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.     

Modifizierte Restgliedabschätzungen für Quadraturformeln

Summary Introduction of the projection operatorP leads to a modification of the Peano representation of remainders for quadrature rules. Combining the different versions of the remainders yields additional estimates.

     

A note on the optimal quadrature inH p

Summary The existence of optimal nodes with preassigned multiplicities is proved for the Hardy spacesH ^p (1<p<). This is then used to show that the exact order of convergence for the optimal qudrature formula withN nodes (including multiplicity) is where 1/p+1/q=1 and 1p.     

Sharp estimates in terms of moduli of smoothness for compound cubature rules

Summary In 1980 Dahmen-DeVore-Scherer introduced a modulus of continuity which turns out to reflect invariance properties of compound cubature rules effectively. Accordingly, sharp error bounds are derived, the existence of relevant counterexamples being a consequence of a quantitative resonance principle, established previously.     

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     

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.     

Optimal addition of knots to cubature formulae for planar regions

Summary A method is described to add knots to a cubature formula of degree 2k–1 for an integral over a symmetric region, to obtain a cubature formula of degree 2k+1. This method is used to construct cubature formulae for the square, the circle, the hexagon and the entire plane.     

Optimal quadrature formula of Markov's and Locher's type with weight function

Summary A quadrature formula of Markov's type with a weight functionx (1–x), which has properties of formulas exact for polynomials of a given degree and properties of optimal formulas on some sets of functions, is given. The particular case of formula (where ==p=q=0) is the formula of Locher [1, 2].     

Parameter tuning and repeated application of the IMT-type transformation in numerical quadrature

Summary The IMT rule, which is especially suited for the integration of functions with end-point singularities, is generalized by introducing parameters and also by repeatedly applying the parametrized IMT transformation. The quadrature formulas thus obtained are improved considerably both in efficiency and in robustness against end-point singularities. Asymptotic error estimates and numerical results are also given.     

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.     

On the construction of multi-dimensional embedded cubature formulae

Summary In order to compute an integralI[f], one needs at least two cubature formulaeQ _j ,j{1, 2}. |Q ₁[f]–Q ₂[f]| can be used as an error estimate for the less precise cubature formula. In order to reduce the amount of work, one can try to reuse some of the function evaluations needed forQ ₁, inQ ₂. The easiest way to construct embedded cubature formulae is: start with a high degree formulaQ ₁, drop (at least) one knot and calculate the weights such that the new formulaQ ₂ is exact for as much monomials as possible. We describe how such embedded formulae with positive weights can be found. The disadvantage of such embedded cubature formulae is that there is in general a large difference in the degree of exactness of the two formulae. In this paper we will explain how the high degree formula can be chosen to obtain an embedded pair of cubature formulae of degrees 2m+1/2m–1. The method works for all regions ⁿ ,n2. We will also show the influence of structure on the cubature formulae.     

Bases de type schauder deC[0,1] et formules de quadrature associées

Résumé On caractérise deux familles de bases deC[0,1] et l'on étudie les formules de quadrature associées. On montre en particulier que les formules de quadrature de Romberg proviennent d'une suite de bases engendrées par des polynômes.

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Bases of schauder type inC[0, 1] and associated quadrature formulas

Summary We characterize two families of bases ofC[0,1] and we study the associated quadrature formulae. In particular, we prove that the Romberg quadrature formulae come from a sequence of bases generated by polynomials.

     

Product integration with the Clenshaw-Curtis points: Implementation and error estimates

Summary This paper is concerned with the practical implementation of a product-integration rule for approximating , wherek is integrable andf is continuous. The approximation is , where the weightsw _ni are such as to make the rule exact iff is any polynomial of degree n. A variety of numerical examples, fork(x) identically equal to 1 or of the form |–x| with >–1 and ||1, or of the form cosx or sinx, show that satisfactory rates of convergence are obtained for smooth functionsf, even ifk is very singular or highly oscillatory. Two error estimates are developed, and found to be generally safe yet quite accurate. In the special casek(x)1, for which the rule reduces to the Clenshaw-Curtis rule, the error estimates are found to compare very favourably with previous error estimates for the Clenshaw-Curtis rule.     

Monotonicity of quadrature formulae of Gauss type and comparison theorems for monosplines

Summary This paper deals with quadrature formulae of Gauss type corresponding to subspaces of spline functionsS _{m–1, k} of degreem–1 withk fixed knots. We shall show monotonicity of the quadrature formulae for functions which are contained in the so-called convexity cone ofS _m–1,k Moreover, we apply these results to monosplines and establish comparison theorems for these splines.     

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.     

Convergence of product formulas for the numerical evaluation of certain two-dimensional Cauchy principal value integrals

Summary We consider product rules of interpolatory type for the numerical approximation of certain two-dimensional Cauchy principal value integrals. We present convergence results which generalize those known in the one-dimensional case.Work sponsored by the Ministero della Pubblica Istruzione of Italy     

Suboptimal Kronrod extension formulae for numerical quadrature

Summary We consider cases where the Stieltjes polynomial associated with a Gaussian quadrature formula has complex zeros. In such cases a Kronrod extension of the Gaussian rule does not exist. A method is described for modifying the Stieltjes polynomial so that the resulting polynomial has no complex zeros. The modification is performed in such a way that the Kronrod-type extension rule resulting from the addition of then+1 zeros of the modified Stieltjes polynomial to the original knots of the Gaussian rule has only slightly lower degree of precision than normally achieved when the Kronrod extension rule exists. As examples of the use of the method, we present some new formulae extending the classical Gauss-Hermite quadrature rules. We comment on the limited success of the method in extending Gauss-Laguerre rules and show that several modified extensions of the Gauss Gegenbauer formulae exist in cases where the standard Kronrod extension does not.     

Error estimates for spline interpolants on equidistant grids

Summary For oddm, the error of them-th-degree spline interpolant of power growth on an equidistant grid is estimated. The method is based on a decomposition formula for the spline function, which locally can be represented as an interpolation polynomial of degreem which is corrected by an (m+1)-st.-order difference term.Dedicated to Prof. Dr. Karl Zeller on the occasion of his 60th birthday     

Über optimale Quadraturformeln mit freien Knoten zu Hilberträumen periodischer Funktionen

Summary For some special Hilbert-spaces of periodic analytic functions it is known that quadrature formulae of minimal norm with preassigned equidistant nodes are even so-called Wilf-formulae, i.e. they satisfy necessary conditions for minimal norm with respect to their nodes. By simple examples, however, it can be shown that equidistant Wilf-formulae are not necessarily optimal. In this paper the question of optimality of equidistant nodes in quadrature formulae for rather general Hilbert-spaces of periodic analytic functions is answered by giving sufficient conditions which can be interpreted as conditions on the size of the regularity-regions of the functions belonging to the Hilbert-spaces under consideration. Examples prove these conditions to be quite sharp.In addition the trapezoidal-rule is shown to be only optimal formula (with respect to the nodes and coefficients) of orderk.Finally the trapezoidal-rule is shown to be asymptotically optimal for wide classes of Hilbert-spaces of periodic functions.

     

Error bounds and estimates for eigenvalues of integral equations

Summary Approximate solutions of the linear integral equation eigenvalue problem can be obtained by the replacement of the integral by a numerical quadrature formula and then collocation to obtain a linear algebraic eigenvalue problem. This method is often called the Nyström method and its convergence was discussed in [7]. In this paper computable error bounds and dominant error terms are derived for the approximation of simple eigenvalues of nonsymmetric kernels.