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.     

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.     

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.     

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.     

An asymptotic analysis of two algorithms for certain Hadamard finite-part integrals

Two algorithms for the approximate evaluation of certain Hadamardfinite-part integrals, which date back to Grünwald (1867),have been analysed. It is shown that the quadrature sums areclosely related to the finite-part integrals of the Bernsteinpolynomials. The stability of the algorithms is considered andtwo 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     

The superconvergence of Newton–Cotes rules for the Hadamard finite-part integral on an interval

We study the general (composite) Newton–Cotes rules for the computation of Hadamard finite-part integral with the second-order singularity and focus on their pointwise superconvergence phenomenon, i.e., when the singular point coincides with some a priori known point, the convergence rate is higher than what is globally possible. We show that the superconvergence rate of the (composite) Newton–Cotes rules occurs at the zeros of a special function and prove the existence of the superconvergence points. Several numerical examples are provided to validate the theoretical analysis. The work of J. Wu was partially supported by the National Natural Science Foundation of China (No. 10671025) and a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (No. CityU 102507). The work of W. Sun was supported in part by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (No. City U 102507) and the National Natural Science Foundation of China (No. 10671077).     

Hadamard expansions for integrals with saddles coalescing with an endpoint

Hadamard expansions are constructed for Laplace-type integrals containing a parameter and an asymptotic variable $x$, which may be real or complex. These expansions yield a method of hyperasymptotic evaluation that remains valid throughout a range of the parameter corresponding to coalescence of a saddle point with an endpoint of the integration path. Numerical examples are given to illustrate the practical aspects of the computations.     

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     

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.     

Generalized compound quadrature formulae for finite-part integrals

Received on 31 July 1995. Revised on 19 August 1996. We investigate the error term of the dth degree compound quadratureformulae for finite-part integrals of the form where and p 1.We are mainly interested in error bounds of the form with best possible constants c. Itis shown that, for and n uniformlydistributed nodes, the error behaves as O(n^p–s–1for , p–1 <s d+1.In a previous paper we have shown that this is not true for As an improvement, we consider the case of non-uniformly distributednodes. Here, we show that for all p I and , an O(n^–s) error estimate can be obtainedin theory by a suitable choice of the nodes. A set of nodeswith this property is staled explicitly. In practice, this gradedmesh causes stability problems which are computationally expensiveto overcome. E-mail address: diethelm{at}informatik.uni-hildesheim.de     

Modified Gauss rules for approximate calculation of some strongly singular integrals

The approach we follow consists in transforming the numerical evaluation of hyper-singular integrals into the calculation of a nearly singular integral whose mass is distributed according to a positive parameter ε. To evaluate the latter we apply a Gauss quadrature formula associated with a nearly singular weight function. It is estimated the error in terms of ε. Some numerical results are presented.     

Asymptotic expansions for oscillatory integrals using inverse functions

We treat finite oscillatory integrals of the form ∫ _a ^b F(x)e ^ikG(x) dx in which both F and G are real on the real line, are analytic over the open integration interval, and may have algebraic singularities at either or both interval end points. For many of these, we establish asymptotic expansions in inverse powers of k. No appeal to the theories of stationary phase or steepest descent is involved. We simply apply theory involving inverse functions and expansions for a Fourier coefficient ∫ _a ^b φ(t)e ^ikt dt. To this end, we have assembled several results involving inverse functions. Moreover, we have derived a new asymptotic expansion for this integral, valid when , −1<σ ₁<σ ₂<⋅⋅⋅. The authors were supported by the Office of Advanced Scientific Computing Research, Office of Science, US Department of Energy, under Contract DE-AC02-06CH11357.     

Asymptotic expansions for trapezoidal type product integration rules

This paper is concerned with the numerical integration of functions by piecewise polynomial product integration rules followed by application of extrapolation procedures. The studied rules can be considered as generalizations of the conventional trapezoidal rule. Euler-MacLaurin type asymptotic expansions are obtained with only even powers. Furthermore, numerical examples are given in order to show the effectiveness of these methods and a comparison with rules of similar characteristics is also made.     

The numerical evaluation of principal value integrals by finite-part integration

Summary We give a numerical formula for the evaluation of finite-part integrals of the form This method is very convenient for computational purposes since mere scalar products of certain weights and function values have to be calculated. Iff ^(2m-1) (s)=0,m=1,2, ..., [k/2],k>1 the above integral reduces to a generalized principal value integral.     

Asymptotic error expansions for hypersingular integrals

This paper presents quadrature formulae for hypersingular integrals $\int_a^b\frac{g(x)}{|x-t|^{1+\alpha }}\mathrm{d}x$ , where a?<?t?<?b and 0?<?α?≤?1. The asymptotic error estimates obtained by Euler–Maclaurin expansions show that, if g(x) is 2m times differentiable on [a,b], the order of convergence is O(h ^2μ ) for α?=?1 and O(h ^2μ???α ) for 0?<?α?<?1, where μ is a positive integer determined by the integrand. The advantages of these formulae are as follows: (1) using the formulae to evaluate hypersingular integrals is straightforward without need of calculating any weight; (2) the quadratures only involve g(x), but not its derivatives, which implies these formulae can be easily applied for solving corresponding hypersingular boundary integral equations in that g(x) is unknown; (3) more precise quadratures can be obtained by the Richardson extrapolation. Numerical experiments in this paper verify the theoretical analysis.     

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.     

Some expansions for integrals with weight functions

Two classes of expansions for integrals with arbitrary weight functions are derived. As one special case is obtained a generalization of Hermite's expansion. As a possible application is indicated the calculation of integrals with arbitrary weight functions.