UDC 518 : 517.392

ESTIMATING THE UPPER BOUND OF THE ERROR OF QUADRATURE FORMULAS ON A CLASS OF PERIODIC FUNCTIONS

V. T. KALAIDA, V. K. YASEL’SKII

(Presented by the scientific seminar of the Faculty of Physics and Technology)

In solving physical problems by methods of numerical analysis, along with estimating the lower bound of the error of numerical integration formulas, in order to choose the type of quadrature formula correctly it is necessary to estimate the integration error from above; this estimate depends on the coefficients of the quadrature formulas, the nodes, and the number of integration points.

Consider an arbitrary quadrature formula

\[ \int_{0}^{1} f(x)\,dx=\sum_{K=1}^{N} C_K f(x_K), \tag{1} \]

where \(C_K\) are the coefficients of the quadrature formula;
\(x_K\) are the nodes of the quadrature formula;
\(N\) is the number of nodes.

Of special interest for practice is the case when the function \(f(x)\) is periodic, since in applications the representation of a function in the form of trigonometric series is often used.

The aim of the present work is to determine an expression for the upper bound of the error of numerical integration for the case of periodic functions.

Let \(W_2^m\) be the class of one-periodic functions defined on the interval \([0,1]\), having an absolutely continuous derivative of order \(m-1\) and a derivative of order \(m\) integrable with square.

As follows from [1], the upper bound of the error of the quadrature formula \(R(C_K,x_K,N)\), defined by the relation

\[ R(C_K,x_K,N)=\sup_{\|f\|_{W_2^m}=1}|(l,f)|, \tag{2} \]

where

\[ (l,f)=\int_{0}^{1} f(x)\,dx-\sum_{K=1}^{N} C_K f(x_K), \]

is attained on the function

\[ u_2=\frac{u_1}{\sqrt{d}}\in W_2^m. \tag{3} \]

Here \(u_1\) is a periodic solution of the equation

\[ \frac{d^{2m}u_1}{dx^{2m}} = (-1)^m \left[ 1-\sum_{K=1}^{N} C_K \delta(x-x_K) \right], \tag{4} \]

where \(\delta(x-x_K)\) is the Dirac delta function, and

\[ d=\sum_{K=1}^{N} C_K u_1(x_K)-\int_{0}^{1} u_1\,dx . \tag{5} \]

Let us consider the solution of equation (4). Denoting by \(u_l\) the periodic solution of the equation

\[ \frac{d^{2m}u_l}{dx^{2m}} = (-1)^m[1-\delta(x-x_l)] \tag{6} \]

and taking into account that \(\sum_{l=1}^{N} C_l=1\), we may write

\[ u_1=\sum_{l=1}^{N} C_l u_l . \tag{7} \]

Since the delta function \(\delta(x-x_l)\) on the interval \([0,1]\) admits a Fourier-series expansion according to [2],

\[ \delta(x-x_l)=\sum_{\gamma=-\infty}^{\infty} e^{2\pi i(x-x_l)\gamma}, \tag{8} \]

equation (6), in view of (8), may be written in the form

\[ \frac{d^{2m}u_l}{dx^{2m}} = (-1)^m \sum_{\substack{\gamma=-\infty\\ \gamma\ne 0}}^{\infty} e^{2\pi i(x-x_l)\gamma}. \tag{9} \]

Taking into account that, in the periodic case, the boundary-value problem (9) has no boundary conditions, we obtain the solution for \(u_l\) by integrating the right-hand side of this expression \(2m\) times. Noting that

\[ \sum_{\substack{\gamma=-\infty\\ \gamma\ne 0}}^{\infty} e^{2\pi i(x-x_l)\gamma} = \sum_{\gamma=1}^{\infty} e^{2\pi i(x-x_l)\gamma} + \sum_{\gamma=1}^{\infty} e^{-2\pi i(x-x_l)\gamma} = \]

\[ = 2\sum_{\gamma=1}^{\infty}\cos 2\pi(x-x_l)\gamma, \tag{10} \]

and taking relation (10) into account, we find for \(u_l\)

\[ u_l= \frac{2}{(2\pi)^{2m}} \sum_{\gamma=1}^{\infty} \frac{\cos 2\pi(x-x_l)\gamma}{\gamma^{2m}} . \tag{11} \]

