Numerical analysis of a fast integration method for highly oscillatory functions

The integration of systems containing Bessel functions is a central point in many practical problems in physics, chemistry and engineering. This paper presents a new numerical analysis for the collocation method presented by Levin for and gives more accurate error analysis about the integration of systems containing Bessel functions. The effectiveness and accuracy of the quadrature is tested for Bessel functions with large arguments. AMS subject classification (2000)  65D32, 65D30     

Uniqueness and computation of Gaussian interval quadrature formula for Jacobi weight function

Summary. In this paper we prove the uniqueness of the Gaussian interval quadrature formula for the Jacobi weight function. An algorithm for numerical construction has also investigated and some suitable solutions are proposed. For the special case of Chebyshev weight of the first kind and the special set of lengths we give an analytic solution.Mathematics Subject Classification (2000): 65D32, 65D30, 41A55The authors were supported in part by the Serbian Ministry of Science and Environmental Protection (Project #2002: Applied Orthogonal Systems, Constructive Approximation and Numerical Methods).     

Computation of Bessel and Airy Functions and of Related Gaussian Quadrature Formulae

Procedures are described for the high-precision calculation of the modified Bessel function K (x), 0 < < 1, and the Airy function Ai(x), for positive arguments x, as pre-requisites for generating Gaussian quadrature rules having these functions as weight function.     

An efficient and reliable quadrature algorithm for integration with respect to binomial measures

In this work numerical methods for integration with respect to binomial measures are considered. Binomial measures are examples of fractal measures and arise when multifractal properties are investigated. Interpolatory quadrature rules are considered. An automatic integrator with local quadrature rules that generalize the five points Newton Cotes formula and error estimates based on null rules is then described. Numerical tests are performed to verify the efficiency and accuracy of the method. These tests confirm that the automatic integrator turns out to be as good as one of the best known quadrature algorithms with respect to the Lebesgue measure. AMS subject classification (2000)  28A25, 60G18, 65D30, 65D32, 68M15     

对振荡函数数值积分方法的进一步探讨

本文在 [1 ]等成果的基础上 ,对振荡函数数值积分的方法做了进一步的探讨 ,给出了一种代数精确度更高、具有函数振荡越剧烈求积结果越精确的特点的、优于 [1 ]的新的对振荡函数的 Gauss型积分 .     

Integration of Polyharmonic Functions

The results in this paper are motivated by two analogies. First, -harmonic functions in are extensions of the univariate algebraic polynomials of odd degree . Second, Gauss' and Pizzetti's mean value formulae are natural multivariate analogues of the rectangular and Taylor's quadrature formulae, respectively. This point of view suggests that some theorems concerning quadrature rules could be generalized to results about integration of polyharmonic functions. This is done for the Tchakaloff-Obrechkoff quadrature formula and for the Gaussian quadrature with two nodes.

     

Why Gaussian quadrature in the complex plane?

This paper synthesizes formally orthogonal polynomials, Gaussian quadrature in the complex plane and the bi-conjugate gradient method together with an application. Classical Gaussian quadrature approximates an integral over (a region of) the real line. We present an extension of Gaussian quadrature over an arc in the complex plane, which we call complex Gaussian quadrature. Since there has not been any particular interest in the numerical evaluation of integrals over the long history of complex function theory, complex Gaussian quadrature is in need of motivation. Gaussian quadrature in the complex plane yields approximations of certain sums connected with the bi-conjugate gradient method. The scattering amplitude c ^T A ^–1 b is an example where A is a discretization of a differential–integral operator corresponding to the scattering problem and b and c are given vectors. The usual method to estimate this is to use c ^T x ^(k). A result of Warnick is that this is identically equal to the complex Gaussian quadrature estimate of 1/. Complex Gaussian quadrature thereby replaces this particular inner product in the estimate of the scattering amplitude.     

Sensitivity of the Lanczos recurrence to Gaussian quadrature data: How malignant can small weights be?

Stability of passing from Gaussian quadrature data to the Lanczos recurrence coefficients is considered. Special attention is paid to estimates explicitly expressed in terms of quadrature data and not having weights in denominators. It has been shown that the recent approach, exploiting integral representation of Hankel determinants, implies quantitative improvement of D. Laurie’s constructive estimate.It has also been demonstrated that a particular implementation on the Hankel determinant approach gives an estimate being unimprovable up to a coefficient; the corresponding example involves quadrature data with a small but not too small weight. It follows that polynomial increase of a general case upper bound in terms of the dimension is unavoidable.     

Error estimates for Gaussian quadrature

This paper is concerned with estimating the Gaussian quadrature error in the numerical integration of an analytic function$\text{?(x)}$ over the interval?1<x<1. An approximate expression for the quadrature error is given in terms of a contour integral in the complex plane. Known techniques can be applied directly to this contour integral to obtain quadrature error estimates. This approach has the advantage of avoiding the computation of high order derivatives as required in classical Gaussian quadrature error representations.     

Derivatives of a finite class of orthogonal polynomials related to inverse Gamma distribution

In this work, we consider derivatives of a finite class of orthogonal polynomials with respect to weight function which is related to the probability density function of the inverse gamma distribution over the positive real line. General properties for this derivative class such as orthogonality, Rodrigues’ formula, recurrence relation, generating function and various other related properties such as self-adjoint form and normal form are indicated. The corresponding Gaussian quadrature formulae are introduced with examples. These examples are provided to support the advantages of considering the derivatives class of the finite class of orthogonal polynomials related to inverse gamma distribution. The orthogonality property related to the Fourier transform of the derivative class under discussion is also given.     

