UDC 681.172.088

DETERMINATION OF THE MAXIMUM EXTRAPOLATION INTERVAL

V. S. MOSKVIN, V. K. YASEL’SKII, E. M. BELOV

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

To determine the magnitude of the interval \(\Delta t_{\mathrm{e}}\) over which the error in computing the extrapolated value of a function does not exceed the admissible value \(\varphi_{\max}\) (Fig. 1), one may use estimates of the interpolation error [1].

The use of Stirling’s and Bessel’s interpolation formulas for solving the indicated problem should be regarded as expedient, since in the case under consideration forward extrapolation is involved. The most acceptable is Newton’s formula.

Fig. 1. Curves of variation of \(f(t)\) and \(f_{\mathrm{e}}(t)\)

Let us consider the case in which the interpolation and extrapolation intervals are not equal to one another.

Denoting the values of the function at the interpolation nodes by \(f(t_i)\), where \(i=0,1,2,\ldots\), the difference between the exact value of the function \(f(t)\) and its approximate (extrapolated) value \(f_{\mathrm{e}}(t)\) can be represented in the form

\[ \varphi_n(t)=f(t)-\sum_{\nu=0}^{n}\prod_{\kappa=0}^{\nu-1}(t-t_\kappa)f(t_0,t_1,\ldots,t_\nu). \tag{1} \]

Analysis of the expression given shows that

\[ \varphi_n(t)=\sum_{\kappa=0}^{n}(t-t_\kappa)f(t,t_0,t_1,\ldots,t_n). \tag{2} \]

Since for the interpolation nodes \(\varphi_n(t_0)=0,\ldots,\varphi_n(t_n)=0\), then, using Rolle’s theorem and carrying out certain transformations, one may write

\[ f(t,t_0,\ldots,t_n)=\frac{f^{(n+1)}(\xi)}{(n+1)!}, \tag{3} \]

where \(\xi\) denotes an intermediate value of the argument lying between the numbers \(t,t_0,t_1,\ldots,t_n\).

On this basis we obtain that

\[ \varphi_n(t)=\frac{f^{(n+1)}(\xi)}{(n+1)!}\prod_{k=0}^{n}(t-t_k). \tag{4} \]

Since, in the case under consideration, linear extrapolation takes place \((n=1)\), expression (4) should be rewritten in the form

\[ \varphi_n(t)=\frac{f^{(2)}(\xi)}{2!}\prod_{k=0}^{1}(t-t_k). \tag{5} \]

The absolute error \(\varphi_{\mathrm{e}}(t)\) in determining the extrapolated value of the function beyond the time interval \(\Delta t_{\mathrm{e}}\) can be represented by the relation

\[ \varphi_{\mathrm{e}}(t)=\frac{\varphi_{\mathrm{rel}}\cdot |f(t)|_{\max}}{100}, \tag{6} \]

where \(\varphi_{\mathrm{rel}}\) is the relative reduced error in determining the extrapolated value of the parameter;
\(|f(t)|_{\max}\) is the maximum value of the function being extrapolated.

Analysis of expressions (5) and (6) shows that the maximum value of the extrapolation interval satisfying the imposed accuracy requirements can be determined from the solution of an equation of the form

\[ \frac{|f^{(2)}(t)|_{\max}}{2!}\prod_{k=0}^{1}(t-t_k) -\frac{\varphi_{\mathrm{rel}}|f(t)|_{\max}}{100}=0. \tag{7} \]

REFERENCES

1. B. P. Demidovich, I. A. Maron. Foundations of Computational Mathematics. Moscow, “Nauka,” 1966.