Gaussian Quadrature Rules with Simple Node-Weight Relations

In this paper, we consider the symmetric Gaussian and L-Gaussian quadrature rules associated with twin periodic recurrence relations with possible variations in the initial coefficient. We show that the weights of the associated Gaussian quadrature rules can be given as rational functions in terms of the corresponding nodes where the numerators and denominators are polynomials of degree at most 4. We also show that the weights of the associated L-Gaussian quadrature rules can be given as rational functions in terms of the corresponding nodes where the numerators and denominators are polynomials of degree at most 5. Special cases of these quadrature rules are given. Finally, an easy to implement procedure for the evaluation of the nodes is described.     

Numerical solution of a Fredholm integro-differential equation modelling neural networks

We compare piecewise linear and polynomial collocation approaches for the numerical solution of a Fredholm integro-differential equations modelling neural networks. Both approaches combine the use of Gaussian quadrature rules on an infinite interval of integration with interpolation to a uniformly distributed grid on a bounded interval. These methods are illustrated by numerical experiments on neural networks equations.     

The Construction of Reducible Quadrature Rules for Volterra Integral and Integro-differential Equations

A formal relationship between quadrature rules and linear multistepmethods for ordinary differential equations is exploited forthe generation of quadrature weights. Employing the quadraturerules constructed in this way, step-by-step methods for secondkind Volterra integral equations and integro-differential equationsare defined and convergence and stability results are presented. The construction of the quadrature rules generated by the backwarddifferentiation formulae is discussed in detail. The use ofthese rules for the solution of Volterra type equations is proposedand their good performance is demonstrated by numerical experiments.     

Structured eigenvalue problems for rational gauss quadrature

The connection between Gauss quadrature rules and the algebraic eigenvalue problem for a Jacobi matrix was first exploited in the now classical paper by Golub and Welsch (Math. Comput. 23(106), 221–230, 1969). From then on many computational problems arising in the construction of (polynomial) Gauss quadrature formulas have been reduced to solving direct and inverse eigenvalue problems for symmetric tridiagonals. Over the last few years (rational) generalizations of the classical Gauss quadrature formulas have been studied, i.e., formulas integrating exactly in spaces of rational functions. This paper wants to illustrate that stable and efficient procedures based on structured numerical linear algebra techniques can also be devised for the solution of the eigenvalue problems arising in the field of rational Gauss quadrature.     

Solving multivariable mathematical models by the quadrature and cubature methods

The utilization and generalization of quadrature and cubature approximations for numerical solution of mathematical models of multivariable transport processes involving integral, differential, and integro-differential operators, and for numerical interpolation and extrapolation, are presented. The methodology for determination of the quadrature and cubature weights for composite operators is developed to accommodate for general functional representations. Application of these methods is demonstrated by solving two-dimensional steady-state and one-dimensional transient-state problems. The solutions are compared with exact-analytical solutions to evaluate the performance of these methods. It is demonstrated that the quadrature and cubature approximations are simple and universal; i.e., the same formula is applicable irrespective of the order of accuracy of the numerical approximation, the type of linear operator, and the number of temporal and/or spatial variables. Since the quadrature and cubature methods can produce solutions with sufficient accuracy even when using fewer discrete points, both the programming task and computational effort are reduced considerably. Therefore, the quadrature and cubature methods appear to be very practical in solving the mathematical models of a variety of transport processes. © 1994 John Wiley & Sons, Inc.     

Stable high-order quadrature rules with equidistant points

Newton-Cotes quadrature rules are based on polynomial interpolation in a set of equidistant points. They are very useful in applications where sampled function values are only available on a regular grid. Yet, these rules rapidly become unstable for high orders. In this paper we review two techniques to construct stable high-order quadrature rules using equidistant quadrature points. The stability follows from the fact that all coefficients are positive. This result can be achieved by allowing the number of quadrature points to be larger than the polynomial order of accuracy. The computed approximations then implicitly correspond to the integral of a least squares approximation of the integrand. We show how the underlying discrete least squares approximation can be optimised for the purpose of numerical integration.     

