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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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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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RATIONAL FUNCTIONS AND QUADRATURE FORMULAE
Analysis, 1988Summary: In the theory of classical, strong and trigonometric moment problems quadrature formulae for the linear functional, defined by the moments, can be obtained by using ordinary or modified approximants of certain continued fractions as intermediaries.
Njåstad, Olav, Thron, W. J.
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Mathematical Proceedings of the Cambridge Philosophical Society, 1950
1. It is frequently required to find the numerical value of the definite integralIt is, however, often found that even if the analytical expression off(x) is given, it cannot be integrated in terms of known elementary functions. The elliptic integrals are perhaps the best known examples of functions of this type; and more common are cases wheref(x) is ...
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1. It is frequently required to find the numerical value of the definite integralIt is, however, often found that even if the analytical expression off(x) is given, it cannot be integrated in terms of known elementary functions. The elliptic integrals are perhaps the best known examples of functions of this type; and more common are cases wheref(x) is ...
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New symmetric interpolatory quadrature formulas
Calcolo, 1995zbMATH Open Web Interface contents unavailable due to conflicting licenses.
FAVATI P, LOTTI G, ROMANI, FRANCESCO
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ON THE ČEBYŠEV QUADRATURE FORMULA
Mathematics of the USSR-Izvestiya, 1969In this paper we examine several weight functions for which the Cebysev quadrature formula is valid. A method is given in the general case by means of which the degree of precision of the formula may be estimated.
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Characterization of Quadrature Formula II
SIAM Journal on Mathematical Analysis, 1984This paper is concerned with interpolatory quadrature formulas of the type \[ (1)\quad \int^{+1}_{- 1}f(x)w(x)dx=\sum^{n}_{i=1}\lambda_ i\quad f(x_ i)+R_ n(f) \] where ...
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Ratio asymptotics and quadrature formulas
Constructive Approximation, 1997Suppose \(p_n\) \((n=0,1,2,\ldots)\) is a sequence of orthogonal polynomials on the real line, satisfying a three-term recurrence relation \(tp_n(t) = a_{n+1}p_{n+1}(t)+b_np_n(t)+a_np_{n-1}(t)\). The author gives a method for obtaining the asymptotic behaviour of the ratio \(s_n(z)/p_n(z)\) for a comparison sequence \(s_n\) \((n=0,1,2,\ldots)\) of ...
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Characterization of Positive Quadrature Formulas
SIAM Journal on Mathematical Analysis, 1981We give a complete description of those numerical integration formulas based on n nodes which have positive weights and are exact for polynomials of degree equal or less than $2n - 1 - m$, where $0 \leqq m \leqq n$.
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On chebyshev quadrature and variance of quadrature formulas
BIT, 1988The 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.
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On Quadrature Formulae Near Gaussian Quadrature
1992In this paper, for product integration on the finite interval [a, b], we consider the class of n-point quadrature formulae Q n of at least algebraic degree 2n – 3. We study a new approach for their characterization using the simple fact that such a quadrature formula is uniquely determined by one node y and its associated weight b. For a given node y ∈
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