ON A SCHEME FOR APPROXIMATING FUNCTIONS BY CHEBYSHEV POLYNOMIALS

A. S. Gitman, E. S. Gitman

(Presented by the scientific seminar of the Departments of Electrical Machines and General Electrical Engineering)

When preparing problems for solution on electronic digital computers (EDCs), a known difficulty is the approximation of functions specified in tabular or graphical form. From the standpoint of using an EDC, it is desirable to approximate functions by polynomials. Although there are no rules for choosing a method of approximation by polynomials, one of the important features that facilitate the task is the character of the original function. If the function is “pointwise,” it is expedient to obtain such an approximation in which the errors are distributed uniformly over the entire interval. In this sense, the orthogonal Chebyshev polynomials of the first kind \(T_n(x)=\cos(n\arccos x)\), defined on the interval \([-1,+1]\), are of fundamental importance. For a continuous original function, finding the coefficients by the formula

\[ c_n=\frac{2}{\pi}\int_{-1}^{1} f(x)T_n(x)\frac{dx}{\sqrt{1-x^2}}, \tag{1} \]

one can obtain an approximation \(\varphi(x)\) in the form

\[ \varphi(x)=\frac{1}{2}c_0+c_1T_1(x)+c_2T_2(x)+\ldots \tag{2} \]

It is therefore clear that the bulk of the computations is associated with finding the coefficients \(c_n\). Excluding from consideration the rare possibility of obtaining an antiderivative for the integrals (1), we shall assume that numerical methods must be used to find them. For relatively large values of the degrees of the polynomials \(T_n(x)\), the use of known methods of numerical integration leads to substantial errors. This is connected with the presence in the integrand of (1) of the rapidly oscillating function \(T_n(x)\), with the oscillations becoming denser toward the ends of the interval. In view of the difficulty of computing the integrals (1), one usually introduces a change of variable in order to transform the original function into a periodic one and then uses trigonometric approximation [1]. There is also another way. Using ideas related to those underlying Filon’s method [2], one can obtain a series of analogues of quadrature formulas and, in particular, an analogue of the trapezoidal formula, which will be found below.

Of greatest practical interest is the case of a uniform distribution of the support points of the function \(f(x)\) on the interval \([a,b]\). Applying the linear transformation

\[ z=\frac{a+b}{a-b}+\frac{2x}{b-a} \tag{3} \]

to the original interval \([a,b]\), we obtain the function \(f(x)\), specified on the interval \([-1,+1]\). By successively joining the ordinates of neighboring points (Fig. 1), we find the equation of the lines between them.

For segment \(i\), on the interval of variation of the abscissa

\[ [-1+ih,\,-1+(i+1)h] \]

\[ f(x)=\frac{y_{i+1}-y_i}{h}(x+1-ih)+y_i \quad (i=0,1,2\ldots N-1), \tag{4} \]

where \(h=\dfrac{2}{N}\) is the length of the interval for a number of points \(N+1\). Let us represent expression (1) in the form

\[ c_{\mathrm{n}}=\frac{2}{\pi}\sum_{i=0}^{N-1}\int_{-1+ih}^{-1+(i+1)h} f(x)T_{\mathrm{n}}(x)\frac{dx}{\sqrt{1-x^2}} \tag{5} \]

and, using (4), find the values of the integrals standing under the summation sign:

Fig. 1

\[ \int_{-1+ih}^{-1+(i+1)h} f(x)T_{\mathrm{n}}(x)\frac{dx}{\sqrt{1-x^2}} = \int_{-1+ih}^{-1+(i+1)h} \frac{(y_{i+1}-y_i)x\cdot \cos(\mathrm{n}\arccos x)\,dx}{h\sqrt{1-x^2}} + \]

\[ +\int_{-1+ih}^{-1+(i+1)h} \left[ \frac{y_{i+1}-y_i}{h}(1-ih)+y_i \right] \frac{\cos(\mathrm{n}\arccos x)\,dx}{\sqrt{1-x^2}}. \tag{6} \]

