A new representation of generalized averaged Gauss quadrature rules

Gauss quadrature rules associated with a nonnegative measure with support on (part of) the real axis find many applications in Scientific Computing. It is important to be able to estimate the quadrature error when replacing an integral by an l-node Gauss quadrature rule in order to choose a suitable number of nodes. A classical approach to estimate this error is to evaluate the associated (2l + 1)-node Gauss-Kronrod rule. However, Gauss-Kronrod rules with 2l + 1 real nodes might not exist. The (2l + 1)-node generalized averaged Gauss formula associated with the l-node Gauss rule described in Spalevic (2007) [16] is guaranteed to exist and provides an attractive alternative to the (2l + 1)-node Gauss-Kronrod rule. This paper describes a new representation of generalized averaged Gauss formulas that is cheaper to evaluate than the available representation

A new representation of generalized averaged Gauss quadrature rules

Gauss quadrature rules associated with a nonnegative measure with support on (part of) the real axis find many applications in Scientific Computing. It is important to be able to estimate the quadrature error when replacing an integral by an l-node Gauss quadrature rule in order to choose a suitable number of nodes. A classical approach to estimate this error is to evaluate the associated (2l + 1)-node Gauss-Kronrod rule. However, Gauss-Kronrod rules with 2l + 1 real nodes might not exist. The (2l + 1)-node generalized averaged Gauss formula associated with the l-node Gauss rule described in Spalevic (2007) [16] is guaranteed to exist and provides an attractive alternative to the (2l + 1)-node Gauss-Kronrod rule. This paper describes a new representation of generalized averaged Gauss formulas that is cheaper to evaluate than the available representation

Rational Averaged Gauss Quadrature Rules

It is important to be able to estimate the quadrature error in Gauss rules. Several approaches have been developed, including the evaluation of associated Gauss-Kronrod rules (if they exist), or the associated averaged Gauss and generalized averaged Gauss rules. Integrals with certain integrands can be approximated more accurately by rational Gauss rules than by Gauss rules. This paper introduces associated rational averaged Gauss rules and rational generalized averaged Gauss rules, which can be used to estimate the error in rational Gauss rules. Also rational Gauss-Kronrod rules are discussed. Computed examples illustrate the accuracy of the error estimates determined by these quadrature rules

Error Estimates for Certain Cubature Formulae

We estimate the errors of selected cubature formulae constructed by the product of Gauss quadrature rules. The cases of multiple and (hyper-)surface integrals over n-dimensional cube, simplex, sphere and ball are considered. The error estimates are obtained as the absolute value of the difference between cubature formula constructed by the product of Gauss quadrature rules and cubature formula constructed by the product of corresponding Gauss-Kronrod or corresponding generalized averaged Gaussian quadrature rules. Generalized averaged Gaussian quadrature rule (G) over cap (2l+1) is (2l + 1)-point quadrature formula. It has 2l + 1 nodes and the nodes of the corresponding Gauss rule G(l) with l nodes form a subset, similar to the situation for the (2l + 1)-point Gauss-Kronrod rule H2l+1 associated with G(l). The advantages of (G) over cap (2l+1) are that it exists also when H2l+1 does not, and that the numerical construction of (G) over cap (2l+1), based on recently proposed effective numerical procedure, is simpler than the construction of H2l+1

Rational Averaged Gauss Quadrature Rules

It is important to be able to estimate the quadrature error in Gauss rules. Several approaches have been developed, including the evaluation of associated Gauss-Kronrod rules (if they exist), or the associated averaged Gauss and generalized averaged Gauss rules. Integrals with certain integrands can be approximated more accurately by rational Gauss rules than by Gauss rules. This paper introduces associated rational averaged Gauss rules and rational generalized averaged Gauss rules, which can be used to estimate the error in rational Gauss rules. Also rational Gauss-Kronrod rules are discussed. Computed examples illustrate the accuracy of the error estimates determined by these quadrature rules

Derivatives of a finite class of orthogonal polynomials defined on the positive real line related to F-distribution

AbstractAmong the six classes of classical orthogonal polynomials, three of them are infinite, namely Jacobi, Hermite and Laguerre and the remaining three are finite and characterized by Masjed Jamei (2002) [5]. In this work, we consider derivatives of one such finite class of orthogonal polynomials that are orthogonal with respect to the weight function which is related to the probability density function of the F distribution. For this derivative class, besides orthogonality we find various other related properties such as the normal form and the self adjoint form. The corresponding Gaussian quadrature formulae are also given. Examples are provided to support the advantages of considering this derivative class of the finite class of orthogonal polynomials

A Simultaneous Numerical Integration Routine for the Fast Calculation of Similar Integrations

In this paper, a fast and simultaneous integration routine tailored for obtaining results of multiple numerical integrations is introduced. In the routine, the same nodes are used when integrating different functions along the same integration path. In the paper it is demonstrated by several examples that if the integrands of interest are similar on the integration path, then using the same nodes decreases the computational costs dramatically. While the method is introduced by updating the popular Gauss-Kronrod quadrature rule, the same steps given in the paper can be applied to any other numerical integration rule

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

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

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