scholarly journals Closed expressions for coefficients in weighted Newton-Cotes quadratures

Filomat ◽  
2013 ◽  
Vol 27 (4) ◽  
pp. 649-658 ◽  
Author(s):  
Mohammad Masjed-Jamei ◽  
Gradimir Milovanovic ◽  
M.A. Jafari
Keyword(s):  
Short Note ◽  
Stirling Numbers ◽  
Quadrature Formulas ◽  
Numerical Examples ◽  
Quadrature Formulae ◽  
Equidistant Nodes

In this short note, we derive closed expressions for Cotes numbers in the weighted Newton-Cotes quadrature formulae with equidistant nodes in terms of moments and Stirling numbers of the first kind. Three types of equidistant nodes are considered. The corresponding program codes in Mathematica Package are presented. Finally, in order to illustrate the application of the obtained quadrature formulas a few numerical examples are included.

scholarly journals Generalized Gaussian quadratures for integrals with logarithmic singularity

Filomat ◽  
2016 ◽  
Vol 30 (4) ◽  
pp. 1111-1126 ◽  
Author(s):  
Gradimir Milovanovic
Keyword(s):  
Logarithmic Singularity ◽  
Gaussian Quadrature ◽  
Quadrature Rules ◽  
Quadrature Formulas ◽  
Short Account ◽  
Numerical Examples ◽  
Gaussian Quadrature Formulas ◽  
Gaussian Quadratures ◽  
Logarithmic Singularities

A short account on Gaussian quadrature rules for integrals with logarithmic singularity, as well as some new results for weighted Gaussian quadrature formulas with respect to generalized Gegenbauer weight x |? |x|(1-x2)?, ? > -1, on (-1,1), which are appropriated for functions with and without logarithmic singularities, are considered. Methods for constructing such kind of quadrature formulas and some numerical examples are included.

A SHORT NOTE ON THE BOUNDEDNESS ANALYSIS AND CONTROL OF THE SPATIAL FRACTAL SET FROM A KIND OF CHAIN COUPLING LOGISTIC TYPE MAP

Fractals ◽  
2020 ◽  
Vol 28 (04) ◽  
pp. 2050060
Author(s):  
DA WANG ◽  
YANG ZHAO ◽  
YI ZHANG ◽  
XIYU LIU
Keyword(s):  
Short Note ◽  
Upper Bounds ◽  
Coupled Systems ◽  
Strongly Coupled ◽  
Numerical Examples ◽  
Fractal Set ◽  
Coupling System ◽  
Preliminary Study ◽  
System Variables ◽  
And Control

This paper presents a preliminary study about a kind of chain coupling system, which we hope could have some enlightened effect for the research on the spatial fractal set of more strongly coupled systems. By analyzing the trajectories of system variables and applying the magnifying or reducing method, the upper bounds of the original and controlled Julia sets from the proposed system are given. Numerical examples are also included to verify the conclusions.

scholarly journals Orthogonal polynomials and generalized Gauss-Rys quadrature formulae

2021 ◽  
Vol 49 (1) ◽  
Author(s):  
Gradimir V. Milovanovic ◽  
◽  
Nevena Vasovic ◽  
Keyword(s):  
Orthogonal Polynomials ◽  
Quadrature Rules ◽  
Quadrature Formulas ◽  
Numerical Construction ◽  
Quadrature Formulae ◽  
Gaussian Type ◽  
Modified Moments

Orthogonal polynomials and the corresponding quadrature formulas of Gaussian type concerni λ ng > the 1 e / v 2 en wei x gh > t f 0 unction ω(t; x) = exp λ (−= xt 1 2) / ( 2 1 − t2)−1/2 on (−1, 1), with parameters − and , are considered. For these quadrature rules reduce to the socalled Gauss-Rys quadrature formulas, which were investigated earlier by several authors, e.g., Dupuis at al 1976 and 1983; Sagar 1992; Schwenke 2014; Shizgal 2015; King 2016; Milovanovic ´ 2018, etc. In this generalized case, the method of modified moments is used, as well as a transformation of quadratures on (−1, 1) with N nodes to ones on (0, 1) with only (N + 1)/2 nodes. Such an approach provides a stable and very efficient numerical construction.

