Symmetric gaussian quadrature formulae for tetrahedronal regions

Quadrature formulae of degrees 2 to 6 are presented for the numerical integration of a function over tetrahedronal regions. The formulae presented are of Gaussian type and fully symmetric with respect to the four vertices of the tetrahedron.     

On Chebyshev quadrature for ultraspherical weight functions

It is well-known that for the ultraspherical weight function (1-x²)^λ-1/2 there exist no Chebyshev quadrature formulae in the strict sense having n nodes, where n is sufficiently large and λ>0, whereas on the other hand for λ=0 every Gaussian quadrature formulae is a Chebyshev formula in the strict sense. In this paper we study the open question of Chebyshev quadrature for λ <0. It is shown that there exists no Chebyshev quadrature formula in the strict sense having more than two nodes for λ≤λ₀=-.30056... (for definition of λ₀ see (1.8) below). Moreover, results are obtained for Chebyshev-type formulae and Chebyshev formulae of closed type. For the remaining values of λ (λ₀<λ<0) numerical results are given.     

On the variance of the Gaussian quadrature rule

Denote by the Gaussian quadrature rule for the integral . We give a simple explicit expression for the “variance”. The method can be used to obtain similar results for the Lobatto rule. Received: June 1997 / Accepted: August 1997     

On the definiteness of quadrature formulae of Clenshaw-Curtis type

The nondefiniteness of the Clenshaw-Curtis and related quadrature formulae having more than three nodes is shown.     

On the existence of optimal quadrature formulae for smooth functions

A general method showing the existence of optimal quadrature formulae with preassigned multiplicities of the nodes for classes of smooth functions is demonstrated. The main result is applied to the Hardy spaceH _∞ of analytic functions.     

Estimating errors of certain Gauss quadrature formulae

Using Cauchy's integral formula (as in [8]) we obtain the closed form error-estimates for the general Gauss-Chebyshev quadrature formulas. Some of the estimates of the errors of these quaratdures are excellent in the sense of Chawla [3].     

Integration points for triangles and tetrahedrons obtained from the gaussian quadrature points for a line

A new set of numerical integration points for triangles and tetrahedrons, derived from basic Gaussian quadrature points for line integrals, are presented.     

On some weight functions admitting (0, 2) quadrature formulae with a high algebraic degree of precision

The purpose of this paper is to find a class of weight functions μ for which there exist quadrature formulae of the form (1) $$\int_{ - 1}^1 {\mu (x) f(x) dx \approx \sum\limits_{k = 1}^n {(a_k f(x_k ) + b_k f''(x_k ))} }$$ , which are precise for every polynomial of degree 2n.     

On the weights of Sard's quadrature formulas

We consider the weights of quadrature formulas which are optimal in the sense of Sard for the evaluation of weighted integrals. The nodes of the quadrature formulas have to obey certain restrictions, but all nodes common in quadrature theory are possible choices (e.g. the nodes of the Gauss-Legendre formulas). In this context, we give an estimate of the size of the weights, and we prove generalizations of the first and second conjecture of Meyers and Sard. As a consequence of the generalized first conjecture, Sard's quadrature formulas satisfy the Trapezoid and Circle Theorem, which were proven for Gauss-type quadrature formulas by Davis and Rabinowitz.     

On the evaluation of correction terms in Gauss-Legendre quadrature

In the numerical integration of analytic functions, the singularities of the integrand affect the rate of convergence of the quadrature. This convergence can be improved significantly by adding the residue correction terms for the poles of the integrand. But this needs the evaluation of the basis function and its corresponding second kind function with complex arguments. We indicate a simple and accurate method to evaluate the correction term involving the basis and its second kind functions in the case of Gauss-Legendre quadrature. This approach does not call for the evaluation of the hypergeometric functions.     

Extension of numerical quadrature formulae to cater for end point singular behaviours over finite intervals

The N-point Gaussian quadrature method is generalised to cater for various possible singular behaviours at the end points of the interval of integration at the expense of being algebraically exact for a polynomial of lower order than usual. Weights and abscissae are chosen to exactly integrate an integrand which is the sum of the singular functions and an arbitrary polynomial. This allows us to cater for several different end-point singularities in the same quadrature formula and in this way differs from published quadratures where a singular behaviour is incorporated in a weight function that multiplies an arbitrary polynomial. We present tables of weights and abscissae that cater for (i) logarithmic end point singularities and (ii) logarithmic plus inverse square root singular behaviours. Also a 10-point quadrature is presented that exactly caters for log(x), x^-1/4,x^-1/2,x^-3/4 singular behaviours and is recommended for programmable calculator use. Finally a brief comparison study of the various (10-point) quadratures herein considered is made.     

On the determination of the weights in multi-dimensional numerical quadrature

Using the concept of the generalized inverse of a bounded linear transformation between ? ⁿ andl ₂, a method is given for constructing quadrature rules for integration over an arbitrary boundedm-dimensional regionB?? ^m with the property that the average error over the prescribed familyF of the functions continuous inB as well as the variance of the rounding errors according to Sard [8] are minimal. Then we specializeF to the weighted monomials and treat as an example integration on the surface of them-sphere.     

On an interesting variant of the IMT quadrature

We indicate an interesting variant of the IMT quadrature. The IMT quadrature is known to converge exponentially in the presence of an end-point integrable singularity. The variant of IMT quadrature we indicate here retains this property and in addition it has exponential convergence for an integrand which has poles lying just below or above the mid-point of the integration interval.     

Minimal quadrature formulae for the spaces H 2 R and L 2 R . A unified interpolatory approach

We consider minimal quadrature formulae for the Hilbert spacesH ₂ ^R andL ₂ ^R consisting of functions which are analytical on the open disc with radiusR and centre at the origin; the inner products are the boundary contour integral forH ₂ ^R and the area integral over the disc forL ₂ ^R . Such formulae can be viewed as interpolatory, generalizing—in two ways—Markoff's idea to construct the classical Gaussian quadratur formulae. This can be done simultaneously for both spaces using the same Hermitian interpolating operator. The advantage of this approach to minimal formulae is that we get a nonlinear system of equations for the nodes of the minimal formulae alone, in contrast to the coupled system for nodes and weights which arises from the minimality conditions. The uncoupled system that we obtain is numerically solvable for reasonable numbers of nodes and numerical tests show that the resulting minimal formulae are very well suited for the integration of functions with boundary singularities.     

On the positivity of quadrature formulas with Jacobi abscissas

This note is dedicated to the study ofS _m , the set of (α,β) for which the interpolatory quadrature formula based on the zeroes ofP _m ^(α,β)(t) has positive weights. In contract to the results published by Lether, Wilhelmsen and Frazier [5], we show thatS _m behaves very regularly. The point is that the casesm odd andm even must be distinguished. Furthermore, informations on the exact number of negative weights for values outside ofS _m are obtained.