Errors of gauss-radau and gauss-lobatto quadratures with double end point

Starting from the explicit expression of the corresponding kernels, derived by Gautschi and Li (W. Gautschi, S. Li: The remainder term for analytic functions of Gauss-Lobatto and Gauss-Radau quadrature rules with multiple end points, J. Comput. Appl. Math. 33 (1990) 315-329), we determine the exact dimensions of the minimal ellipses on which the modulus of the kernel starts to behave in the described way. The effective error bounds for Gauss-Radau and Gauss-Lobatto quadrature formulas with double end point(s) are derived. The comparisons are made with the actual errors

Errors of gauss-radau and gauss-lobatto quadratures with double end point

Starting from the explicit expression of the corresponding kernels, derived by Gautschi and Li (W. Gautschi, S. Li: The remainder term for analytic functions of Gauss-Lobatto and Gauss-Radau quadrature rules with multiple end points, J. Comput. Appl. Math. 33 (1990) 315-329), we determine the exact dimensions of the minimal ellipses on which the modulus of the kernel starts to behave in the described way. The effective error bounds for Gauss-Radau and Gauss-Lobatto quadrature formulas with double end point(s) are derived. The comparisons are made with the actual errors

Error Estimations of Turan Formulas with Gori-Micchelli and Generalized Chebyshev Weight Functions

S. Li in [Studia Sci. Math. Hungar. 29 (1994) 71-83] proposed a Kronrod type extension to the well-known Turan formula. He showed that such an extension exists for any weight function. For the classical Chebyshev weight function of the first kind, Li found the Kronrod extension of Turan formula that has all its nodes real and belonging to the interval of integration, [-1, 1]. In this paper we show the existence and the uniqueness of the additional two cases - the Kronrod exstensions of corresponding Gauss-Turan quadrature formulas for special case of Gori-Micchelli weight function and for generalized Chebyshev weight function of the second kind, that have all their nodes real and belonging to the integration interval [-1, 1]. Numerical results for the weight coefficients in these cases are presented, while the analytic formulas of the nodes are known

Error Bounds for Gauss-Lobatto Quadrature Formula with Multiple End Points with Chebyshev Weight Function of the Third and the Fourth Kind

For analytic functions the remainder terms of quadrature formulae can be represented as a contour integral with a complex kernel. We study the kernel, on elliptic contours with foci at the points -/+ 1, for Gauss-Lobatto quadrature formula with multiple end points with Chebyshev weight function of the third and the fourth kind. Starting from the explicit expression of the corresponding kernel, derived by Gautschi and Li, we determine the locations on the ellipses where maximum modulus of the kernel is attained. The obtained values confirm the corresponding conjectured values given by Gautschi and Li in paper [The remainder term for analytic functions of Gauss-Radau and Gauss-Lobatto quadrature rules with multiple end points, Journal of Computational and Applied Mathematics 33 (1990) 315-329.

The remainder term of certain types of Gaussian quadrature formulae with specific classes of weight functions.

Integracija ima xiroku primenu prilikom matematiqkog mode- lovanja mnogih pojava koje se javljaju u prirodnim, tehniqkim naukama, ekonomiji i drugim oblastima. Kada se vrednost integrala ne moe analitiqki izraqunati, potrebno je kon- struisati formulu koja aproksimira njegovu vrednost sa prih- vatljivom taqnoxu. Pored tradicionalnih formula koje se ko- riste, tendencije u razvoju ove oblasti odnose se na poveanje taqnosti formule i ocenu grexke nastale kada se integral za- meni konaqnom sumom...Mathematical modeling of many phenomena which occur in the natural, technical sciences, economy requires signicant knowledge of the theory of numerical integration. In the situations where the integral cannot be determined analytically, it is necessary to construct the for- mula which approximates its value with acceptable error. Besides the traditional formulae, the tendencies in the development of this area refer to increment of algebraic degree of precision of the quadrature formula and its error estimation..

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

Error bounds for gaussian quadrature formulae with legendre weight function for analytic integrands

In this paper we are concerned with a method for the numerical evaluation of the error terms in Gaussian quadrature formulae with the Legendre weight function. Inspired by the work of H. Wang and L. Zhang [J. Sci. Comput., 75 (2018), pp. 457-477] and applying the results of S. Notaris [Math. Comp., 75 (2006), pp. 1217-1231], we determine an explicit formula for the kernel. This explicit expression is used for finding the points on ellipses where the maximum of the modulus of the kernel is attained. Effective error bounds for this quadrature formula for analytic integrands are derived

The remainder term of certain types of Gaussian quadrature formulae with specific classes of weight functions.

Integracija ima xiroku primenu prilikom matematiqkog mode- lovanja mnogih pojava koje se javljaju u prirodnim, tehniqkim naukama, ekonomiji i drugim oblastima. Kada se vrednost integrala ne moe analitiqki izraqunati, potrebno je kon- struisati formulu koja aproksimira njegovu vrednost sa prih- vatljivom taqnoxu. Pored tradicionalnih formula koje se ko- riste, tendencije u razvoju ove oblasti odnose se na poveanje taqnosti formule i ocenu grexke nastale kada se integral za- meni konaqnom sumom...Mathematical modeling of many phenomena which occur in the natural, technical sciences, economy requires signicant knowledge of the theory of numerical integration. In the situations where the integral cannot be determined analytically, it is necessary to construct the for- mula which approximates its value with acceptable error. Besides the traditional formulae, the tendencies in the development of this area refer to increment of algebraic degree of precision of the quadrature formula and its error estimation..