On Some Gauss and Lobatto Based Integration Formulae
نویسنده
چکیده
1. Introduction. The economy of the Gaussian quadrature formulae for carrying out numerical integration is to some extent reduced by the fact that an increase in the order of the formulae makes no use of previous integrand evaluations. Kronrod [1] has shown how the Gauss formula of degree 2n — 1 can be extended to one of degree 3rc + 2 by making use of the original n Gauss points and an additional set of n points. However, it is not possible to proceed further than this without using an entirely new set of points with a resulting waste of computational labor. It may be noted that due to the absence of a convenient error estimate for the Gaussian formulae it is usually necessary to carry out a quadrature using more than one order of formulae to check the convergence. In this paper a set of integration formulae is derived based on a set of 2r + 1 Gauss or Lobatto points, where r is an integer. If the original points are denoted by Xj, j = 1, 2, • •-, (2r + 1), then r subsets of points xitu_l)+v j = 1, 2, • •- ,
منابع مشابه
Generalized Gauss – Radau and Gauss – Lobatto Formulae ∗
Computational methods are developed for generating Gauss-type quadrature formulae having nodes of arbitrary multiplicity at one or both end points of the interval of integration. Positivity properties of the boundary weights are investigated numerically, and related conjectures are formulated. Applications are made to moment-preserving spline approximation. AMS subject classification: 65D30.
متن کاملOn Gautschi's conjecture for generalized Gauss-Radau and Gauss-Lobatto formulae
Recently, Gautschi introduced so-called generalized Gauss-Radau and Gauss-Lobatto formulae which are quadrature formulae of Gaussian type involving not only the values but also the derivatives of the function at the endpoints. In the present note we show the positivity of the corresponding weights; this positivity has been conjectured already by Gautschi. As a consequence, we establish several ...
متن کاملOn the error term of symmetric Gauss-Lobatto quadrature formulae for analytic functions
Gauss-Lobatto quadrature formulae associated with symmetric weight functions are considered. The kernel of the remainder term for classes of analytic functions is investigated on elliptical contours. Sufficient conditions are found ensuring that the kernel attains its maximal absolute value at the intersection point of the contour with either the real or the imaginary axis. The results obtained...
متن کاملA novel modification of decouple scaled boundary finite element method in fracture mechanics problems
In fracture mechanics and failure analysis, cracked media energy and consequently stress intensity factors (SIFs) play a crucial and significant role. Based on linear elastic fracture mechanics (LEFM), the SIFs and energy of cracked media may be estimated. This study presents the novel modification of decoupled scaled boundary finite element method (DSBFEM) to model cracked media. In this metho...
متن کامل