Fast Construction of the Fejér and Clenshaw–Curtis Quadrature Rules

We present an elegant algorithm for stably and quickly generating the weights of Fejér’s quadrature rules and of the Clenshaw–Curtis rule. The weights for an arbitrary number of nodes are obtained as the discrete Fourier transform of an explicitly defined vector of rational or algebraic numbers. Since these rules have the capability of forming nested families, some of them have gained renewed interest in connection with quadrature over multi-dimensional regions. AMS subject classification (2000) 65D32, 65T20, 65Y20     

A nodal spline interpolant for the Gregory rule of even order

Summary The Gregory rule is a well-known example in numerical quadrature of a trapezoidal rule with endpoint corrections of a given order. In the literature, the methods of constructing the Gregory rule have, in contrast to Newton-Cotes quadrature,not been based on the integration of an interpolant. In this paper, after first characterizing an even-order Gregory interpolant by means of a generalized Lagrange interpolation operator, we proceed to explicitly construct such an interpolant by employing results from nodal spline interpolation, as established in recent work by the author and C.H. Rohwer. Nonoptimal order error estimates for the Gregory rule of even order are then easily obtained.     

A remark on the application of interpolatory quadrature rules to the numerical solution of singular integral equations

A Cauchy type singular integral equation of the first or the second kind can be numerically solved either directly or after its reduction (by the usual regularization procedure) to an equivalent Fredholm integral equation of the second kind. The equivalence of these two methods (that is, the equivalence both of the systems of linear algebraic equations to which the singular integral equation is reduced and of the natural interpolation formulae) is proved in this paper for a class of Cauchy type singular integral equations of the first kind and of the second kind (but with constant coefficients) for general interpolatory quadrature rules under sufficiently mild assumptions. The present results constitute an extension of a series of previous results concerning only Gaussian quadrature rules, based on the corresponding orthogonal polynomials and their properties.     

Convergence of the piecewise linear approximation and collocation method for a hypersingular integral equation on a closed surface

We study the numerical solution of a linear hypersingular integral equation arising when solving the Neumann boundary value problem for the Laplace equation by the boundary integral equation method with the solution represented in the form of a double layer potential. The integral in this equation is understood in the sense of Hadamard finite value. We construct quadrature formulas for the integral occurring in this equation based on a triangulation of the surface and an application of the linear approximation to the unknown function on each of the triangles approximating the surface. We prove the uniform convergence of the quadrature formulas at the interpolation nodes as the triangulation size tends to zero. A numerical solution scheme for this integral equation based on the suggested quadrature formulas and the collocation method is constructed. Under additional assumptions about the shape of the surface, we prove a uniform estimate for the error in the numerical solution at the interpolation nodes.     

Numerical implementations of Cauchy-type integral equations

In this paper, a good interpolation formulae are applied to the numerical solution of Cauchy integral equations of the first kind with using some Chebyshev quadrature rules. To demonstrate the effectiveness of the Radau-Chebyshev with respect to the olds, [6], [7], [8] and [12], some examples are given.     

On the positivity of linear weights in WENO approximations

High order accurate weighted essentially non-oscillatory （WENO） schemes have been used extensively in numerical solutions of hyperbolic partial differential equations and other convection dominated problems. However the WENO procedure can not be applied directly to obtain a stable scheme when negative linear weights are present. In this paper, we first briefly review the WENO framework and the role of linear weights, and then present a detailed study on the positivity of linear weights in a few typical WENO procedures, including WENO interpolation, WENO reconstruction and WENO approximation to first and second derivatives, and WENO integration. Explicit formulae for the linear weights are also given for these WENO procedures. The results of this paper should be useful for future design of WENO schemes involving interpolation, reconstruction, approximation to first and second derivatives, and integration procedures.     

