An optimal quadrature formula in the Sobolev space

2021 ◽  
Vol 65 (3) ◽  
pp. 46-59
Keyword(s):  
Sobolev Space ◽  
Quadrature Formula ◽  
Approximate Calculation ◽  
Order Of Convergence ◽  
Quadrature Formulas ◽  
Explicit Formulas ◽  
Optimal Quadrature Formula ◽  
Optimal Quadrature ◽  
Optimal Coefficients ◽  
Optimal Quadrature Formulas

This paper studies the problem of construction of optimal quadrature formulas for approximate calculation of integrals with trigonometric weight in the L(2m)(0, 1) space for any ω ൐= 0, ω ∈ R. Here explicit formulas for the optimal coefficients are obtained. We study the order of convergence of the optimal formulas for the case m = 1, 2. The obtained optimal quadrature formulas are exact for Pm−1(x), where Pm−1(x) is a polynomial of degree (m − 1).

scholarly journals Optimal quadrature formula in the sense of Sard in K2(P3) space

2014 ◽  
Vol 95 (109) ◽  
pp. 29-47 ◽  
Author(s):  
Abdullo Hayotov ◽  
Gradimir Milovanovic ◽  
Kholmat Shadimetov
Keyword(s):  
Hilbert Space ◽  
Sobolev Space ◽  
Quadrature Formula ◽  
Asymptotic Optimality ◽  
Trigonometric Functions ◽  
Numerical Examples ◽  
Optimal Quadrature Formula ◽  
Optimal Quadrature ◽  
Optimal Coefficients ◽  
Optimal Formula

We construct an optimal quadrature formula in the sense of Sard in the Hilbert space K2(P3). Using Sobolev?s method we obtain new optimal quadrature formula of such type and give explicit expressions for the corresponding optimal coefficients. Furthermore, we investigate order of the convergence of the optimal formula and prove an asymptotic optimality of such a formula in the Sobolev space L (3)2 (0, 1). The obtained optimal quadrature formula is exact for the trigonometric functions sin x, cos x and for constants. Also, we include a few numerical examples in order to illustrate the application of the obtained optimal quadrature formula.

Optimal quadrature formulas with derivatives in the Sobolev space

2018 ◽  
Vol 2018 (3) ◽  
pp. 18-28
Author(s):  
D.M. Akhmedov ◽  
N.H. Mamatova ◽  
F.A. Nuraliyev
Keyword(s):  
Sobolev Space ◽  
Quadrature Formulas ◽  
Optimal Quadrature ◽  
Optimal Quadrature Formulas

scholarly journals The order of convergence of an optimal quadrature formula with derivative in the space W(2,1)2

Filomat ◽  
2020 ◽  
Vol 34 (11) ◽  
pp. 3835-3844
Author(s):  
A.R. Hayotov ◽  
R.G. Rasulov
Keyword(s):  
Quadrature Formula ◽  
Order Of Convergence ◽  
Discrete Analog ◽  
Trapezoidal Rule ◽  
Optimal Quadrature Formula ◽  
Optimal Quadrature ◽  
First Derivative ◽  
Error Functional ◽  
The Error Functional ◽  
Norm Of The Error

The present work is devoted to extension of the trapezoidal rule in the space W(2,1)2. The optimal quadrature formula is obtained by minimizing the error of the formula by coefficients at values of the first derivative of an integrand. Using the discrete analog of the operator d2/dx2-1 the explicit formulas for the coefficients of the optimal quadrature formula are obtained. Furthermore, it is proved that the obtained quadrature formula is exact for any function of the set F = span{1,x,ex,e-x}. Finally, in the space W(2,1) 2 the square of the norm of the error functional of the constructed quadrature formula is calculated. It is shown that the error of the obtained optimal quadrature formula is less than the error of the Euler-Maclaurin quadrature formula on the space L(2)2 .

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