A quadrature rule for finite-part integrals

A quadrature rule is described for evaluating finite-part (f.p.) integrals of the form , witha 1. The rule is obtained by interpolating tof by a polynomial which coincides withf at the shifted zeros of a Legendre polynomial.     

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.     

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     

On the error of quadrature formulae for Cauchy principal value integrals based on piecewise interpolation

We consider the computation of the Cauchy principal value integral by quadrature formulae Q _n ^F [f] of compound type, which are obtained by replacing f by a piecewise defined function F_n[f]. The behaviour of the constants k_{i, n} in the estimates [R _n ^F [f]] |⩽K _i,n ‖f ⁽ⁱ⁾‖_∞ (where R _n ^F [f] is the quadrature error) is determined for fixed i and n→∞, which means that not only the order, but also the coefficient of the main term of k_{i, n} is determined. The behaviour of these error constants k_{i, n} is compared with the corresponding ones obtained for the method of subtraction of the singularity. As it turns out, these error constants have, in general, the same asymptotic behaviour.     

A note on spline approximation for Hadamard finite-part integrals

In this paper we construct product quadrature rules, based on spline interpolation, for the numerical evaluation of singular integrals in the sense of Hadamard. We give a convergence result and examine the behaviour of the stability factor. We also present some numerical tests.     

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.     

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 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.     

Definitions,properties and applications of finite-part integrals

By using standard calculus, we discuss some definitions and properties of finite-part integrals, point out the essence of this concept and show how these integrals naturally arise in some integral equation applications. A couple of new examples are also described.     

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.     

On the approximate evaluation of Hadamard finite-part integrals

Grnwald's algorithms for the numerical evaluation of Hadamardfinite-part integrals with non-integer exponent are extendedto the case of integer exponent. These algorithms are basedon the use of Bernstein polynomials and it is shown how, byan appropriate modification of the first algorithm, a convergencerate of order 1/N² may be obtained, where N is the number offunction evaluations.     

A new algorithm for Cauchy principal value and Hadamard finite-part integrals

An algorithm for the approximate evaluation of integrals defined by Cauchy principal value or by Hadamard finite part has been proposed. The convergence of the procedure is proved. The stability of the algorithm is considered and some numerical examples are given.     

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 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.     

A superconvergence result for the second-order Newton-Cotes formula for certain finite-part integrals

In this paper we investigate the superconvergence phenomenonof the second-order quadrature formula of Newton–Cotestype for the computation of finite-part integrals with a second-ordersingularity on an interval. Superconvergence points are foundand a superconvergence estimate is obtained. The validity ofthe theoretical result is demonstrated by numerical experiments.     

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.     

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.