Quadrature rules based on partial fraction expansions

Quadrature rules are typically derived by requiring that all polynomials of a certain degree be integrated exactly. The nonstandard issue discussed here is the requirement that, in addition to the polynomials, the rule also integrates a set of prescribed rational functions exactly. Recurrence formulas for computing such quadrature rules are derived. In addition, Fejér's first rule, which is based on polynomial interpolation at Chebyshev nodes, is extended to integrate also rational functions with pre-assigned poles exactly. Numerical results, showing a favorable comparison with similar rules that have been proposed in the literature, are presented. An error analysis of a representative test problem is given. This revised version was published online in June 2006 with corrections to the Cover Date.     

On modified Gregory rules based on a generalised mixed interpolation formula

A mixed interpolation function in its generalised form is used to derive the generalised modified Gregory formulae These formulae are expressed in the form of the classical rules along with two correction terms. The error terms are briefly discussed. The newly derived quadrature formulae are tested with certain numerical examples, which shows the efficiency of the generalised modified rules over classical Gregory rules, as well as the modified rules based on the mixed trigonometric interpolation. The importance of the error terms are also discussed.     

Properties of interpolatory quadrature with equidistant nodes on the unit circle

In this paper, we study the computation of the moments associated to rational weight functions given as a power spectrum with known or unknown poles of any order in the interior of the unit disc. A recursive algebraic procedure is derived that computes the moments in a finite number of steps. We also study the associated interpolatory quadrature formulas with equidistant nodes on the unit circle. Explicit expressions are given for the positive quadrature weights in the case of a polynomial weight function. For rational weight functions with simple poles, mostly real or uniformly distributed on a circle in the open unit disc, we also obtain expressions for the quadrature weights and sufficient conditions that guarantee that they are positive. The Poisson kernel is a simple example of a rational weight function, and in the last section, we derive an asymptotic expansion of the quadrature error.     

Symmetric quadrature rules for tetrahedra based on a cubic close-packed lattice arrangement

A family of quadrature rules for integration over tetrahedral volumes is developed. The underlying structure of the rules is based on the cubic close-packed (CCP) lattice arrangement using 1, 4, 10, 20, 35, and 56 quadrature points. The rules are characterized by rapid convergence, positive weights, and symmetry. Each rule is an optimal approximation in the sense that lower-order terms have zero contribution to the truncation error and the leading-order error term is minimized. Quadrature formulas up to order 9 are presented with relevant numerical examples.     

A product integration method for the approximation of the early exercise boundary in the American option pricing problem

Integral equation methods are now becoming well‐established tools in the study of financial models used in theory and practice. In this paper, we investigate the fully nonlinear weakly singular nonstandard Volterra integral equations representing the early exercise boundary of American option contracts, which gained popularity in recent years. We propose a product integration approach based on linear barycentric rational interpolation to solve the problem. The price of the option will then be computed using the obtained approximation of the early exercise boundary and a barycentric rational quadrature. The convergence of the approximation scheme will also be analyzed. Finally, some numerical experiments based on the introduced method are presented and compared with some exiting approaches.     

Two new formulas for the numerical evaluation of the Hilbert Transform

We develop two algorithms for the numerical evaluation of the semi-infinite Hilbert Transform of functions with a given algebraic behaviour at the origin and at infinity. The first algorithm is connected with a Gauss-Jacobi type quadrature formula for unbounded intervals; the second is based on a rational Bernstein-type operator. Error estimates for different classes of functions are shown. Finally numerical examples are given, comparing the rules among themselves.     

Computational aspects of Boolean cubature

Boolean methods of interpolation [1,4] have been applied to construct multivariate quadrature rules for periodic functions of Korobov classes which are comparable with lattice rules of numerical integration [6,7]. In particular, we introducedd-variate Boolean trapezoidal rules [3,4] andd-variate Boolean midpoint rules [2,4]. The basic tools for constructing Boolean midpoint rules are Boolean midpoint sums. It is the purpose of this paper to use a modification of these Boolean midpoint sums to compute Boolean trapezoidal rules in an efficient way.     

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.