Finding a function from its series equivalent

I would think you'd find the coefficients of the Taylor series by multiplying the k-th term by k-factorial, not dividing. E.g., the coefficients of the power series for e^x are 1, 1, 1/2, 1/6, ..., while the Taylor coefficients, the coefficients of (x^n)/(n!), are

Dividing plots beautifully. just checked it. You basically have a ramp that gets VERY steep near a=0.

Also, multiplying -1517.83 by 20! might end up a tad on the largish side of humongous. In any sane function, the higher power terms should be quite small.

There are apparently two or three summation terms to the real answer. One is an easily found and apparently fully determninistic linear ramp. The second is an ever increasing "bent" variable that may or may not need some extra help above 0.9. It may or may not have its own small ramp component.

I suspect it does.

Question about terminology here. By the above, are you saying that

-0.00245142 = a2 * x^2 etc.

Or are you saying that 0.103691 is the coefficient of x^1, -0.00245142 is the coefficient of x^2, etc. so

f(x) = (0.103691 * x^1) + (-0.00245142 * x^2) + .....

BTW, you posted a similar, but slightly different problem a half hour earlier with terms starting with x^0 as opposed to x^1 (above). Which is correct?

Cool. But the function you're plotting isn't the one you're asking about. The power series a_1 x + a_2 x^2 + a_3 x^3 + ... is (obviously!) equivalent to the Taylor series a_1 x + (2 a_2) (x^2 / 2) + (6 a_3) (x^3 / 6) + ... and not equivalent to a_1 x + (a_2 / 2)(x^2 / 2) + (a_3 / 6) (x^3 / 6) + ....

So either you're not doing what you say you're doing, or else the function you're plotting has nothing to do with the function you say interests you.

Is the function known to be periodic?

A Fourier series expansion, if so, would be more illuminating than a power series.

regards, chip

I have very positive experiences in finding good approximating functions. First a set of polynomials is calculated approximating the given data. The most suitable is selected. In most cases the one with least variance. Then a response function is calculated using a function tree structure to find the function combination with a taylor series fitting as well as possible to the approximating polynomial.

Some of the best alternatives are stored. The least squares approximation can be found using nonlinear estimation techniques.

In the one variable case it is practical to divide the data into several "wavelets" and use piecewise approximation.

In practise this method has worked well with 1 - 4 independent variable cases. 5 variables are possible, but never tried.

The method is slow, but quite well behaving. One more layer to the function net should demand grid computation.

On grid alternatives, please contact snipped-for-privacy@estlab.com

its First Few Terms" by Bergeron and Plouffe. I dont have the URL, but it is a free download. It doesn't provide a bullet proof method, but gives some suggestions.

This is the art of curve fitting. You need one additional piece of information, the error (tolerance) of each of the measurements, to do it properly.

Since you have specified this over the range 0 to 1, the classical approach would be to use a set of functions that are orthogonal over this range. Since you already have a polynomial, a kind of Legendre polynomial is your starting point (I think the Legendre polynomials are orthogonal over (-1, +1), so there's some scaling and shifting involved).

Your goal is, from that starting point, to find a simpler expression that holds the same data (i.e. reproduces the same function to the given tolerance). Sine/Cosine, polynomial, Bessel, there are lots of sets of orthogonal functions to choose from, and one of the sets might have all-but-one-term-vanishes character.

If there is any OTHER information about the boundary conditions, (if it's periodic, use sines; if it doesn't blow up at infinity, don't use polynomials) that can help, you need to consider that now. Finally, there's a theorem, the Wiener-Hopf theorem, that states that a set of fit functions is optimal if its autocorrelation is the same as that of the data. That means that data with big ripples doesn't fit well with sinewaves with little tiny ripples, but you can find that kind of thing out the hard way, too.

For orthogonal functions, the fit procedure is simple: there's a unique right set of coefficients, and a procedure to find it (inner product of your data and the function). For other kinds of functions, like wavelets and fractals and such, it gets ... challenging.

Your starting point has 20 coefficients, so anything that expresses the data with 19 or less, and hits within the measurement tolerance, is some kind of success.

Finally, I should note that the 'measurement tolerance' defines a kind of weight, a statistical weight function, that cannot be ignored. The orthogonal polynomials (Legendre polynomials) with constant weight are very different from the orthogonal polynomials (Chebyshev polynomials) with (x(1-x)) weight, and the weight is part of that 'inner product' step as well.

Usually, one starts with a graph of the function, makes a guess as to its form, then looks at a graph of the difference-from-the-guess. If you get lucky with a guess, and it was a natural measurement you started with, you've just created a scientific theory. Kudos!