scholarly journals A Note on Finite Quadrature Rules with a Kind of Freud Weight Function

2009 ◽  
Vol 2009 ◽  
pp. 1-8
Author(s):  
Kamal Aghigh ◽  
M. Masjed-Jamei
Keyword(s):  
Weight Function ◽  
Quadrature Rule ◽  
Quadrature Rules ◽  
Zeros Of Polynomials ◽  
Freud Weight ◽  
Weighted Quadrature

We introduce a finite class of weighted quadrature rules with the weight function on as , where are the zeros of polynomials orthogonal with respect to the introduced weight function, are the corresponding coefficients, and is the error value. We show that the above formula is valid only for the finite values of . In other words, the condition must always be satisfied in order that one can apply the above quadrature rule. In this sense, some numerical and analytic examples are also given and compared.

Weighted quadrature rules with weight function on [0,∞)

2006 ◽  
Vol 180 (1) ◽  
pp. 1-6 ◽  
Author(s):  
Mehdi Dehghan ◽  
M. Masjed-Jamei ◽  
M.R. Eslahchi
Keyword(s):  
Weight Function ◽  
Quadrature Rules ◽  
Weighted 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 Explicit forms of weighted quadrature rules with geometric nodes

2011 ◽  
Vol 53 (5-6) ◽  
pp. 1133-1139 ◽  
Author(s):  
Mohammad Masjed-Jamei ◽  
Gradimir V. Milovanović ◽  
M.A. Jafari
Keyword(s):  
Quadrature Rules ◽  
Weighted Quadrature

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 

scholarly journals One-dimensional adaptive grid generation

1997 ◽  
Vol 20 (3) ◽  
pp. 577-584 ◽  
Author(s):  
Ahmed M. M. Khodier ◽  
Adel Y. Hassan
Keyword(s):  
Weight Function ◽  
Numerical Solution ◽  
Numerical Solutions ◽  
Single Point ◽  
Grid Generation ◽  
Quadrature Rule ◽  
Adaptive Grid ◽  
Solution Function ◽  
One Dimensional ◽  
Adaptive Grid Generation

In this work, we give an adaptive grid generation method which allows a single point to be added in the regions of large variation. This method uses a quadrature rule as a weight function. Our weight function measures the variation of the solution function on each subinterval of the solution domain. The method is applied to obtain the numerical solutions of some differential equations. A comparison of the numerical solution obtained by this method and other methods is given.

A unified representation for some interpolation formulas

Analysis ◽  
2020 ◽  
Vol 40 (3) ◽  
pp. 113-125
Author(s):  
Mohammad Masjed-Jamei ◽  
Zahra Moalemi ◽  
Wolfram Koepf
Keyword(s):  
Existence And Uniqueness ◽  
Lagrange Interpolation ◽  
Quadrature Rules ◽  
Numerical Examples ◽  
Weighted Quadrature

AbstractAs an extension of Lagrange interpolation, we introduce a class of interpolation formulas and study their existence and uniqueness. In the sequel, we consider some particular cases and construct the corresponding weighted quadrature rules. Numerical examples are finally given and compared.

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