V. I. Romanovskiy Institute of Mathematics, Tashkent, Uzbekistan

Farxod Nuraliyev^1,2, Shahobiddin Qo’ziyev<sup>2


1</sup>Tashkent State Transport University,Tashkent, Uzbekistan,
²V.I.Romanovskiy Institute of Mathematics, Tashkent, Uzbekistan
[e-mail], [e-mail]
Abstract: Numerical integration methods can generally be described as combining evaluations of the integrand to get an approximation to the integral. The integrand is evaluated at a finite set of points called integration points and a weighted sum of these values is used to approximate the integral. The integration points and weights depend on the specific method used and the accuracy required from the approximation.
An important part of the analysis of any numerical integration method is to study the behavior of the approximation error as a function of the number of integrand evaluations. A method that yields a small error for a small number of evaluations is usually considered superior. Reducing the number of evaluations of the integrand reduces the number of arithmetic operations involved, and therefore reduces the total round-off error. Also, each evaluation takes time, and the integrand may be arbitrarily complicated.
Keywords: Optimal quadrature formulas, the error functional, the extremal function, the Sobolev space, the optimal coefficients.
Now we consider the following quadrature formula of the Euler Maclaurin type
\[\int\limits_{0}^{1}{\varphi (x)dx\cong }\sum\limits_{\beta =0}^{N}{{{C}_{0}}\left[ \beta \right]}\varphi (h\beta )+\frac{{{h}^{2}}}{12}(\varphi '(0)-\varphi '(1))+\sum\limits_{\beta =0}^{N}{{{C}_{1}}\left[ \beta \right]}\varphi ''(h\beta )\] (1)
with the error functional
\[{{\ell }_{N}}(x)={{\varepsilon }_{\left[ 0,1 \right]}}(x)-\sum\limits_{\beta =0}^{N}{{{C}_{0}}\left[ \beta \right]}\delta (x-h\beta )+\frac{{{h}^{2}}}{12}(\delta '(x)-\delta '(x-1))-\sum\limits_{\beta =0}^{N}{{{C}_{1}}\left[ \beta \right]}\delta ''(x-h\beta )\] (2)
The aim of this paper is to construct the Euler–Maclaurin type optimal quadrature formulas of the form (1) in the sense of Sard in the space \[L_{2}^{(m)}(0,1)\], i.e. to get explicit expressions for the coefficients \[{{C}_{1}}\left[ \beta \right]\]of the optimal quadrature formulas (1) which are very useful in applications and to study order of convergence of the obtained optimal quadrature formulas.
The error of the quadrature formula (1) is the following difference

(3)
For the error functional (2) to be defined on the space \[L_{2}^{(m)}(0,1)\] it is necessary to impose the following conditions (see [37])
\[({{\ell }_{N}},{{x}^{\alpha }})=0,\,\,\,\,\,\,\alpha =0,1,...,m-1.\] (4)
Hence it is clear that for existence of the quadrature formulas of the form (1) the condition \[N+3\ge m\] has to be met.
Note that here in after \[{{\ell }_{N}}\] means the functional (2).
As was noted above by the Cauchy–Schwarz inequality, the error of the formula (1) is estimated by the norm \[\left\| {{\ell }_{N}}\left| L{{_{2}^{(m)}}^{*}}(0,1) \right. \right\|\] of the error functional (2). Furthermore the norm of the error functional (2) depends on the coefficients \[{{C}_{1}}\left[ \beta \right]\]. We minimize the norm of the error functional (2) by the coefficients \[{{C}_{1}}\left[ \beta \right]\], i.e., we find
(5)
The coefficients which satisfy the equality (5) are called the optimal coefficients and are denoted by and the corresponding quadrature formula is called the optimal quadrature formula in the sense of Sard. In the sequel, for the purposes of convenience the optimal coefficients will be denoted as .
Thus to construct optimal quadrature formulas in the form (1) in the sense of Sard we have to consequently solve the following problems.
Problem 1. Find coefficients which satisfy the equality (5).
Problem 2. Find the solution of the equation
(6)
having the form:


(7)
where and are unknown polynomials of degree .
If we find and
(8)
(9)
Unknowns , can be found from the discrete analogue of the differential operator , using the discrete argument function . Then we can obtain the explicit form of the function and respectively we can find the optimal coefficients . But here we will not find , . Instead, using and the form (6) of the discrete argument function , taking into account , we find the expressions for the optimal coefficients when . We introduce the following notations

(10)


(11)
(12)
where , is the Euler–Frobenius polynomial of degree , note that because of the series in the (10) and (11) are convergent.
The following holds
Theorem 1. The coefficients , of the optimal quadrature formulas of the form (4) in the space , have the following form
(13)
where , are defined by (10) and (11).