Nonexistence of extended Gauss-Laguerre and Gauss-Hermite quadrature rules with positive weights

1978 ◽  
Vol 29 (6) ◽  
pp. 983-986 ◽  
Author(s):  
D. K. Kahaner ◽  
G. Monegato
Keyword(s):  
Quadrature Rules ◽  
Positive Weights ◽  
Gauss Hermite Quadrature

scholarly journals Generating Nested Quadrature Rules with Positive Weights based on Arbitrary Sample Sets

2020 ◽  
Vol 8 (1) ◽  
pp. 139-169 ◽  
Author(s):  
Laurent van den Bos ◽  
Benjamin Sanderse ◽  
Wim Bierbooms ◽  
Gerard van Bussel
Keyword(s):  
Quadrature Rules ◽  
Positive Weights

scholarly journals Weighted approximate Bayesian computation via Sanov’s theorem

2021 ◽  
Author(s):  
Cecilia Viscardi ◽  
Michele Boreale ◽  
Fabio Corradi
Keyword(s):  
Large Deviations ◽  
Posterior Distribution ◽  
Approximate Bayesian Computation ◽  
Bayesian Computation ◽  
Information Theoretic ◽  
Discrete Random Variables ◽  
Positive Weights ◽  
Approximate Bayesian ◽  
Information Theoretic Method ◽  
Computational Resources

AbstractWe consider the problem of sample degeneracy in Approximate Bayesian Computation. It arises when proposed values of the parameters, once given as input to the generative model, rarely lead to simulations resembling the observed data and are hence discarded. Such “poor” parameter proposals do not contribute at all to the representation of the parameter’s posterior distribution. This leads to a very large number of required simulations and/or a waste of computational resources, as well as to distortions in the computed posterior distribution. To mitigate this problem, we propose an algorithm, referred to as the Large Deviations Weighted Approximate Bayesian Computation algorithm, where, via Sanov’s Theorem, strictly positive weights are computed for all proposed parameters, thus avoiding the rejection step altogether. In order to derive a computable asymptotic approximation from Sanov’s result, we adopt the information theoretic “method of types” formulation of the method of Large Deviations, thus restricting our attention to models for i.i.d. discrete random variables. Finally, we experimentally evaluate our method through a proof-of-concept implementation.

scholarly journals On Appel-Type Quadrature Rules

2002 ◽  
Vol 9 (3) ◽  
pp. 405-412
Author(s):  
C. Belingeri ◽  
B. Germano
Keyword(s):  
Remainder Term ◽  
Quadrature Rule ◽  
Bernoulli Numbers ◽  
Quadrature Rules ◽  
Rule Based ◽  
Numerical Example ◽  
The Given ◽  
Derivatives Of

Abstract The Radon technique is applied in order to recover a quadrature rule based on Appel polynomials and the so called Appel numbers. The relevant formula generalizes both the Euler-MacLaurin quadrature rule and a similar rule using Euler (instead of Bernoulli) numbers and even (instead of odd) derivatives of the given function at the endpoints of the considered interval. In the general case, the remainder term is expressed in terms of Appel numbers, and all derivatives appear. A numerical example is also included.

The equal coefficients quadrature rules and their numerical improvement

2005 ◽  
Vol 171 (2) ◽  
pp. 1331-1351 ◽  
Author(s):  
M.R. Eslahchi ◽  
Mehdi Dehghan ◽  
M. Masjed-Jamei
Keyword(s):  
Quadrature Rules

scholarly journals Revisited Optimal Error Bounds for Interpolatory Integration Rules

2016 ◽  
Vol 2016 ◽  
pp. 1-8 ◽  
Author(s):  
François Dubeau
Keyword(s):  
Error Bounds ◽  
Error Term ◽  
Quadrature Rules ◽  
Optimal Error ◽  
Integration By Parts ◽  
Integration Rules

We present a unified way to obtain optimal error bounds for general interpolatory integration rules. The method is based on the Peano form of the error term when we use Taylor’s expansion. These bounds depend on the regularity of the integrand. The method of integration by parts “backwards” to obtain bounds is also discussed. The analysis includes quadrature rules with nodes outside the interval of integration. Best error bounds for composite integration rules are also obtained. Some consequences of symmetry are discussed.

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