Error estimates of gaussian quadrature formulae with the third class of bernstein-szego weights

For analytic functions we study the remainder terms of Gauss quadrature rules with respect to Bernstein-Szego weight functions w(t) = w(alpha,beta,delta)(t) = root 1+t/1-t/beta(beta-2 alpha)t(2) + 2 delta(beta-alpha)t+alpha(2) + delta(2) , t epsilon(-1,1), where 0 lt alpha lt beta, beta not equal 2 alpha, vertical bar delta vertical bar lt beta-alpha, and whose denominator is an arbitrary polynomial of exact degree 2 that remains positive on [-1,1]. The subcase alpha = 1, beta = 2/(1 + gamma), -1 lt gamma lt 0 and delta = 0 has been considered recently by M. M. Spalevie, Error bounds of Gaussian quadrature formulae for one class of Bernstein-Szego weights, Math. Comp., 82 (2013), 1037-1056

Error estimates of gaussian quadrature formulae with the third class of bernstein-szego weights

For analytic functions we study the remainder terms of Gauss quadrature rules with respect to Bernstein-Szego weight functions w(t) = w(alpha,beta,delta)(t) = root 1+t/1-t/beta(beta-2 alpha)t(2) + 2 delta(beta-alpha)t+alpha(2) + delta(2) , t epsilon(-1,1), where 0 lt alpha lt beta, beta not equal 2 alpha, vertical bar delta vertical bar lt beta-alpha, and whose denominator is an arbitrary polynomial of exact degree 2 that remains positive on [-1,1]. The subcase alpha = 1, beta = 2/(1 + gamma), -1 lt gamma lt 0 and delta = 0 has been considered recently by M. M. Spalevie, Error bounds of Gaussian quadrature formulae for one class of Bernstein-Szego weights, Math. Comp., 82 (2013), 1037-1056

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

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 bounds of Micchelli-Rivlin quadrature formula for analytic functions

We consider the well known Micchelli-Rivlin quadrature formula, of highest algebraic degree of precision, for the Fourier-Chebyshev coefficients. 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 -/+ 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 locations on the ellipses where maximum modulus of the kernel is attained. So we derive effective L-infinity-error bounds for this quadrature formula. Complex-variable methods are used to obtain expansions of the error in the Micchelli-Rivlin quadrature formula over the interval [-1, 1]. Finally, effective L-1-error bounds are also derived for this quadrature formula

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.

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