A note on the numerical evaluation of finite integrals of oscillatory functions

A method is described for the numerical evaluation of integrals of the form ∫ _?1 ¹ f(x)K(m,x)dx, wheref(x) is smooth in [?1,1], whileK(m,x) is highly oscillatory for large values ofm.     

On numerical evaluation of integrals involving oscillatory Bessel and Hankel functions

Numerical Algorithms - A mixed-type formulation composed of Gauss-Laguerre quadrature and meshless collocation is presented for approximation of oscillatory integrals containing Hankel function of...     

A quadrature method for a class of strongly oscillatory infinite integrals

A constructive process is presented which leads to a class of quadrature formulas with any preassigned compound precision for the numerical integration of rapidly oscillating functions on (0, ) of the forme ^–x F(x, x), where is a large parameter andF(·,y) is periodic of period unity iny.1980 Mathematics Subject Classification, Primary 41A55, Scondary 41A60     

On rates of acceleration of extrapolation methods for oscillatory infinite integrals

The purpose of this work is to complement and expand our knowledge of the convergence theory of some extrapolation methods for the accurate computation of oscillatory infinite integrals. Specifically, we analyze in detail the convergence properties of theW- and -transformations of the author as they are applied to three integrals, all with totally different behavior at infinity. The results of the analysis suggest different convergence and acceleration of convergence behavior for the above mentioned transformations on the different integrals, and they improve considerably those that can be obtained from the existing convergence theories.     

A three-point formula for numerical quadrature of oscillatory integrals with variable frequency

A simple three-point formula is constructed for the evaluation of general oscillatory integrals.A rigorous derivation of the local error term is presented, and the implications to high frequency oscillations are discussed.Simple examples given include integrals with variable frequency for which the usual Filon formula would be inappropriate. For cases where Filon's formula is appropriate, the new formula appears to be computationally more efficient.The main application of the formula is to an example chosen from a class of integrals arising in the theory of water waves on a sloping beach. Comparison with exact results is possible from the work of Stoker [16] for a case which, whilst special in the physical sense, does not simplify the integral involved.In all cases the implementation of the formula is as straightforward as the implementation of the ordinary Simpson Rule.     

An automatic integration procedure for infinite range integrals involving oscillatory kernels

Let the real functionsK(x) andL(x) be such thatM(x)=K(x)+iL(x)=e^ix g(x), whereg(x) is infinitely differentiable for all largex and is non-oscillatory at infinity. We develop an efficient automatic quadrature procedure for numerically computing the integrals _a K(t)f(t) and _a L(t)f(t)dt, where the functionf(t) is smooth and nonoscillatory at infinity. One such example for which we also provide numerical results is that for whichK(x)=J (x) andL(x)=Y (x), whereJ (x) andY (x) are the Bessel functions of order . The procedure involves the use of an automatic scheme for Fourier integrals and the modified W-transformation which is used for computing oscillatory infinite integrals.     

A note on an extension of extrapolative techniques for a class of infinite oscillatory integrals

Recent work on extrapolative processes for a class of infinite oscillatory integrals is extended by a procedure which involves the unknown abscissae at which the remainder integral vanishes identically. The asymptotic proximity of these abscissae to a known set is used as a basis for further extrapolation.Results show that the method can be useful for the very slowly converging integrals whose amplitudes decay only algebraically.     

Efficient evaluation of highly oscillatory acoustic scattering surface integrals

We consider the approximation of some highly oscillatory weakly singular surface integrals, arising from boundary integral methods with smooth global basis functions for solving problems of high frequency acoustic scattering by three-dimensional convex obstacles, described globally in spherical coordinates. As the frequency of the incident wave increases, the performance of standard quadrature schemes deteriorates. Naive application of asymptotic schemes also fails due to the weak singularity. We propose here a new scheme based on a combination of an asymptotic approach and exact treatment of singularities in an appropriate coordinate system. For the case of a spherical scatterer we demonstrate via error analysis and numerical results that, provided the observation point is sufficiently far from the shadow boundary, a high level of accuracy can be achieved with a minimal computational cost.     

