Results 181 to 190 of about 789 (216)

Optimal quadrature formulae

Russian Mathematical Surveys, 2005
In this survey we consider results and open problems related to two major ideas in the theory of optimal quadrature formulae: the ideas of Gauss and Kolmogorov.
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The Coefficients of Optimal Quadrature Formulas

2006
2 (R) are studied by means of a variational method. Here w(x) is a weight function, χΩ(x) is the characteristic function of the interval Ω, and c(β) are the coefficients of the quadrature formula. The results generalize some results by A. Sard, L. F. Meyers, I. J. Schoenberg, S. D. Silliman (1-4), and others derived by the method of splines.
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Optimal quadrature formulas for a sphere

USSR Computational Mathematics and Mathematical Physics, 1977
Abstract The problem of finding the statistically optimal quadrature formula for a sphere is posed. A system of linear algebraic equations satisfied by weights of the quadrature formula is written down. Two examples are given. Asymptotic estimates of the relative error are given.
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Optimal quadrature formulae for differentiable functions

Calcolo, 1986
The existence of an optimal quadrature formula of the form \(\int^{b}_{a}f| x| dx\approx \sum^{n}_{k=1}\sum^{\nu_ k-1}_{\lambda =0}a_{k\lambda}f^{(\lambda)}(xk)\) with preassigned multiplicities \((\nu_ k)_ 1^ n\) in the classes \(LW_ q^ r:=\{f\in C^{(r-1)}:f^{(r-1)}\)- abs.
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A WEIGHTED OPTIMAL QUADRATURE FORMULA WITH DERIVATIVE

UZBEK MATHEMATICAL JOURNAL
This article focuses on the derivation and analysis of a weighted optimal quadra- ture formula in the Hilbert space W (2,1) 2 (0, 1). The formula is expressed as a linear combination of function values and its first-order derivatives at equidistant nodes in the interval [0, 1].
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Error inequalities for an optimal quadrature formula

Journal of Applied Mathematics and Computing, 2007
An optimal 3-point quadrature formula of closed type is derived. It is shown that the optimal quadrature formula has a better error bound than the well-known Simpson's rule. A corrected formula is also considered. Various error inequalities for these formulas are established. Applications in numerical integration are given.
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Optimal stochastic quadrature formulas for convex functions

BIT, 1994
Optimal stochastic (Monte Carlo) quadrature formulas for defined classes of convex functions are studied. Specifically, non-adaptive Monte Carlo methods are seen to be no better than deterministic methods, but adaptive Monte Carlo methods are shown to exhibit a superior performance.
Novak, E., Petras, K.
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Asymptotic Properties of Optimal Quadrature Formulas

1979
In [9], Sard introduced a notion of “best” for quadrature formulas which may be described as follows. Let 0 = t0 < t1 < ... < tN = 1 be fixed points, and consider the formula $${Q_N}(f) \equiv \sum\limits_{i = 0}^N {{c_i}f({t_i}) \simeq \int_0^1 f (\tau )} d\tau \equiv I(f)$$ (1) .
David L. Barrow, Philip W. Smith
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An Optimal Quadrature Formula With Derivative for Weakly Singular Integrals

Mathematical Methods in the Applied Sciences
ABSTRACT This article presents the derivation and analysis of an optimal quadrature formula for the numerical integration of fractional integrals in the Hilbert space . In this space, functions satisfy certain smoothness conditions.
Abdullo Hayotov, Samandar Babaev
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THE EXISTENCE OF OPTIMAL QUADRATURE FORMULAS WITH GIVEN MULTIPLICITIES OF NODES

Mathematics of the USSR-Sbornik, 1978
Suppose that is the error of the best method of integration in the class with nodes of multiplicities , i.e. . It is then shown that for and for every system of multiplicities with for , the lower bound is attained for some nodes with exactly the multiplicities . Moreover, and .Bibliography: 20 titles.
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