28 research outputs found
The Error Estimates of Kronrod Extension for Gauss-Radau and Gauss-Lobatto Quadrature with the Four Chebyshev Weights
In this paper, we consider the Kronrod extension for the Gauss-Radau and Gauss-Lobatto quadrature consisting of any one of the four Chebyshev weights. The main purpose is to effectively estimate the error of these quadrature formulas. This estimate needs a calculation of the maximum of the modulus of the kernel. We compute explicitly the kernel function and determine the locations on the ellipses where a maximum modulus of the kernel is attained. Based on this, we derive effective error bounds of the Kronrod extensions if the integrand is an analytic function inside of a region bounded by a confocal ellipse that contains the interval of integration
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
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]
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..
Quadrature Strategies for Constructing Polynomial Approximations
Finding suitable points for multivariate polynomial interpolation and
approximation is a challenging task. Yet, despite this challenge, there has
been tremendous research dedicated to this singular cause. In this paper, we
begin by reviewing classical methods for finding suitable quadrature points for
polynomial approximation in both the univariate and multivariate setting. Then,
we categorize recent advances into those that propose a new sampling approach
and those centered on an optimization strategy. The sampling approaches yield a
favorable discretization of the domain, while the optimization methods pick a
subset of the discretized samples that minimize certain objectives. While not
all strategies follow this two-stage approach, most do. Sampling techniques
covered include subsampling quadratures, Christoffel, induced and Monte Carlo
methods. Optimization methods discussed range from linear programming ideas and
Newton's method to greedy procedures from numerical linear algebra. Our
exposition is aided by examples that implement some of the aforementioned
strategies
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