Chebyshev approximation - from any equation

Question

So let's say I have a function:

$f(x) = {e^x}$

How do I use Chebyshev polynomials up to order 4, to find the corresponding coefficients? how do I make an approximation equation using these 4 coefficients?

I am sorry if this question is basic part, I just don't know how Chebyshev works and how to use it.

I have done taylor/maclaurin expansion before but i cant find any simple material to follow for chebyshev to get an answer.

Answer 1

Hint

To make it simple, remember that Chebyshev polynomials are just ... polynomials $$T_0=1 \qquad T_1=x \qquad T_2=-1+2x^2 \qquad T_3=-3x+4x^3 \qquad T_4=1-8x^2+8x^4$$ So, write $$e^x=\sum_{i=0}^4 a_i Ti$$ Replace the lhs by the usual Taylor expansion and identify the $a_i$.

Answer 2

how do I make an approximation equation

The way to go is Remez' Algorithm. It starts with an (informed) guess where the maxima of the error (in terms of the maximum norm $\|\cdot\|_\infty$) are located, and from there solve a linear system in $n+2$ variables for an approx of order $n$. One variable is the "expected max error", and $n+1$ variables are the coefficients of a polynomial of degree $n$. Then iterate that.

The result will depend on the interval over which the function is to be approxed, of course.

Specifically in your case of $e^x$ and an approximation of order 4, the results would be something like

$p_{[-1,1]}(x) = 1.00009 + 0.997309252 x + 0.49883511 x^2 + 0.177345274 x^3 + 0.0441555176 x^4$

$p_{[0,1]}(x) = 1.000027162 + 0.9986854006 x + 0.5101394602 x^2 + 0.1396981485 x^3 + 0.06970449423 x^4$

with absolute errors $5.46668\cdot10^{-4}$ resp. $2.71624\cdot10^{-5}$ for approximations over the intervals $[-1,1]$ and $[0,1]$, respectively.

For minimal relative errors however, the results will be

$q_{[-1,1]}(x) = 0.999627896 + 0.997938729 x + 0.502898651 x^2 + 0.176486232 x^3 + 0.0399629142 x^{4}$

$q_{[0,1]}(x) = 1.000016135 + 0.9990684905 x + 0.5081199094 x^2 + 0.1430489414 x^3 + 0.06798449148 x^4$

with relative errors $5.03041\cdot10^{-4}$ and $1.61353\cdot10^{-5}$, respectively.

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Notice that expressing the Taylor series in terms of Chebyshev polynomials $T_n$ will get you nowhere: That's just a change of basis, and all you get is that the resulting expressions will be harder to evaluate.