From expressions (7) and (11), the solution for the function \(u_1\) is easily obtained:

\[ u_1= \frac{2}{(2\pi)^{2m}} \sum_{l=1}^{N} C_l \sum_{\gamma=1}^{\infty} \frac{\cos 2\pi(x-x_l)\gamma}{\gamma^{2m}} . \tag{12} \]

From (1) it follows that

\[ R(C_K,x_K,N)=(l,u_2)=-\frac{1}{\sqrt{d}}(l,u_1)=\sqrt{d}. \tag{13} \]

Taking into account that \(\int_0^1 u_1\,dx=0\) for \(R(C_K,x_K,N)\), we may write

\[ \begin{aligned} R^2(C_K,x_K,N)=d &=\sum_{K=1}^{N} C_K u_1(x_K) \\ &=\frac{2}{(2\pi)^{2m}}\sum_{K=1}^{N} C_K \sum_{l=1}^{N} C_l \sum_{\gamma=1}^{\infty} \frac{\cos 2\pi(x_K-x_l)\gamma}{\gamma^{2m}} . \end{aligned} \tag{14} \]

The expression obtained in (14) is inconvenient for the practical estimation of errors, since it contains a sum with an infinite number of terms. We transform (14) according to the relation [3]

\[ \sum_{\gamma=1}^{\infty} \frac{\cos 2\pi(x_K-x_l)\gamma}{\gamma^{2m}} = (-1)^{m-1}\frac{(2\pi)^{2m}}{(2m)!}\,B_{2m}(x_K-x_l), \tag{15} \]

where \(B_{2m}(x)\) are the Bernoulli polynomials.

Taking into account that \(B_{2m}(0)=B_{2m}\) (\(B_{2m}\) are the Bernoulli numbers), and separating all terms with equal indices \((K=l)\), equality (14) can be written in the form

\[ R^2(C_K,x_K,N) = \frac{1}{(2m)!}\,|B_{2m}|\sum_{K=1}^{N} C_K^2 + \frac{2(-1)^{m-1}}{(2m)!} \sum_{K=1}^{N-1} C_K \sum_{l=K+1}^{N} C_l B_{2m}(x_l-x_K). \tag{16} \]

For Bernoulli polynomials the relation [3] holds

\[ B_n(x)=\sum_{r=0}^{n}\binom{n}{r}B_r x^{\,n-r}, \tag{17} \]

where \(\binom{n}{r}\) is the number of combinations of \(n\) taken \(r\) at a time.

Then the final expression for the upper bound of the error of quadrature formulas on the class of one-periodic functions can be written as

\[ R^2(C_K,x_K,N) = \frac{1}{(2m)!}\,|B_{2m}|\sum_{K=1}^{N} C_K^2 + \frac{2(-1)^{m-1}}{(2m)!} \sum_{K=1}^{N-1} C_K \sum_{l=K+1}^{N} C_l \sum_{r=0}^{2m}\binom{2m}{r}B_r(x_K-x_l)^{2m-r}. \tag{18} \]

The expression obtained is convenient for practical use.

References

1. S. L. Sobolev. On formulas of mechanical cubature in \(n\)-dimensional space. Doklady Akademii nauk SSSR, 1961, vol. 137, No. 3.

2. I. M. Gel'fand, G. E. Shilov. Generalized Functions and Operations on Them. Moscow, Fizmatgiz, 1959.

3. I. S. Gradshteyn and I. M. Ryzhik. Tables of Integrals, Sums, Series, and Products. Moscow, “Nauka,” 1971.