The numerical evaluation of Hadamard finite-part integrals

Summary A quadrature rule is described for the numerical evaluation of Hadamard finite-part integrals with a double pole singularity within the range of integration. The rule is based upon the observation that such an integral is the derivative of a Cauchy principal value integral.     

On the uniform convergence of Gaussian quadrature rules for Cauchy principal value integrals

Summary In a previous paper the authors proposed a modified Gaussian rule _* ^m (wf;t)to compute the integral (wf;t) in the Cauchy principal value sense associated with the weightw, and they proved the convergence in closed sets contained in the integration interval. The main purpose of the present work is to prove uniform convergence of the sequence { _* ^m (wf;t)} on the whole integration interval and to give estimates for the remainder term. The same results are shown for particular subsequences of the Gaussian rules _m (wf;t) for the evaluation of Cauchy principal value integrals. A result on the uniform convergence of the product rules is also discussed and an application to the numerical solution of singular integral equations is made.     

Comparison theorems for compound quadrature formulas

Summary The aim of this note is to compare the rates of convergence of quadrature processes. To this end, representation formulas for the remainders of one process in terms of a second one are developed. The main tool is the Möbius inversion. It turns out that for a large class of compound quadrature processes the rates of convergence are essentially the same.     

Gauss-Tchebycheff quadrature formulas

Summary It is well known that the Tchebycheff weight function (1-x ²)^–1/2 is the only weight function (up to a linear transformation) for which then point Gauss quadrature formula has equal weights for alln. In this paper we describe explicitly all weight functions which have the property that then _k-point Gauss quadrature formula has equal weights for allk, where (n _k),n ₁<n ₂<..., is an arbitrary subsequence of . Furthermore results on the possibility of Tchebycheff quadrature on several intervals are given.     

Stochastic properties of quadrature formulas

Summary The average error of suitable quadrature formulas and the stochastic error of Monte Carlo methods are both much smaller than the worst case error in many cases. This depends, however, on the classF of functions which is considered and there are counterexamples as well.Nonlinear methods, adaptive methods, or even methods with varying cardinality are not significantly better (with respect to certain stochastic error bounds) than the simplest linear methods .     

Comparison of optimal quadrature formulas

Summary We prove the monotonicity of the error of the optimal quadrature formula of a given quasi-Hermitian type inW _q ^r [0.1] (1<q) with respect to the order of the derivatives appearing in the end point terms.     

Calculating singular integrals as an ill-posed problem

Summary Applying numerical quadrature to singular integrals in a straightforward way leads to uncontrolled instability with respect to data errors. In this paper we describe how to control this instability.Dedicated to Professor G. Hämmerlin on the occasion of his 60th birthday.     

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.     

An analysis of Ralston's quadrature

Summary Ralston's quadrature achieves higher accuracy in composite rules than analogous Newton-Cotes or Gaussian formulas. His rules are analyzed, computable expressions for the weights and knots are given, and a more suitable form of the remainder is derived.     

On Gauss quadrature formulas with equal weights

Summary It is well known that the Chebyshev weight function (1–x ²)^–1/2 is the only weight function (up to a linear transformation) for which then point Gauss quadrature formula has equal weights for alln. In this paper we describe all weight functions for which thenm point Gauss quadrature formula has equal weights for alln, wherem is fixed.     

A probabilistic theory for error estimation in automatic integration

Summary A probabilistic theory for derivation and analysis of error criteria for automatic quadrature is presented. In particular, conditional average error criteria are derived for quadratures which have derivative-bound error estimates. These probabilistic error criteria are compared to variations of heuristic error criteria derived by discretizing the derivative in the original error bound. It is shown that the theory provides a mathematical foundation and a quantitative model for these discrete error criteria. It is also shown that estimating the conditional average error is equivalent to testing error with the spline interpolation as a sample integrand, and that this process can be made implicit by using appropriate error criteria with local error-checks.This paper is based on the author's Ph.D. thesis in computational complexity and numerical analysis, completed at the University of California, Berkeley     

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.     

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.     

Numerical cubature formulae with preassigned knots

Summary In this paper we give using the orthogonal polynomial theory, conditions ensuring the existence of cubature formulae with weight function on compact subsetsK in ² which have some given knots. The formulae are exact on the space of polynomials of two variablesx ₁ ,x ₂ with a degree not greater thanm _i +k _i ,i=1, 2.     

Cubature formulae which are exact on spacesP,intermediate betweenP k andQ k′

Summary In this paper we search, from the orthogonal polynomial theory, for conditions which allow to obtain cubature formulae on compacts of ⁿ , with weight function, and which are exact on the spaceR( _{k 1}, k₂, ..., k_n) of all polynomials of degree k _i respectively to each variablex _i , 1in.     

Sur les formules de quadrature numérique à nombre minimal de noeuds d'intégration

Résumè Cet article a pour objet la recherche, à partir de la théorie des polynômes orthogonaux, de conditions permettant l'obtention de formules de quadrature numérique sur des domaines de ⁿ, avec fonction poids, à nombre minimal de noeuds et exactes sur les espacesQ _k de polynômes de degré k par rapport à chacune de leurn variables. Ces résultats, complétés par des exemples numériques originaux dans ², adaptent à ces espacesQ _k ceux démontréq par H.J. Schmid [14] dans le cadre des espacesP _k de polynômes.

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About Cubature formulas with a minimal number of knots

Summary In this paper we search, from the orthogonal polynomial theory, for conditions which allow to obtain cubature formulas on sets of ⁿ, with weight function. which have a minimal number of knots and which are exact on the spaceQ _k of all polynomials of degree k with respect to each variablex _i, 1in.These results, completed by original numerical examples in ², adapt to the spacesQ _k those proved by H.J. Schmid [14] in the case of polynomial spacesP _k.

     

Das asymptotische Verhalten der Peanokerne einiger interpolatorischer Quadraturverfahren

Summary Interpolatory quadrature formulae consist in replacing by wherep _f denotes the interpolating polynomial off with respect to a certain knot setX. The remainder may in many cases be written as wherem=n resp. (n+1) forn even and odd, respectively. We determine the asymptotic behaviour of the Peano kernelP _X (t) forn for the quadrature formulae of Filippi, Polya and Clenshaw-Curtis.

     

A Monte Carlo method for high dimensional integration

Summary A new method for the numerical integration of very high dimensional functions is introduced and implemented based on the Metropolis' Monte Carlo algorithm. The logarithm of the high dimensional integral is reduced to a 1-dimensional integration of a certain statistical function with respect to a scale parameter over the range of the unit interval. The improvement in accuracy is found to be substantial comparing to the conventional crude Monte Carlo integration. Several numerical demonstrations are made, and variability of the estimates are shown.     

Product integration for the linear transport equation in slab geometry

Summary In this paper we present a product quadrature rule for the discretization of the well-known linear transport equation in slab geometry. In particular we give an algorithm for constructing the weights of the rule and prove that the order of convergence isO(n ^–3+ ), >0 small as we like. Numerical examples are given, and our formula is also compared with product Simpson rules. Finally, we examine a Nyström method based on our quadrature.Work sponsored by the Ministero della Pubblica Instruzione of Italy