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 Evaluation of real denite integrals by using of mixed quadrature rules over a Triangular Domain

2016 ◽  
Vol 12 (2) ◽  
pp. 5945-5948
Author(s):  
Rajani Ballav Dash ◽  
Pritikanta Patra ◽  
Dwitikrushna Behera
Keyword(s):  
Quadrature Rule ◽  
Quadrature Rules ◽  
Two Dimensional ◽  
Approximate Evaluation ◽  
Suitable Test ◽  
Triangular Domain ◽  
Definite Integrals

A mixed quadrature rule of higher precision for approximate evaluation of real definite integrals over a triangular domain has been constructed.The relative ecienciesof the proposed mixed quadrature rule has been veried by using suitable test inte-grals.In this paper we present a mixed quadrature i.e. mixed quadrature of anti-Lobatto rule and Fejer's rst rule in one variable.For real denite integral over the triangular surface : f(x; y)j0 x; y 1; x + y 1g in the Cartesian two dimensional (x,y) space.Mathematical transformation from (x,y) space to (; ) space maps the standard triangle in (x,y) space to a standard 2-square in (:) space: f(; )j 1 ; 1g.

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.

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 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.

scholarly journals Application of MATLAB Symbolic Maths with Variable Precision Arithmetic (VPA) to Compute Some High Order Gauss Legendre Quadrature Rules

1970 ◽  
Vol 29 ◽  
pp. 117-125
Author(s):  
HT Rathod ◽  
RD Sathish ◽  
Md Shafiqul Islam ◽  
Arun Kumar Gali
Keyword(s):  
Gaussian Quadrature ◽  
High Order ◽  
Quadrature Rules ◽  
Zeros Of Polynomials ◽  
Matlab Program ◽  
Present Context ◽  
First Time ◽  
Legendre Quadrature ◽  
Weight Coefficients ◽  
Integration Rules

Gauss Legendre Quadrature rules are extremely accurate and they should be considered seriously when many integrals of similar nature are to be evaluated. This paper is concerned with the derivation and computation of numerical integration rules for the three integrals: (See text for formulae) which are dependent on the zeros and the squares of the zeros of Legendre Polynomial and is quite well known in the Gaussian Quadrature theory. We have developed the necessary MATLAB programs based on symbolic maths which can compute the sampling points and the weight coefficients and are reported here upto 32 – digits accuracy and we believe that they are reported to this accuracy for the first time. The MATLAB programs appended here are based on symbolic maths. They are very sophisticated and they can compute Quadrature rules of high order, whereas one of the recent MATLAB program appearing in reference [21] can compute Gauss Legendre Quadrature rules upto order twenty, because the zeros of Legendre polynomials cannot be computed to desired accuracy by MATLAB routine roots (……..). Whereas we have used the MATLAB routine solve (……..) to find zeros of polynomials which is very efficient. This is worth noting in the present context. GANIT J. Bangladesh Math. Soc. (ISSN 1606-3694) 29 (2009) 117-125  DOI: http://dx.doi.org/10.3329/ganit.v29i0.8521 

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