Further properties of inner product quadrature formulas

An inner product quadrature formula is of the form $$\int_{ - 1}^1 {w(x)f(x)g(x)dx \cong \sum\limits_{i = 0}^m {\sum\limits_{j = 0}^n {f(x_i )a_{ij} g(y_j ) = f^T Ag.} } } $$ Conditions are established for which these quadrature formulas are exact whenf andg are polynomials of degree not greater thanm+k andn?l (also, analogouslym?k andn+l) respectively. The structure and properties of the matrixA are also considered.     

Interpolatory quadrature formulas on the unit circle for Chebyshev weight functions

Summary. In this paper, interpolatory quadrature formulas based upon the roots of unity are studied for certain weight functions. Positivity of the coefficients in these formulas is deduced along with computable error estimations for analytic integrands. A comparison is made with Szeg? quadrature formulas. Finally, an application to the interval [-1,1] is also carried out. Received February 29, 2000 / Published online August 17, 2001     

Inner product quadrature formulas exact on maximal product spaces of functions

This work extends and complements earlier work of the author [1]. An Inner Product Quadrature Formula (I P Q F) is used when approximating the definite integral of the product of two (or more) funcitons, i.e. σ_?1¹w(x) f(x) g(x) dx, where w is a weight function. The functions f and g are approximated by interpolatory functions f_φ? span φ_{ii = 0}^γ, g_ψ? span ψj_{j = 0}^δ, and the integral σ_?1¹w(x)f_φ(x)g_ψ (x) dx is evaluated exactly. Maximal values which the numbers γ and δ may take are investigated. Numerical examples of I P Q F are given. Also, the applicaitons of I P Q F in higher dimensions are commented on.     

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.     

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.     

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.      

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.     

The numerical solution of Fredholm integral equations using product type quadrature formulas

Product type quadrature formulas are applied to obtain approximate solutions of Fredholm integral equations. A convergence theorem, and several numerical examples which demonstrate the efficacy of the technique, are presented.     

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.     

Derivative corrections for quadrature formulas

In this paper, we develop corrected quadrature formulas by approximating the derivatives of the integrand that appear in the asymptotic error expansion of the quadrature, using only the function values in the original quadrature rule. A higher order convergence is achieved without computing additional function values of the integrand.This author is in part supported by National Science Foundation under grant DMS-9504780 and by NASA-OAI Summer Faculty Fellowship (1995).     

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 .     

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.     

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.     

Error estimates for quadrature formulas

We construct two-sided polynomials of collocation type of the same order as a given system of basis functions according to a given ordered system of nodes of arbitrary multiplicity and according to a system of nodes displaced to the right (or to the left) at one position. Numerical estimates are given for the remaining terms of the quadrature formulas.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 70, pp. 21–31, 1990.     

Comparison of optimal quadrature formulas

Summary We prove the monotonicity of the error of the optimal quadrature formula of a given quasi-Hermitian type inW _q ^r [0.1] (1<q) with respect to the order of the derivatives appearing in the end point terms.     

Ratio asymptotics and quadrature formulas

A technique to find the asymptotic behavior of the ratio between a polynomialss _n and thenth orthonormal polynomial with respect to a positive measureμ is shown. Using it, some new results are found and a very simple proof for other classics is given.