Taylor Expansion

TV Taylor expansion is approximating a function F (normally from |R ---> |R) by a 'Taylor' polynomial T of degree N which is based on the first N derivatives at a point a of the domain of F.

The idea is to make a polynomial by looking at the function value, and make it follow the first N derivatives at a point a as an approximation of F around F(a).

This works so well that a lot of relevant theory is based on the main theorem:

                                        2          (n)         n
t (x) = f(a) + f'(a)(x-a) + f''(a) (x-a)  + ... + f   (a) (x-a)
 n                          ------                -------
                              2!                     n!

It can be proven that the absolute error in the approximation is upper bounded by the next term of the expansion [no, but it's not far off] (TV I put the correct equation back in).

In various theoretical mathematical proofs, the theorem has a prominent place (also in engineering because of its practical value) mainly because of its general applicability: any continuous, bounded function [not true, says CL; it's "any analytic function", although that's far from the most useful form of the theory] can be fully represented, without error by an infinite order (or as many orders as the function has) Taylor expansion in one point only.

It would be interesting to have a math lib which can differentiate on lists in Tcl, but at this point, I didn't write one, or know of one. Wolfram licences are expensive (at least they were 10 years ago). [ CL speculates that this paragraph is about symbolic calculation [1 ], which has been done in Tcl. Results are certainly available without paying Wolfram.]

Why are pages like these being added to the Tclers Wiki? -jcw


MS does not know the answer to jcw's question. He does know that the difference between analytic (ie, approximated by a Taylor series in a neighborhood of a point) and infinitely differentiable hides some nice surprises. As an example,

f(x) = exp(-1/(x**2))   if x !=0
f(0) = 0

is infinitely differentiable at x=0 - all derivatives exist, and are zero. It's Taylor series exists, and is zero throughout. But the function is obviously not zero everywhere ...

The thing gets really interesting for complex functions, of course. There differentiability is a deeper concept, and all differentiable functions are analytic. For a wide class of other functions, there are approximating power series expansions even around singularities of the function - the Laurent series.

TV Apart from that more can be done with this, I'll look up some things here, I think I wrote most things right. going to fetch a book.