scholarly journals New error bounds for Gauss-Legendre quadrature rules

Filomat ◽  
2014 ◽  
Vol 28 (6) ◽  
pp. 1281-1293 ◽  
Author(s):  
Mohammad Masjed-Jamei
Keyword(s):  
Quadrature Formula ◽  
Error Bounds ◽  
Quadrature Rule ◽  
Order Derivative ◽  
Quadrature Rules ◽  
Linear Kernel ◽  
Large N ◽  
Interpolatory Quadrature ◽  
Integrand Function ◽  
Legendre Quadrature

It is well-known that the remaining term of any n-point interpolatory quadrature rule such as Gauss-Legendre quadrature formula depends on at least an n-order derivative of the integrand function, which is of no use if the integrand is not smooth enough and requires a lot of differentiation for large n. In this paper, by defining a specific linear kernel, we resolve this problemand obtain new bounds for the error of Gauss-Legendre quadrature rules. The advantage of the obtained bounds is that they do not depend on the norms of the integrand function. Some illustrative examples are given in this direction.

Unified error bounds for all Newton–Cotes quadrature rules

2015 ◽  
Vol 23 (1) ◽  
Author(s):  
Mohammad Masjed-Jamei
Keyword(s):  
Error Bound ◽  
Error Bounds ◽  
Order Derivative ◽  
Quadrature Rules ◽  
Stringent Condition ◽  
Large N ◽  
Almost All ◽  
Integrand Function

AbstractIt is well-known that the remaining term of a classical n-point Newton-Cotes quadrature depends on at least an n-order derivative of the integrand function. Discounting the fact that computing an n-order derivative requires a lot of differentiation for large n, the main problem is that an error bound for an n-point Newton-Cotes quadrature is only relevant for a function that is n times differentiable, a rather stringent condition. In this paper, by defining two specific linear kernels, we resolve this problem and obtain new error bounds for all closed and open types of Newton-Cotes quadrature rules. The advantage of the obtained bounds is that they do not depend on the norms of the integrand function and are very general such that they cover almost all existing results in the literature. Some illustrative examples are given in this direction.

scholarly journals A Mixed Quadrature Rule by Blending Clenshaw-Curtis and Gauss-Legendre Quadrature Rules for Approximate Evaluation of Real Definite Integrals

2012 ◽  
Vol 2 ◽  
pp. 10-15
Author(s):  
Anasuya Pati ◽  
Rajani B. Dash ◽  
Pritikanta Patra
Keyword(s):  
Quadrature Rule ◽  
Quadrature Rules ◽  
Approximate Evaluation ◽  
Definite Integrals ◽  
Legendre Quadrature

A mixed quadrature rule blending Clenshaw-Curtis five point rule and Gauss-Legendre three point rule is formed. The mixed rule has been tested and found to be more effective than that of its constituent Clenshaw-Curtis five point rule.

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 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.

scholarly journals Spectral Limitations of Quadrature Rules and Generalized Spherical Designs

2019 ◽  
Author(s):  
Stefan Steinerberger
Keyword(s):  
Quadrature Rule ◽  
Quadrature Rules ◽  
Dimensional Manifold ◽  
Spherical Designs

Abstract We study manifolds $M$ equipped with a quadrature rule \begin{equation*} \int_{M}{\phi(x)\,\mathrm{d}x} \simeq \sum_{i=1}^{n}{a_i \phi(x_i)}.\end{equation*}We show that $n$-point quadrature rules with nonnegative weights on a compact $d$-dimensional manifold cannot integrate more than at most the 1st $c_{d}n + o(n)$ Laplacian eigenfunctions exactly. The constants $c_d$ are explicitly computed and $c_2 = 4$. The result is new even on $\mathbb{S}^2$ where it generalizes results on spherical designs.

scholarly journals Variational Integrators in Holonomic Mechanics

Mathematics ◽  
2020 ◽  
Vol 8 (8) ◽  
pp. 1358
Author(s):  
Shumin Man ◽  
Qiang Gao ◽  
Wanxie Zhong
Keyword(s):  
Variational Principle ◽  
Dynamic Systems ◽  
Time Integration ◽  
Quadrature Rule ◽  
Quadrature Rules ◽  
Variational Integrators ◽  
Lagrange Polynomials ◽  
Holonomic Constraints ◽  
Long Time ◽  
Long Time Integration

Variational integrators for dynamic systems with holonomic constraints are proposed based on Hamilton’s principle. The variational principle is discretized by approximating the generalized coordinates and Lagrange multipliers by Lagrange polynomials, by approximating the integrals by quadrature rules. Meanwhile, constraint points are defined in order to discrete the holonomic constraints. The functional of the variational principle is divided into two parts, i.e., the action of the unconstrained term and the constrained term and the actions of the unconstrained term and the constrained term are integrated separately using different numerical quadrature rules. The influence of interpolation points, quadrature rule and constraint points on the accuracy of the algorithms is analyzed exhaustively. Properties of the proposed algorithms are investigated using examples. Numerical results show that the proposed algorithms have arbitrary high order, satisfy the holonomic constraints with high precision and provide good performance for long-time integration.

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