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.     

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 numerical evaluation of singular integrals

Product-integration rules of the form _–1 ¹ k(x)f(x)dx _i ⁼¹ⁿ w _ni f(x _ni ) are studied, with the points {w _ni } chosen to be the zeros of certain orthogonal polynomials, and the weights {w _ni } chosen to make the rule exact iff is any polynomial of degree less thann. If, in particular, the points are the Chebyshev points, and ifk L _p [–1, 1] for somep>1, then it is shown that the rule converges to the exact result for all continuous functionsf. With this choice of points, the practical application of the rule is shown to be straightforward in many cases, and to yield satisfactory rates of convergence. The casek(x)=|–x|, >–1, is studied in detail. Results of a similar, but weaker, kind are also obtained for other choices of the points {x _ni }.     

A recursive algorithm by the moments method to evaluate a class of numerical integrals over an infinite interval

A special recursive algorithm is built by a three-term recursive formula with coefficients evaluated by the moments method.A new functionalc(·) is studied over any function space that contains the polynomial space and it is shown that such a functional is positive definite, enabling us to use the advantages of such a property on the zeros of orthogonal polynomials for such a functional. A comparison is presented of the numerical advantages of such a method with respect to the Laguerre polynomials.     

Efficient numerical methods for Cauchy principal value integrals with highly oscillatory integrands

Numerical Algorithms - This paper focus on the numerical evaluation of the Cauchy principal value integrals with oscillatory integrands where α, β > ??1,??1...     

A class of infinite sums and integrals

In this paper closed-form sums are given for various slowly-convergent infinite series which arise essentially from the differentiation of Dirichlet -series. Some associated integrations are also considered. A small number of the results appear in standard tables, but most seem to be new.

     

The numerical evaluation of certain multivariate normal integrals

We show that a multivariate normal integral with tridiagonal covariance matrix can be computed efficiently using iterated integration.     

The numerical evaluation of singular integrals with coth-kernel

Singular integrals with hyperbolic cotangent kernel present their own numerical problems because of the poles of the kernel located in the complex plane. We write such integrals as ordinary Cauchy principal value integrals involving an appropriate (nonclassical) weight function and apply quadrature methods of Gaussian and interpolatory type. The most accurate one is based on Gauss-Christoffel quadrature relative to the weight function in question. Its error is studied both by real-and complex-variable techniques. Numerical examples are given to illustrate the theory.The work of the first author was supported in part by the National Science Foundation under grant DCR-8320561.     

On the evaluation of Cauchy principal value integrals of oscillatory functions

The problem of the numerical evaluation of Cauchy principal value integrals of oscillatory functions , where −1<τ<1, has been discussed. Based on analytic continuation, if f is analytic in a sufficiently large complex region G containing [−1, 1], the integrals can be transformed into the problems of integrating two integrals on [0,+∞) with the integrand that does not oscillate, and that decays exponentially fast, which can be efficiently computed by using the Gauss-Laguerre quadrature rule. The validity of the method has been demonstrated in the provision of two numerical experiments and their results.     

Some phases of oscillatory integrals

The first example of a phase is presented for which Arhold’s conjecture on the validity of uniform estimates for oscillatory integrals with maximal singularity index is true, while his conjecture on the semicontinuity of the singularity index is false. A rough upper bound for the Milnor number such that the latter conjecture fails is obtained. The corresponding counterexample is simpler than Varchenko’s well-known counterexample to Arnold’s conjecture on the semicontinuity of the singularity index. This gives hope to decrease codimension and the Milnor number for which the conjecture on the semicontinuity of the singularity index fails.     

The numerical evaluation of cauchy principal values of integrals by Romberg integration

The problem considered is that of evaluating numerically an integral of the form where the integrand has one or more simple poles in the interval (O,p). Modified forms of the trapezoidal and mid-ordinate rules, taking account of the singularities, are obtained; it is then shown that the resulting approximations can be extrapolated by Romberg's method. Further modifications to deal with the case when the integrand has an integrable branch singularity at one or both ends of the interval of integration are also briefly discussed.