Some quadrature formulae with nonstandard weights (original) (raw)

Weighted quadrature rules with weight function on [0,∞)

Applied Mathematics and Computation, 2006

In this paper, we introduce an analytical class of weighted quadrature rules whose weight function is as W ðx; pÞ ¼ x Àp e À 1 x on [0, 1). The formulas given in this work generally obey the following form: Z 1 0 x Àp e À 1 x f ðxÞdx ¼ X n i¼1 w i f ðx i Þ þ R n ½f ; in which x i are the zeros of polynomials orthogonal with respect to the introduced weight function, w i are the corresponding coefficients and R n [f] is the related error. It should be pointed out that the above formula is valid only for the finite values of n. In other words, the condition p > 2{max n} + 1 is essential in order to be applicable the above quadrature rule. Some analytical examples are finally given and compared.

Gauss-Radau Quadrature Rule Using Special Class of Polynomials

A form of Gauss-Quadrature rule over [0,1] has been investigated that involves the derivative of the integrand at the pre-assigned left or right end node. This situation arises when the underlying polynomials are orthogonal with respect to the weight function ( ): 1 x x ω = − over [0,1] . Along the lines of Golub's work, the nodes and weights of the quadrature rule are computed from a Jacobi-type matrix with entries related to simple rational sequences. The structure of these sequences is based on some characteristics of the identity-type polynomials recently developed by one of the authors. The devised rule has a slight advantage over that subject to the weight function ( ): 1. x ω = Index Terms-Gauss-Radau quadrature rule, Jacobi-matrix, Hypergeometric series, Identity-type polynomials, 3-term recurrence relation.

On numerical improvement of Gauss–Lobatto quadrature rules

Applied Mathematics and Computation, 2005

It is well known that Gauss-Lobatto quadrature rule Z 1 À1 f ðxÞ dx ' X n i¼1 w i f ðx i Þ þ pf ðÀ1Þ þ qf ð1Þ; is exact for polynomials of degree at most 2n + 1. In this paper we are going to find a formula which is approximately exact for monomials x j , j = 0,1,.. ., 2n + 3 instead of being analytically exact for monomials x j , j = 0,1,.. ., 2n + 1. We also consider a class of functions for which the new formula produces better results.

On the Gauss-Kronrod quadrature formula for a modified weight function of Chebyshev type

Numerical Algorithms

In this paper, we consider the Gauss-Kronrod quadrature formulas for a modified Chebyshev weight. Efficient estimates of the error of these Gauss–Kronrod formulae for analytic functions are obtained, using techniques of contour integration that were introduced by Gautschi and Varga (cf. Gautschi and Varga SIAM J. Numer. Anal. 20, 1170–1186 1983). Some illustrative numerical examples which show both the accuracy of the Gauss–Kronrod formulas and the sharpness of our estimations are displayed. Though for the sake of brevity we restrict ourselves to the first kind Chebyshev weight, a similar analysis may be carried out for the other three Chebyshev type weights; part of the corresponding computations are included in a final appendix.

On numerical improvement of Gauss–Legendre quadrature rules

Applied Mathematics and Computation, 2005

Among all integration rules with n points, it is well-known that n-point Gauss-Legendre quadrature rule Z 1 À1 f ðxÞ dx ' X n i¼1 w i f ðx i Þ has the highest possible precision degree and is analytically exact for polynomials of degree at most 2n À 1, where nodes x i are zeros of Legendre polynomial P n ðxÞ, and w i 's are corresponding weights. In this paper we are going to estimate numerical values of nodes x i and weights w i so that the absolute error of introduced quadrature rule is less than a preassigned tolerance 0 , say 0 ¼ 10 À8 , for monomial functions f ðxÞ ¼ x j ; j ¼ 0; 1;. .. ; 2n þ 1: (Two monomials more than precision degree of Gauss-Legendre quadrature rules.) We also consider some conditions under which the new rules act, numerically, more q Research supported in part by MIM Grant no. A82-109.

Quadrature formulas for semi-infinite integrals

Mathematics of Computation, 1974

Polynomials orthogonal on [ 0 , ∞ ] [0,\infty ] with regard to the weight function w ( x ) = x α ( 1 + x ) − β w(x) = {x^\alpha }{(1 + x)^{ - \beta }} are obtained, recurrence relations are found and the differential equation, which is satisfied by them, is given. Formulas for weights and abscissas in the corresponding quadrature formula are given.

On Appel-Type Quadrature Rules

2002

The Radon technique is applied in order to recover a quadrature rule based on Appel polynomials and the so called Appel numbers. The relevant formula generalizes both the Euler-MacLaurin quadrature rule and a similar rule using Euler (instead of Bernoulli) numbers and even (instead of odd) derivatives of the given function at the endpoints of the considered interval. In the general case, the remainder term is expressed in terms of Appel numbers, and all derivatives appear. A numerical example is also included. operators.

Chebyshev series method for computing weighted quadrature formulas

Applied Mathematics and Computation, 2011

In this paper we study convergence and computation of interpolatory quadrature formulas with respect to a wide variety of weight functions. The main goal is to evaluate accurately a definite integral, whose mass is highly concentrated near some points. The numerical implementation of this approach is based on the calculation of Chebyshev series and some integration formulas which are exact for polynomials. In terms of accuracy, the proposed method can be compared with rational Gauss quadrature formula.