scholarly journals Double condensation of singularities for product-quadrature formulas with differentiable functions

2012 ◽  
Vol 28 (1) ◽  
pp. 83-91
Author(s):  
ALEXANDRU I. MITREA ◽  
Keyword(s):  
Quadrature Formulas ◽  
Triangular Matrices ◽  
Differentiable Functions ◽  
Equidistant Nodes ◽  
The Matrix

The main goal of this paper is to emphasize the phenomenon of double condensation of singularities for spaces of differentiable functions with respect to product-quadrature formulas associated to a class of node triangular matrices in [−1, 1], including a Gegenbauer node matrix and the matrix of equidistant nodes in [−1, 1].

P-SV reflection moveouts for transversely isotropic media with a vertical symmetry axis

Geophysics ◽  
1991 ◽  
Vol 56 (8) ◽  
pp. 1271-1274 ◽  
Author(s):  
A. J. Seriff ◽  
K. P. Sriram
Keyword(s):  
Symmetry Axis ◽  
Short Note ◽  
Transversely Isotropic ◽  
Isotropic Layer ◽  
Numerical Examples ◽  
The Real ◽  
Polar Anisotropy ◽  
Transversely Isotropic Media ◽  
Isotropic Media ◽  
Transversely Isotropic Layer

In a recently published short note, F. K. Levin (1989) discusses the relation between the “moveout velocities” of P-P, P-SV, and SV-SV reflections from the bottom of a transversely isotropic layer with a vertical symmetry axis. We refer to such a medium as one exhibiting “polar anisotropy.” Levin’s note was prompted by a paper of Tessmer and Behle (1988), and it is relevant to a paper by Iverson and others (1989), both of which discuss the computation of shear velocities from moveout velocities obtained with P-P and P-S reflections. Levin’s note addresses the practically important question of the use of this method in the presence of polar anisotropy, a phenomenon which we believe occurs almost universally in the sedimentary layers of the real earth. Levin suggests that polar anisotropy of “typical” magnitude must be considered in this problem. He uses as an estimate of typical magnitudes data given by Thomsen (1986) and concludes from numerical examples that the method of estimating shear velocities proposed by Tessmer and Behle and by Iverson may be subject to unacceptably large errors in many real cases. Moreover, Levin suggests that the source of these errors is mysterious.

scholarly journals Comparing the Performance of Four Sinc Methods for Numerical Indefinite Integration

2021 ◽  
pp. 7-41
Author(s):  
Richard Olatokunbo Akinola
Keyword(s):  
Numerical Approximation ◽  
Quadrature Formulas ◽  
Numerical Examples ◽  
Sinc Method ◽  
Cpu Time ◽  
The Third ◽  
Sinc Methods ◽  
Quadrature Methods ◽  
Double Exponential ◽  
Single Exponential

Aims/ Objectives: To compare the performance of four Sinc methods for the numerical approximation of indefinite integrals with algebraic or logarithmic end-point singularities. Methodology: The first two quadrature formulas were proposed by Haber based on the sinc method, the third is Stengers Single Exponential (SE) formula and Tanaka et al.s Double Exponential (DE) sinc method completes the number. Furthermore, an application of the four quadrature formulas on numerical examples, reveals convergence to the exact solution by Tanaka et al.s DE sinc method than by the other three formulae. In addition, we compared the CPU time of the four quadrature methods which was not done in an earlier work by the same author. Conclusion: Haber formula A is the fastest as revealed by the CPU time.

scholarly journals Ostrowski type inequalities and some selected quadrature formulae

2020 ◽  
pp. 54-54
Author(s):  
Gradimir Milovanovic
Keyword(s):  
Numerical Integration ◽  
Error Term ◽  
Quadrature Formulas ◽  
Quadrature Formulae ◽  
Survey Paper ◽  
Ostrowski’S Inequality ◽  
Basic Facts ◽  
Peano Kernel

