrated by two numerical examples.     

On certain definite quadrature formulae

Gauss and Lobatto quadrature formulae related to spaces of cubic splines with double and equidistant knots are constructed. Such quadrature formulae are known as asymptotically optimal definite formulae of order 4. Some monotonicity results concerning the associated quadrature processes are established.     

Asymptotic minimum norm quadrature formulae

The asymptotic limit of minimum norm quadrature formulae for some Hilbert spaces of functions regular and analytic in a domainB is studied, asB expands.     

Convergence of osculatory quadrature formulae

The regions of analyticity of functions to be integrated using equally spaced osculatory quadrature formulae are obtained. As a by-product it is noted that the asymptotic forms used are applicable to estimating or placing bounds on errors.     

Fast evaluation of quadrature formulae on the sphere

Recently, a fast approximate algorithm for the evaluation of expansions in terms of standard -orthonormal spherical harmonics at arbitrary nodes on the sphere has been proposed in [S. Kunis and D. Potts. Fast spherical Fourier algorithms. J. Comput. Appl. Math., 161:75-98, 2003]. The aim of this paper is to develop a new fast algorithm for the adjoint problem which can be used to compute expansion coefficients from sampled data by means of quadrature rules.

We give a formulation in matrix-vector notation and an explicit factorisation of the spherical Fourier matrix based on the former algorithm. Starting from this, we obtain the corresponding factorisation of the adjoint spherical Fourier matrix and are able to describe the associated algorithm for the adjoint transformation which can be employed to evaluate quadrature rules for arbitrary weights and nodes on the sphere. We provide results of numerical tests showing the stability of the obtained algorithm using as examples classical Gauß-Legendre and Clenshaw-Curtis quadrature rules as well as the HEALPix pixelation scheme and an equidistribution.

     

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.     

The error norm of quadrature formulae

In certain spaces of analytic functions the error term of a quadrature formula is a bounded linear functional. We give a survey of the methods used in order to compute explicitly, or in some cases estimate, the norm of the error functional. The results, some of which are fairly recent, cover Gauss, Gauss?CLobatto, Gauss?CRadau, Gauss?CKronrod and Fejér type rules.     

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.     

Some quadrature formulae with nonstandard weights

In this paper the authors study “truncated” quadrature rules based on the zeros of Generalized Laguerre polynomials. Then, they prove the stability and the convergence of the introduced integration rules. Some numerical tests confirm the theoretical results.     

The remainder term in quadrature formulae

Summary It is shown that the remainder term of any quadrature formula has an asymptotic expansion in terms of the step size; the occurrence of remainder terms of formc f ⁽ⁿ⁺¹⁾ () is discussed.