A Survey of Gauss-Christoffel Quadrature Formulae
نویسنده
چکیده
4. 4.1. 4.1.1. 4.1.2. 4.1.3. 4.2. 4.3. Gaussian quadrature with preassigned nodes Christoffel's work and related developments Kronrod's extension of quadrature rules Gaussian quadrature with multiple nodes The quadrature formula of Turan Arbitrary multiplicities and preassigned nodes Power-orthogonal polynomials Constructive aspects and applications Further miscellaneous extensions Product-type quadrature rules Gaussian quadrature involving interval functionals Nonpolynomial Gaussian quadrature
منابع مشابه
Construction of Gauss-Christ of fei Quadrature Formulas
Each of these rules will be called a Gauss-Christoffel quadrature formula if it has maximum degree of exactness, i.e. if (1.1) is an exact equality whenever / is a polynomial of degree 2n — 1. It is a well-known fact, due to Christoffel [3], that such quadrature formulas exist uniquely, provided the weight function w(x) is nonnegative, integrable with /* w(x)dx > 0, and such that all its moments
متن کاملOn sensitivity of Gauss-Christoffel quadrature
In numerical computations the question how much does a function change under perturbations of its arguments is of central importance. In this work, we investigate sensitivity of Gauss-Christoffel quadrature with respect to small perturbations of the distribution function. In numerical quadrature, a definite integral is approximated by a finite sum of functional values evaluated at given quadrat...
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 کاملError Bounds for Gauss-kronrod Quadrature Formulae
The Gauss-Kronrod quadrature formula Qi//+X is used for a practical estimate of the error R^j of an approximate integration using the Gaussian quadrature formula Q% . Studying an often-used theoretical quality measure, for ߣ* , we prove best presently known bounds for the error constants cs(RTMx)= sup \RlK+x[f]\ ll/(l»lloo<l in the case s = "Sn + 2 + tc , k = L^J LfJ • A comparison with the Ga...
متن کاملA note on the Gauss-Jacobi quadrature formulae for singular integral equations of the second kind
A fast and efficient numerical method based on the Gauss-Jacobi quadrature is described that is suitable for solving Fredholm singular integral equations of the second kind that are frequently encountered in fracture and contact mechanics. Here we concentrate on the case when the unknown function is singular at both ends of the interval. Quadrature formulae involve fixed nodal points and provid...
متن کامل