We apply the substitution \(\theta=\arccos x\). Then

\[ x=\cos\theta,\qquad d\theta=-\frac{dx}{\sqrt{1-x^2}} \]

and expression (6), for \(\mathrm{n}>1\), takes the form

\[ \begin{aligned} &\frac{y_{i+1}-y_i}{h} \int_{\arccos[-1+(i+1)h]}^{\arccos(-1+ih)} \cos\theta \cos n\theta\,d\theta +\left[\frac{y_{i+1}-y_i}{h}(1-ih)+y_i\right] \int_{\arccos[-1+(i+1)h]}^{\arccos(-1+ih)} \cos n\theta\,d\theta =\\ &=\frac{y_{i+1}-y_i}{2h(n^2-1)} \left[(n-1)\sin(n+1)\theta+(n+1)\sin(n-1)\theta\right] \Bigg|_{\arccos[-1+(i+1)h]}^{\arccos(-1+ih)} +\\ &\quad+\frac{(y_{i+1}-y_i)(1-ih)+y_i h}{nh}\, \sin\theta \Bigg|_{\arccos[-1+(i+1)h]}^{\arccos(-1+ih)} . \end{aligned} \tag{7} \]

Expression (7) can be simplified. To this end, denote

\[ -1+ih=z_i,\qquad \Delta y_i=y_{i+1}-y_i,\qquad \arccos z_i=\alpha_i, \]

and

\[ \arccos(z_i+h)=\arccos[-1+(i+1)h]=\alpha_i-\delta_i . \]

To determine \(\delta_i\), one can obtain the equation

\[ \cos(\alpha_i-\delta_i)=\cos\alpha_i\cos\delta_i+\sin\alpha_i\sin\delta_i \]

\[ =z_i\cos\delta_i+\sqrt{1-z_i^2}\,\sin\delta_i=z_i+h, \]

whence

\[ \delta_i=\arccos\{z_i(z_i+h)+\sqrt{[1-(z_i+h)^2](1-z_i^2)}\}. \]

With the adopted notation, for any \(m\),

\[ \sin m u\Big|_{\alpha_i-\delta_i}^{\alpha_i} = 2\sin\frac{m\delta_i}{2}\cos\frac{m(2\alpha_i-\delta_i)}{2}. \tag{8} \]

Substituting (8) into (7) and (7) into (5), we find, for \(n>1\),

\[ \begin{aligned} c_n&=\frac{2}{\pi h(n^2-1)} \sum_{i=0}^{N-1}\Delta y_i \left[ (n-1)\sin\frac{(n+1)\delta_i}{2} \cos\frac{(n+1)(2\alpha_i-\delta_i)}{2} +\right.\\ &\qquad\left. +(n+1)\sin\frac{(n-1)\delta_i}{2} \cos\frac{(n-1)(2\alpha_i-\delta_i)}{2} \right] +\\ &\quad+\frac{2(n^2-1)}{n}(y_i h-\Delta y_i z_i) \sin\frac{n\delta_i}{2} \cos\frac{n(2\alpha_i-\delta_i)}{2}. \end{aligned} \tag{9} \]

The coefficients \(c_0\) and \(c_1\), respectively, are equal to

\[ c_0=\frac{2}{\pi h}\sum_{i=0}^{N-1} 2\Delta y_i\sin\frac{\delta_i}{2}\cos\frac{2\alpha_i-\delta_i}{2} +\delta_i(y_i h-\Delta y_i z_i) \tag{10} \]

\[ c_1=\frac{1}{\pi h}\sum_{i=0}^{N-1}\Delta y_i \left[\delta_i+\sin\delta_i\cos(2\alpha_i-\delta_i)\right] + \]

\[ +4(y_i h-\Delta y_i z_i)\sin\frac{\delta_i}{2} \cos\frac{2\alpha_i-\delta_i}{2}. \tag{11} \]

REFERENCES

1. K. Lanczos. Practical Methods of Applied Analysis. Fizmatgiz, Moscow, 1961.
2. K. J. Tranter. Integral Transforms in Mathematical Physics. GITL, Moscow, 1956.