Some selected Ostrowski type inequalities and a connection with numerical integration are studied in this survey paper, which is dedicated to the memory of Professor D. S. Mitrinovic, who left us 25 years ago. His significant inuence to the development of the theory of inequalities is briefly given in the first section of this paper. Beside some basic facts on quadrature formulas and an approach for estimating the error term using Ostrowski type inequalities and Peano kernel techniques, we give several examples of selected quadrature formulas and the corresponding inequalities, including the basic Ostrowski's inequality (1938), inequality of Milovanovic and Pecaric (1976) and its modifications, inequality of Dragomir, Cerone and Roumeliotis (2000), symmetric inequality of Guessab and Schmeisser (2002) and asymmetric in-equality of Franjic (2009), as well as four point symmetric inequalites by Alomari (2012) and a variant with double internal nodes given by Liu and Park (2017).

Remarks on randomization of quasi-random numbers

2018 ◽  
Vol 24 (2) ◽  
pp. 139-145 ◽  
Author(s):  
Sergej M. Ermakov ◽  
Svetlana N. Leora
Keyword(s):  
Monte Carlo ◽  
Quadrature Formulas ◽  
Random Numbers ◽  
Free Node ◽  
Shift Method ◽  
Numerical Examples ◽  
Quasi Monte Carlo ◽  
The Subject ◽  
Halton Sequences ◽  
Special Case

Abstract In this paper we discuss estimation of the quasi-Monte Carlo methods error in the case of calculation of high-order integrals. Quasi-random Halton sequences are considered as a special case. Randomization of these sequences by the random shift method turns out to lead to well-known random quadrature formulas with one free node. Some new properties of such formulas are pointed out. The subject is illustrated by a number of numerical examples.

scholarly journals Trigonometric multiple orthogonal polynomials of semi-integer degree and the corresponding quadrature formulas

2014 ◽  
Vol 96 (110) ◽  
pp. 211-226 ◽  
Author(s):  
Gradimir Milovanovic ◽  
Marija Stanic ◽  
Tatjana Tomovic
Keyword(s):  
Orthogonal Polynomials ◽  
Trigonometric Polynomials ◽  
Quadrature Rules ◽  
Quadrature Formulas ◽  
Numerical Examples ◽  
Multiple Orthogonal Polynomials ◽  
Optimal Set ◽  
Numer Math ◽  
Theoretical Results

An optimal set of quadrature formulas with an odd number of nodes for trigonometric polynomials in Borges? sense [Numer. Math. 67 (1994), 271-288], as well as trigonometric multiple orthogonal polynomials of semi-integer degree are defined and studied. The main properties of such a kind of orthogonality are proved. Also, an optimal set of quadrature rules is characterized by trigonometric multiple orthogonal polynomials of semiinteger degree. Finally, theoretical results are illustrated by some numerical examples.

scholarly journals Construction of Gaussian quadrature formulas for even weight functions

2017 ◽  
Vol 11 (1) ◽  
pp. 177-198 ◽  
Author(s):  
Mohammad Masjed-Jamei ◽  
Gradimir Milovanovic
Keyword(s):  
Weight Function ◽  
Jacobi Matrix ◽  
Gaussian Quadrature ◽  
Quadrature Rule ◽  
Weight Functions ◽  
Quadrature Formulas ◽  
Numerical Examples ◽  
Numerical Instabilities ◽  
Gaussian Quadrature Formulas ◽  
Free Parameters

Instead of a quadrature rule of Gaussian type with respect to an even weight function on (?a, a) with n nodes, we construct the corresponding Gaussian formula on (0, a2) with only [(n+1)/2] nodes. Especially, such a procedure is important in the cases of nonclassical weight functions, when the elements of the corresponding three-diagonal Jacobi matrix must be constructed numerically. In this manner, the influence of numerical instabilities in the process of construction can be significantly reduced, because the dimension of the Jacobi matrix is halved. We apply this approach to Pollaczek?s type weight functions on (?1, 1), to the weight functions on R which appear in the Abel-Plana summation processes, as well as to a class of weight functions with four free parameters, which covers the generalized ultraspherical and Hermite weights. Some numerical examples are also included.

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