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Suppose that we approximate a function $f(x)$ for $x$ near $0$ by a polynomial of degree $n$: $$f(x)\approx P_n(x)=C_0+C_1x+C_xx^2 + \dots + C_{n-1}x^{n-1} +C_nx^n$$ We need to find the values of the constants: $C_0,C_1,C_2,\dots , C_n$. To do this, we require that the function $f(x)$ and each of its first $n$ derivatives agree with those of the polynomial $P_n(x)$ at the point $x=0$. In general, the more derivatives that agree at $x=0$, the larger the interval on which the function and the polynomial remain close to each other.

My questions are:

  1. Why is it a requirement that each $n$ derivatives of $f(x)$ agree with the $P(x)$? Would not following this requirement make $P(x)$ a worse approximation? How can we show this? My feeling is that this is simply an extension from observing that for $P_1(x)$, $P(0)=f(x)$ and $P'(x)=f'(x)$. So, to get an even better approximation, let's just add more terms where each additional term is the $n$ derivative of $f(x)$. Is there more to this than what I just said?

  2. Suppose $P_1(x)$ is already a good approximation to $f(x)$. When we create $P_2(x)$, we have an $x^2$ term. Since $x^2>x$, wouldn't $P_2(x)$ cause a larger difference between $f(x)$ compared to between $P_1(x)$ and $f(x)$, i.e. the higher the degree of the polynomial, the worse the approximation gets.

  3. The last sentence, "The more derivatives that agree...". How can we prove that?

  4. Why use a power series to approximate a function? Are there other series that is not a power series that can that be used to approximate a function?

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Why is it a requirement that each $n$ derivatives of $f(x)$ agree with the $P(x)$

It's not really a requirement. The author is just suggesting this as an approach. As you guessed, there are other possible approaches.

Why use a power series to approximate a function? Are there other series that is not a power series that can that be used to approximate a function?

Sounds like your real question is "why use this technique of matching derivatives at a single point". If you want an approximation that's uniformly good throughout some interval, this is not the best approach. One approach that's better (though not optimal) is to interpolate at abscissae derived from zeros of a Chebyshev polynomial. There is a software system called Chebfun that's very good at constructing these sorts of approximations.

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