The Levin–Stečkin Inequality and Simple Quadrature Rules

2019 ◽  
pp. 583-588
Author(s):  
Peter R. Mercer
Keyword(s):  
Quadrature Rules ◽  
Simple Quadrature

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.

scholarly journals Why simple quadrature is just as good as Monte Carlo

2020 ◽  
Vol 26 (1) ◽  
pp. 1-16
Author(s):  
Kevin Vanslette ◽  
Abdullatif Al Alsheikh ◽  
Kamal Youcef-Toumi
Keyword(s):  
Monte Carlo ◽  
Random Sampling ◽  
Bayesian Probability ◽  
Independent Factor ◽  
Worst Case ◽  
Statistical Equivalence ◽  
Deterministic Sampling ◽  
Quadrature Methods ◽  
Quadrature Sampling ◽  
Simple Quadrature

AbstractWe motive and calculate Newton–Cotes quadrature integration variance and compare it directly with Monte Carlo (MC) integration variance. We find an equivalence between deterministic quadrature sampling and random MC sampling by noting that MC random sampling is statistically indistinguishable from a method that uses deterministic sampling on a randomly shuffled (permuted) function. We use this statistical equivalence to regularize the form of permissible Bayesian quadrature integration priors such that they are guaranteed to be objectively comparable with MC. This leads to the proof that simple quadrature methods have expected variances that are less than or equal to their corresponding theoretical MC integration variances. Separately, using Bayesian probability theory, we find that the theoretical standard deviations of the unbiased errors of simple Newton–Cotes composite quadrature integrations improve over their worst case errors by an extra dimension independent factor {\propto N^{-\frac{1}{2}}}. This dimension independent factor is validated in our simulations.

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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