Gauss-Legendre Quadrature Rule Research Articles

Computing stability of differential equations with bounded distributed delays

This paper deals with the stability analysis of scalar delay integro-differential equations (DIDEs). We propose a numerical scheme for computing the stability determining characteristic roots of DIDEs which involves a linear multistep method as time integration scheme and a quadrature method based on Lagrange interpolation and a Gauss–Legendre quadrature rule. We investigate to which extent the proposed scheme preserves the stability properties of the original equation. We derive and prove a sufficient condition for (asymptotic) stability of a DIDE (with a constant kernel) which we call RHP-stability. Conditions are obtained under which the proposed scheme preserves RHP-stability. We compare the obtained results with corresponding ones using Newton–Cotes formulas. Results of numerical experiments on computing the stability of DIDEs with constant and nonconstant kernel functions are presented.

TWO SHARP OSTROWSKI-LIKE INEQUALITIES AND APPLICATIONS

Two sharp inequalities are derived. The first of them is a sharp inequality which gives an error bound for a Gauss-Legendre quadrature rule. The second is a sharp inequality which gives an error bound for a Radau quadrature rule. These inequalities enlarge the applicability of the corresponding quadrature rules with respect to the obtained error bounds. Applications in numerical integration are also given.

An efficient transformation with Gauss quadrature rule for weakly singular integrals

AbstractThe superiority of the use of non‐linear transformations, in numerical evaluation of the weakly singular integrals, has been recently demonstrated. In this paper, we propose a new non‐linear transformation containing a parameter which has most properties of the sigmoidal transformation. The sigmoidal behaviour of the present transformation, ΩEm(b;x) is governed not only by the order m but also by the parameter b≠0. It is shown that the presented transformation together with the Gauss–Legendre quadrature rule can better the asymptotic truncation error of the approximation effectively by controlling the value of b.Comparison of the present method with a well‐known sigmoidal transformation, in numerical examples, shows that more accurate evaluation of weakly singular integrals can be obtained so long as the number of integration points is moderately large. Copyright © 2001 John Wiley & Sons, Ltd.

Integration of singular Galerkin-type boundary element integrals for 3D elasticity problems

A Galerkin approximation of both strongly and hypersingular boundary integral equation (BIE) is considered for the solution of a mixed boundary value problem in 3D elasticity leading to a symmetric system of linear equations. The evaluation of Cauchy principal values (v. p.) and finite parts (p. f.) of double integrals is one of the most difficult parts within the implementation of such boundary element methods (BEMs). A new integration method, which is strictly derived for the cases of coincident elements as well as edge-adjacent and vertex-adjacent elements, leads to explicitly given regular integrand functions which can be integrated by the standard Gauss-Legendre and Gauss-Jacobi quadrature rules. Problems of a wide range of integral kernels on curved surfaces can be treated by this integration method. We give estimates of the quadrature errors of the singular four-dimensional integrals.

Minimax approximations to the zeros of Pn( x) and Gauss-Legendre quadrature

This paper is concerned with the numerical development of some minimax trigonometric approximations to the positive zeros of the nth Legendre polynomial P n ( x). One of the approximation formulas we derive yields at least 4.2 significant decimal digits of accuracy for any n ⩾ 2, and can be used to furnish initial guesses in an iterative method for the computation of the zeros of P n ( x) to nearly full machine accuracy. This approach avoids some of the computational complexity associated with the selection of appropriate initial guesses for use in a special 5th order scheme previously developed by the first author for the numerical computation of the abscissas required in the n-point Gauss-Legendre quadrature rule.

Gaussian integration of Chebyshev polynomials and analytic functions

Explicit bounds for the quadrature error of thenth Gauss-Legendre quadrature rule applied to themth Chebyshev polynomial are derived. They are precise up to the orderO(m4n−6). As an application, error constants for classes of functions which are analytic in the interior of an ellipse are estimated. The location of the maxima of the corresponding kernel function is investigated.

Asymptotic expansions of an integral of the free-space Green's function

In this paper, we derive the expressions of the three first terms in the asymptotic expansions of the integral Im(, z; ', z') = ∫x o R(') cos(m') d' as k→ ∞. R(') is the distance between point P = (p, z, = 0) and a point P' = (p', z', ') situated on a circular loop of radius ' and axis z. The numerical values of Im derived from these asymptotic expansions are compared with the exact values computed with a Gauss-Legendre quadrature rule. Accurate results are obtained for sufficiently large values of k even when P is close to the loop, and a reliable estimate of the error is defined.

On bicubic transformation for the numerical evaluation of Cauchy principal value integrals

AbstractRecently a bicubic transformation was introduced to numerically compute the Cauchy principal value (CPV) integrals. Numerical results show that this new method converges faster than the conventional Gauss–Legendre quadrature rule when the integrand contains different types of singularity. Assume η is the singular point of a CPV integral. The point η divides the interval [−1, 1] into two parts: [−1, η] and [η, 1]. The bicubic transformation maps the intervals [−1, η] and [η, 1] to the interval [−1, 1] with the following constraints: it maps the point η − ϵ to μn, and η + ϵ to −μn, where μn is the largest Gaussian point of an n‐point Gauss–Legendre quadrature rule, and ϵ is a user‐supplied constant. The n‐point Gauss–Legendre quadruture rule is then applied. In contrast to ordinary expectation, further numerical experiment shows that smaller ϵ does not always produce better results. In this paper we are concerned with the selection of ϵ to yield rapid convergence of numerical integration when the bicubic transformation method is applied.

