Error bounds of the Micchelli-Sharma quadrature formula for analytic functions

Micchelli and Sharma constructed in their paper [On a problem of Turan: multiple node Gaussian quadrature, Rend. Mat. 3 (1983) 529-552] a quadrature formula for the Fourier-Chebyshev coefficients, which has the highest possible precision. For analytic functions the remainder term of this quadrature formula can be represented as a contour integral with a complex kernel. We study the kernel, on elliptic contours with foci at the points not equal 1 and a sum of semi-axes rho > 1, for the quoted quadrature formula. Starting from the explicit expression of the kernel, we determine the location on the ellipses where the maximum modulus of the kernel is attained, and derive effective error bounds for this quadrature formula. Numerical examples are included

Error bounds of the Micchelli-Sharma quadrature formula for analytic functions

Micchelli and Sharma constructed in their paper [On a problem of Turan: multiple node Gaussian quadrature, Rend. Mat. 3 (1983) 529-552] a quadrature formula for the Fourier-Chebyshev coefficients, which has the highest possible precision. For analytic functions the remainder term of this quadrature formula can be represented as a contour integral with a complex kernel. We study the kernel, on elliptic contours with foci at the points not equal 1 and a sum of semi-axes rho > 1, for the quoted quadrature formula. Starting from the explicit expression of the kernel, we determine the location on the ellipses where the maximum modulus of the kernel is attained, and derive effective error bounds for this quadrature formula. Numerical examples are included

Quadrature with multiple nodes, power orthogonality, and moment-preserving spline approximation, part ii

The paper deals with new contributions to the theory of the Gauss quadrature formulas with multiple nodes that are published after 2001, including numerical construction, error analysis and applications. The first part was published in Numerical analysis 2000, Vol. V, Quadrature and orthogonal polynomials (W. Gautschi, F. Marcellan, and L. Reichel, eds.) [J. Comput. Appl. Math. 127 (2001), no. 1-2, 267-286]

Quadrature with multiple nodes, power orthogonality, and moment-preserving spline approximation, part ii

The paper deals with new contributions to the theory of the Gauss quadrature formulas with multiple nodes that are published after 2001, including numerical construction, error analysis and applications. The first part was published in Numerical analysis 2000, Vol. V, Quadrature and orthogonal polynomials (W. Gautschi, F. Marcellan, and L. Reichel, eds.) [J. Comput. Appl. Math. 127 (2001), no. 1-2, 267-286]

Error estimates of gaussian-type quadrature formulae for analytic functions on ellipses-a survey of recent results

This paper presents a survey of recent results on error estimates of Gaussian-type quadrature formulas for analytic functions on confocal ellipses

Error estimates of gaussian-type quadrature formulae for analytic functions on ellipses-a survey of recent results

This paper presents a survey of recent results on error estimates of Gaussian-type quadrature formulas for analytic functions on confocal ellipses

Ocene grešaka kvadraturnih formula Gausovog tipa za analitičke funkcije

The field of research in this dissertation is concerned with numerical integration,i.e. with the derivation of error bounds for Gauss-type quadratures and their generalizations when we use them to approximate integrals of functions which are analytic inside an elliptical contour Eρ with foci at ∓1 and sum of semi-axes ρ > 1. Special attention is given to Gauss-type quadratures with the special kind of weight functions - weight functions of Bernstein–Szeg˝o type. Three kinds of error bounds are considered in the dissertation, which means analysis of kernels of quadratures, i.e. determination of the location of the extremal point on Eρ at which the modulus of the kernels attains its maximum, calculation of the contour integral of the modulus of the kernel, and, also, series expansion of the kernel. Beyond standard, corresponding quadratures for calculation of Fourier expansion coefficients of an analytic function are also analysed in this dissertation

Ocene grešaka kvadraturnih formula Gausovog tipa za analitičke funkcije

The field of research in this dissertation is concerned with numerical integration,i.e. with the derivation of error bounds for Gauss-type quadratures and their generalizations when we use them to approximate integrals of functions which are analytic inside an elliptical contour Eρ with foci at ∓1 and sum of semi-axes ρ > 1. Special attention is given to Gauss-type quadratures with the special kind of weight functions - weight functions of Bernstein–Szeg˝o type. Three kinds of error bounds are considered in the dissertation, which means analysis of kernels of quadratures, i.e. determination of the location of the extremal point on Eρ at which the modulus of the kernels attains its maximum, calculation of the contour integral of the modulus of the kernel, and, also, series expansion of the kernel. Beyond standard, corresponding quadratures for calculation of Fourier expansion coefficients of an analytic function are also analysed in this dissertation

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..