Gaussian Versus Optimal Integration of Analytic Functions

We consider error estimates for optimal and Gaussian quadrature formulas if the integrand is analytic and bounded in a certain complex region. First, a simple technique for the derivation of lower bounds for the optimal error constants is presented. This method is applied to Szeg?-type weight functions and ellipses as regions of analyticity. In this situation, the error constants for the Gaussian formulas are close to the obtained lower bounds, which proves the quality of the Gaussian formulas and also of the lower bounds. In the sequel, different regions of analyticity are investigated. It turns out that almost exclusively for ellipses, the Gaussian formulas are near-optimal. For classes of simply connected regions of analyticity, which are additionally symmetric to the real axis, the asymptotic of the worst ratio between the error constants of the Gaussian formulas and the optimal error constants is calculated. As a by-product, we prove explicit lower bounds for the Christoffel-function for the constant weight function and arguments outside the interval of integration. September 7, 1995. Date revised: October 25, 1996.     

Talbot quadratures and rational approximations

Many computational problems can be solved with the aid of contour integrals containing e ^z in the integrand: examples include inverse Laplace transforms, special functions, functions of matrices and operators, parabolic PDEs, and reaction-diffusion equations. One approach to the numerical quadrature of such integrals is to apply the trapezoid rule on a Hankel contour defined by a suitable change of variables. Optimal parameters for three classes of such contours have recently been derived: (a) parabolas, (b) hyperbolas, and (c) cotangent contours, following Talbot in 1979. The convergence rates for these optimized quadrature formulas are very fast: roughly O(3^-N ), where N is the number of sample points or function evaluations. On the other hand, convergence at a rate apparently about twice as fast, O(9.28903^-N ), can be achieved by using a different approach: best supremum-norm rational approximants to e ^z for z∈(–∞,0], following Cody, Meinardus and Varga in 1969. (All these rates are doubled in the case of self-adjoint operators or real integrands.) It is shown that the quadrature formulas can be interpreted as rational approximations and the rational approximations as quadrature formulas, and the strengths and weaknesses of the different approaches are discussed in the light of these connections. A MATLAB function is provided for computing Cody–Meinardus–Varga approximants by the method of Carathéodory–Fejér approximation. In memory of Germund Dahlquist (1925–2005).AMS subject classification (2000) 65D30, 41A20     

Error bounds for Romberg quadrature

Denote by the error of a Romberg quadrature rule applied to the function f. We determine approximately the constants in the bounds of the types and for all classical Romberg rules. By a comparison with the corresponding constants of the Gaussian rule we give the statement “The Gaussian quadrature rule is better than the Romberg method” a precise meaning. Received September 10, 1997 / Revised version received February 16, 1998     

AN EXTREMAL APPROACH TO BIRKHOFF QUADRATURE FORMULAS

1. Introduction and Main ResultsIn tfor paPer we shaJl use the ddstions and notations of [3l. Let E = (e'k)7t' kt. be anincidence matrir with entries consisting of zeros and ones and satisfying lEl:= Z.,* ei* = n + 1(here we allow a zero row ). Furthermore, in wha follOws we assume that(A) E satisfies the P6lya condition(B) all sequences of E in the interior rows, 0 < i < m + 1, are even.Let Sm denote the set of poiats X = (xo, z1 l "') xm, x.+1) fOr whichand Sm its clOusure. If some O…     

Numerical Quadratures for Hadamard Hypersingular Integrals

In this paper,we develop Gaussian quadrature formulas for the Hadamard fi- nite part integrals.In our formulas,the classical orthogonal polynomials such as Legendre and Chebyshev polynomials are used to approximate the density function f(x)so that the Gaussian quadrature formulas have degree n-1.The error estimates of the formulas are obtained.It is found from the numerical examples that the convergence rate and the accu- racy of the approximation results are satisfactory.Moreover,the rate and the accuracy can be improved by choosing appropriate weight functions.     

A note on generalized averaged Gaussian formulas

We have recently proposed a very simple numerical method for constructing the averaged Gaussian quadrature formulas. These formulas exist in many more cases than the real positive Gauss–Kronrod formulas. In this note we try to answer whether the averaged Gaussian formulas are an adequate alternative to the corresponding Gauss–Kronrod quadrature formulas, to estimate the remainder term of a Gaussian rule.     

Stopping functionals for Gaussian quadrature formulas

Gaussian formulas are among the most often used quadrature formulas in practice. In this survey, an overview is given on stopping functionals for Gaussian formulas which are of the same type as quadrature formulas, i.e., linear combinations of function evaluations. In particular, methods based on extended formulas like the important Gauss–Kronrod and Patterson schemes, and methods which are based on Gaussian nodes, are presented and compared.     

Absorbing boundary conditions and optimized Schwarz waveform relaxation

The numerical evaluation of the transforms in the title, and their inverses, is considered, using a variety of decomposition, truncation, and quadrature methods. Extensive numerical testing is provided and an application given to the numerical evaluation of the kernel of a Fredholm integral equation of interest in mixed boundary value problems on wedge-shaped domains. AMS subject classification (2000) 44A15, 65D30, 65R10     

Dependence structure between LIBOR rates by copula method

This paper discusses the correlation structure between London Interbank Offered Rates (LIBOR) by using the copula function. We start from one simplified model of A. Brace, D. Gatarek, and M. Musiela (1997) and find out that the copula function between two LIBOR rates can be expressed as a sum of an infinite series, where the main term is a distribution function with Gaussian copula. Partial differential equation method is used for deriving the copula expansion. Numerical results show that the copula of the LIBOR rates and Gaussian copula are very close in the central region and differ in the tail, and the Gaussian copula approximation to the copula function between the LIBOR rates provides satisfying results in the normal situation.     

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.