Treatment of the ‘pole’ at infinity in classical numerical integration with computer algebra software

The error in the classical interpolatory (including Gaussian) numerical integration rules for analytic functions can be considered to be due to a ‘pole’ at infinity. The residue of this ‘pole’ is approximately taken into account by using Taylor-Maclaurin series expansions at the origin- and asymptotic expansions at infinity. The computer algebra system Maple V was found appropriate and used in this task. The cases of the Gauss-Chebyshev and the Gauss-Legendre quadrature rules were considered in some detail. Finally, numerical results in applications both for regular and for Cauchy-type principal value integrals are also presented and the strong improvement in the numerical results by the present approach is illustrated.

Numerical integration of functions with a sharp peak at or near one boundary using möbius transformations

A class of coordinate transformations depending on a single parameter is studied as quadrature tool. Working rules for the choice of the parameter are proposed. Numerical tests for the method are presented. They show that these coordinate transformations, when combined with Gauss-Legendre quadrature rules, are well suited for the numerical integration of functions possessing a sharp peak at or near one boundary of the interval of integration. A method to combine the transformations with automatic quadrature routines is also proposed; it seems to be useful for the numerical evaluation of integrals with the same kind of integrand behavior.

Remarks on the gaussian quadrature rule for finite-part integrals with a second-order singularity

The Gaussian quadrature rule for finite-part integrals with a second-order singularity (that is, weight function equal to 1/ y 2 on [0,1]), proposed by Kutt, is revisited here. It is proved that the related orthogonal polynomials can be expressed as a linear combination of three successive shifted Legendre polynomials. The coefficients in this linear combination are expressed in an exact, closed, and simple form either by using the nodes and weights of the classical Gauss-Legendre quadrature rule on [0,1] or by using simple finite sums. Formulae for the weights of the relevant quadrature rule are also proposed. Numerical results, derived in numerical experiments, verify also the convergence of the considered numerical integration rule. A large variety of applications of the present results to applied mechanics and engineering is reported in brief, but the main application is to the hypersingular integral equations of crack problems in the theory of two- and three-dimensional elasticity.

Shortened quadrature rules for finite elements

A numerical method to calculate Rieman's integrals for complete polynomials on R n space is presented. The method is based on the direct elimination of some polynomial tenns, whose contribution to the value of the integral is zero for particular positions of the sampling points. Using the proposed scheme to carry out the integrations occurring in the finite element formulations leads to a remarkable reduction in the computing time, with respect to the employment of the ordinary Gauss-Legendre quadrature rule. Moreover, if this rule is available at least as an alternative, those indeterminations can be removed, that depend on local singularities arising in the transformations performed before the integrations themselves. Tables are reported giving the normalised coordinates and the weighting factors of the sampling points, for various quadrature orders and polynomial degrees. Examples are produced from which an estimate can be drawn of the computing savings obtained by the method.

Some observations on Gauss-Legendre quadrature error estimates for analytic functions

This paper is concerned with estimates for the error when a Gauss-Legendre quadrature rule is used to numerically integrate an analytic function. The error in a standard approximation to the kernel function, which appears in a contour integral representation for the quadrature error, is investigated for this purpose. Applications to meromorphic functions with simple poles are discussed.

Positivity of the weights of extended Gauss-Legendre quadrature rules

We show that the weights of extended Gauss-Legendre quadrature rules are all positive.

Cubature Error Bounds for Gauss-Legendre Product Rules

In this paper we derive cubature error bounds for the product rule generated by two Gauss-Legendre quadrature rules. An analogue to the fundamental theorem of algebra, for monosplines, is used to characterize the kernels in the two-dimensional form of Peano’s theorem.Tables are given which can be used to compute the error coefficients we obtain.

Gauss-Legendre Numerical Integrations over a Quadrilateral Element in Closed Form

In this paper we investigate the stiffness matrix of a general quadrilateral element in closed form using n x n Gauss-Legendre quadrature rule. For this, we propose four types of nodal coordinate transformation. The terms of the matrix are divided into two groups, namely - diagonal and non-diagonal. Only one term (called leading) from each group is computed, and then the remaining fourteen terms are computed from these two leading terms exploited one of the proposed types of coordinate transformation. This leads us a great savings in computational time and memory space. In order to compute the matrix we use these transformations in two ways, and thus two algorithms are given to generate the matrix. Finally, numerical example is given to verify the effectiveness of the present formulation. Keywords: Gauss-Legendre quadrature; Numerical integration; Quadrilateral finite element; Stiffness matrix; Closed form. DOI: http://dx.doi.org/10.3329/bjsir.v46i3.9050 BJSIR 2011; 46(3): 399-405