A sinc quadrature rule for Hadamard finite-part integrals

Summary A Sinc quadrature rule is presented for the evaluation of Hadamard finite-part integrals of analytic functions. Integration over a general are in the complex plane is considered. Special treatment is given to integrals over the interval (–1,1). Theoretical error estimates are derived and numerical examples are included.     

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.     

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     

Monotony in interpolatory quadratures

Summary Let , be holomorphic in an open disc with the centrez ₀=0 and radiusr>1. LetQ _n (n=1, 2, ...) be interpolatory quadrature formulas approximating the integral . In this paper some classes of interpolatory quadratures are considered, which are based on the zeros of orthogonal polynomials corresponding to an even weight function. It is shown that the sequencesQ _n 9f] (n=1, 2, ...) are monotone. Especially we will prove monotony in Filippi's quadrature rule and with an additional assumption onf monotony in the Clenshaw-Curtis quadrature rule.     

Produktintegration mit nicht-äquidistanten Stützstellen

Summary For the numerical evaluation of , 0<<1 andx smooth, product integration rules are applied. It is known that high-order rules, e.g. Gauss-Legendre quadrature, become normal-order rules in this case. In this paper it is shown that the high order is preserved by a nonequidistant spacing. Furthermore, the leading error terms of this product integration method and numerical examples are given.

     

On best cubature formulas and spline interpolation

Summary A general cubature formula with an arbitrary preassigned weight function is derived using monosplines and integration by parts. The problem of determining the best cubature is formulated in terms of monosplines of least deviation and a solution to the problem is given by Theorem 3 below. This theorem may also be viewed as an optimal property of a new kind of two-dimensional spline interpolation.This work was done while the author was working at CERN, Geneva, Switzerland     

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].     

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.     

Zur Interpolation und Integration differenzierbarer periodischer Funktionen

Summary Letx ₀<x ₁<...<x _n–1<x ₀+2 be nodes having multiplicitiesv ₀,...,v _n–1, 1v _k r (0k<n). We approximate the evaluation functional ,x fixed, and the integral respectively by linear functionals of the form and determine optimal weights for the Favard classesW _r C ₂. In the even case of optimal interpolation these weights are unique except forr=1,x(x _k +x _k–1)/2 mod 2. Moreover we get periodic polynomial splinesw _{k, j} (0k<n, 0j<v _k ) of orderr such that are the optimal weights. Certain optimal quadrature formulas are shown to be of interpolatory type with respect to these splines. For the odd case of optimal interpolation we merely have obtained a partial solution.

Bojanov hat in [4, 5] ähnliche Resultate wie wir erzielt. Um Wiederholungen zu vermeiden, werden Resultate, deren Beweise man bereits in [4, 5] findet, nur zitiert     

Monotonie bei den Quadraturverfahren von Gauß und Newton-Cotes

Summary LetQ _n be the quadrature rule of Gauss or Newton-Cotes withn abscissas. It is proven here, thatf ⁽²ⁿ⁾0 impliesQ _n ^G [f]Q _m ^G [f] (for allm>n) andQ _2n–1 ^NC [f]Q _2n ^NC [f]Q _2n+1 ^NC [f]. It follows that the sequenceQ _n[f] (n=1, 2, ...) is monotone, if all derivatives off are positive.

     

Dividierte Differenzen und Monotonie von Quadraturformeln

Summary As a generalisation of divided differences we consider linear functionals vanishing for polynomials of given degree and with discrete support. It is shown that functionals of that type may be uniquely represented by a linear combination of divided differences. On the basis of this representation theorem we introduce the concept of positivity and definiteness of functions and linear functionals. Next we show that in many cases positivity follows from the number of sign changes of the coefficients of the given linear functional. These results may be applied to the problems of nonexistence of Newton-Côtes and Gegenbauer quadrature formulas with positive weights and to the monotony problem of Gauss and Newton Côtes quadrature.

     

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.

---

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.

     

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     

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.     

Fehlerabschätzungen für Gauß-Quadraturformeln

Summary We consider Gauss quadrature formulaeQ _n ,n, approximating the integral ,w an even weight function. Let be analytic inK _r :={z:|z|<r},r>1, and . The error functionalR _n :=I-Q _n is continuous with respect to |·|_r and the relation , q_2k (x):=x ^2k holds.In this paper estimates for R _n are given. To this end we first derive two new representations of R _n which are essential for our further investigations. The R _n =r ² R _n (), with (x):=1/(r ²-x ²), is estimated in various ways by using the best uniform approximation of in P_2n-1, and also the expansion of with respect to Chebyshe polynomials of the first and second kind. Forw(x)=(1-x ²), =±1/2, R _n is calculated. The asymptotic behaviour, forr1+, of R _n and of the derived error bounds is also discussed. Finally, we compare different error bounds and give numerical examples.

     

Product-integration with the Clenshaw-Curtis and related points

Summary This paper is concerned with the theoretical properties of a productintegration method for the integral , wherek is absolutely integrable andf is continuous. The integral is approximated by , where the points are given byx _ni =cos(i/n, 0in, and where the weightsw _ni are chosen to make the rule exact iff is any polynomial of degree n. The principal result is that ifkL _p [–1, 1] for somep>1, then the rule converges to the exact result asn for all continuous (or indeed R-integrable) functionsf, and moreover that the sum of the absolute values of the weights converges to the least possible value, namely . A limiting expression for the individual weights is also obtained, under certain assumptions. The results are exteded to other point sets of a similar kind, including the classical Chebyshev points.     

On quasi degree quadrature rules

Summary In this paper we examine quadrature rules for the integral which are exact for all with +d. We specify three distinct families of solutions which have properties not unlike the standard Gauss and Radau quadrature rules. For each integerd the abscissas of the quadrature rules lie within the closed integration interval and are expressed in terms of the zeros of a polynomialq _d(y). These polynomialsq _d(y), (d=0, 1, ...), which are not orthogonal, satisfy a three term recurrence relation of the type Q_d+1(y)=(y+_d+1)q_d(y)–_d+1yq_d–1(y) and have zeros with the standard interlacing property.This work was supported by the Applied Mathematical Sciences Research Program (KC-04-02) of the Office of Energy Research of the U.S. Department of Energy under Contract W-31-109-Eng-38     

Sigmoidal transformations and the Euler-Maclaurin expansion for evaluating certain Hadamard finite-part integrals

Summary. Starting with some results of Lyness concerning the Euler-Maclaurin expansion of Cauchy principal value integrals over it is shown how, by the use of sigmoidal transformations, good approximations can be found for the Hadamard finite-part integral where The analysis is illustrated by some numerical examples. Received March 13, 1996     

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.     

The Euler-Maclaurin expansion and finite-part integrals

In this paper we compare G(p), the Mellin transform (together with its analytic continuation), and , the related Hadamard finite-part integral of a function g(x), which decays exponentially at infinity and has specified singular behavior at the origin. Except when p is a nonpositive integer, these coincide. When p is a nonpositive integer, is well defined, but G(p) has a pole. We show that the terms in the Laurent expansion about this pole can be simply expressed in terms of the Hadamard finite-part integral of a related function. This circumstance is exploited to provide a conceptually uniform proof of the various generalizations of the Euler-Maclaurin expansion for the quadrature error functional. Received June 11, 1997 / Revised version received December 15, 1997