一种Gauss型求积公式的收敛性

构造一种有理插值型求积公式(RIQFs),并证明其收敛性.该方法是Gauss求积公式在有理函数空间(Γ)2n中的推广.     

On Gauss quadrature formulas with equal weights

Summary It is well known that the Chebyshev weight function (1–x ²)^–1/2 is the only weight function (up to a linear transformation) for which then point Gauss quadrature formula has equal weights for alln. In this paper we describe all weight functions for which thenm point Gauss quadrature formula has equal weights for alln, wherem is fixed.     

Product type quadrature formulas

This paper is concerned with the numerical approximation of integrals of the form _a ^b f(x)g(x)dx by means of a product type quadrature formula. In such a formula the functionf (x) is sampled at a set ofn+1 distinct points and the functiong(x) at a (possibly different) set ofm+1 distinct points. These formulas are a generalization of the classical (regular) numerical integration rules. A number of basic results for such formulas are stated and proved. The concept of a symmetric quadrature formula is defined and the connection between such rules and regular quadrature formulas is discussed. Expressions for the error term are developed. These are applied to a specific example.The work of the first author was supported in part by NIH Grant No. FRO 7129-01 and that of the second author in part by U.S. Army Ballistic Research Laboratories Contract DA-18-001-AMC-876 X.     

On the Gauss type quadrature with parametric weight function

Acta Mathematica Hungarica -     

Optimal quadrature formulas

The problem of finding optimal quadrature formulas of given precision which minimize the sum of the absolute values of the quadrature weights is discussed and some optimal predictor and corrector type quadrature formulas are listed. Alternative derivation of minimum variance and Sard's optimal quadrature formulas is also given.     

Gauss-Tchebycheff quadrature formulas

Summary It is well known that the Tchebycheff weight function (1-x ²)^–1/2 is the only weight function (up to a linear transformation) for which then point Gauss quadrature formula has equal weights for alln. In this paper we describe explicitly all weight functions which have the property that then _k-point Gauss quadrature formula has equal weights for allk, where (n _k),n ₁<n ₂<..., is an arbitrary subsequence of . Furthermore results on the possibility of Tchebycheff quadrature on several intervals are given.     

About some quadrature formulas

In this paper, we studied a class of quadrature formulas obtained by using the connection between the monospline functions and the quadrature formulas. For this class we obtain the optimal quadrature formula with regard to the error and we give some inequalities for the remainder term of this optimal quadrature formula.      

On chebyshev quadrature and variance of quadrature formulas

The purpose of this note is to give an example which demonstrates that one can achieve much higher algebraic precision with a quadrature rule with small but not minimal variance than with a Chebyshev rule with minimal variance.     

Generalized Gaussian quadrature formulas

Existence and uniqueness of canonical points for best L₁-approximation from an Extended Tchebycheff (ET) system, by Hermite interpolating “polynomials” with free nodes of preassigned multiplicities, are proved. The canonical points are shown to coincide with the nodes of a “generalized Gaussian quadrature formula” of the form (*) which is exact for the ET-system. In (*), ∑_{j = 0}^{v_i − 2} ≡ 0 if v_i = 1, the v_i (> 0), I = 1,…, n, are the multiplicities of the free nodes and v₀0, v_{n + 1} 0 of the boundary points in the L₁-approximation problem, ∑_{i = 0}^{n + 1} v_i is the dimension of the ET-system, and σ is the weight in the L₁-norm.The results generalize results on multiple node Gaussian quadrature formulas (v₁,…, v_n all even in (*)) and their relation to best one-sided L₁-approximation. They also generalize results on the orthogonal signature of a Tchebycheff system (v₀ = v_{n + 1} = 0, v_i = 1, I = 1,…, n, in (*)), and its role in best L₁-approximation. Recent works of the authors were the first to treat Gaussian quadrature formulas and orthogonal signatures in a unified way.     

On optimal quadrature formulas

Quadrature formulas with free nodes which are optimal in the norm of a Banach space are studied. It is shown that it is impossible with some reasonable assumptions to increase the accuracy of such a formula by defining the partial derivatives of the integrable function at the nodes.     

Monotonicity of quadrature formulae of Gauss type and comparison theorems for monosplines

Summary This paper deals with quadrature formulae of Gauss type corresponding to subspaces of spline functionsS _{m–1, k} of degreem–1 withk fixed knots. We shall show monotonicity of the quadrature formulae for functions which are contained in the so-called convexity cone ofS _m–1,k Moreover, we apply these results to monosplines and establish comparison theorems for these splines.     

Stochastic properties of quadrature formulas

Summary The average error of suitable quadrature formulas and the stochastic error of Monte Carlo methods are both much smaller than the worst case error in many cases. This depends, however, on the classF of functions which is considered and there are counterexamples as well.Nonlinear methods, adaptive methods, or even methods with varying cardinality are not significantly better (with respect to certain stochastic error bounds) than the simplest linear methods .     

Computation of Gauss-type quadrature formulas

Gaussian formulas for a linear functional L (such as a weighted integral) are best computed from the recursion coefficients relating the monic polynomials orthogonal with respect to L. In Gauss-type formulas, one or more extraneous conditions (such as pre-assigning certain nodes) replace some of the equations expressing exactness when applied to high-order polynomials. These extraneous conditions may be applied by modifying the same number of recursion coefficients. We survey the methods of computing formulas from recursion coefficients, methods of obtaining recursion coefficients and modifying them for Gauss-type formulas, and questions of existence and numerical accuracy associated with those computations.     

Properties of product-type quadrature formulas

In this note some properties of the coefficient matrix associated with a product-interpolatory quadrature formula are determined and certain consequences of exactness of a product-type quadrature rule are deduced. For example, it is shown that the coefficient matrix has maximal rank and it is positive definite when the rule is symmetric. Conditions are stated under which a product-interpolatory rule reduces to a regular quadrature rule. A characterization of the error committed in applying a regular rule to a product of two functions is given.A portion of this research was carried out while the author was a Summer Faculty Research Participant at the Oak Ridge National Laboratory.