# Empirical Formulas and Interpolation

• I. N. Bronshtein
• K. A. Semendyayev

## Abstract

Formulation of the problem. We sometimes encounter a need to choose an analytic expression which would represent approximately a function given only by a table or a graph. A similar problem can arise for a function given by a formula, when this formula is too complicated or not appropriate to a given purpose as, for example, when the function has to be integrated while the integral cannot be expressed in terms of elementary functions. Formulas representing a functional dependence obtained from an experiment in the form of a table or a graph are called empirical formulas. To represent approximately a given function f(x), we usually choose an approximating function φ(x) from among functions of a definite form; for example, we seek for a function φ(x) in the form of a polynomial
$$\varphi (x)=a_{0}+a_{1}x+\cdots +a_{n}x^{n}$$
or in the form
$$\varphi (x)=Ae^{rx}+Be^{sx}+\cdots$$
, requiring that the function φ(x) approaches the function f(x) in a certain interval axb as closely as possible. According to the manner of estimating the approximation of the function f(x) with φ(x), we obtain various systems of parameters of the function φ(x) yielding the best approximation of f(x).

## Keywords

Empirical Formula Uniform Approximation Interpolation Formula Interpolation Node Conditional Equation

## Notes

1. (1).
For example, φ(x) is a polynomial, if φ0 = 1, φ1 = x, …, φn = xn, or is a trigonometric polynomial, if φ0 = 1, φ1 = cos x, φ2 = sin x, …, φ2n-l = cos nx, φ2n = sin nx.Google Scholar
2. (2).
Two examples of orthogonal systems of functionsGoogle Scholar
3. (1).
1, cos x, cos 2x, …, cos nx; sin x, sin 2x, …, sin nx, in the interval (0, 2π).Google Scholar
4. (2).
Legendre’s polynomials Pi(x) in the interval (− 1, +1) (see pp. 551, 552).Google Scholar
5. (1).
The best approximation of f(x) can be defined, as above (see the uniform approximation, p. 754), as a function φ(x) such that the maximum of [f(xi) − φ(xi)] has the least value. However, determining the approximation in this way is, practically, troublesome.Google Scholar
6. (1).
For an example of such a scheme see p. 761.Google Scholar
7. (1).
The decimal point is usually omitted in the table of differences and the difference is expressed in the units of the last significant figure.Google Scholar
8. (1).
After determining a and b, we can choose c equal to the mean value of yaxb.Google Scholar
9. (1).
After determining a and b, we can anew take c equal to the mean value of yacbx.Google Scholar