# Polynomial fitting - how to fit and what is _polynomial fitting_

I don't understand what is polynomial fitting.
Can anyone explain me how to fit a curve to given points?

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If you google for "polynomial fitting" you will find at least a couple of rather informative results. (In particular, after reading at least the wikipedia page it should be clear that one does not "fit a polynomial", so that your third line does not make sense) –  Mariano Suárez-Alvarez Oct 18 '10 at 14:47
Thank you! So can I say "how to fit a curve"? –  rubik Oct 18 '10 at 16:01
More properly, "how to fit a curve to given points". –  Guess who it is. Oct 18 '10 at 16:10
Seriously, edited just for removing thanks? Things certainly have changed since I was away... –  Guess who it is. Mar 23 '13 at 16:30
@J.M. Well said. Nice to see you back. –  Rudy the Reindeer Mar 23 '13 at 16:48

There are two concepts frequently conflated: interpolation and fitting. I'll discuss both since you don't seem to know what you really want.

Let's say you have a set of $n$ points

$$(x_i,y_i)\qquad i=1\dots n$$

Interpolation is the problem of finding the (unique) polynomial $p(x)$ that passes through all your given points (under the assumption that no two points have the same abscissa), i.e. $p(x_i)=y_i \qquad i=1\dots n$. Your resulting polynomial is of degree $n-1$. The usual techniques for finding the interpolating polynomial are the methods of Lagrange, Newton, and Neville-Aitken.

Fitting on the other hand assumes your data is contaminated with error, and you want the polynomial that is the "best approximation" to your data. Here polynomial interpolation does not make much sense since you do not want your function to be reproducing the inherent errors in your data as well. Least-squares is a common technique: it finds the polynomial $f(x)$ such that the quantity

$$\sum_{j=1}^{n}\left(f(x_i)-y_i\right)^2$$

which measures the departure of your polynomial from the ordinates is minimized (here the assumption is that your abscissas are error-free, and the error in your ordinates is normally distributed). The degree of $f(x)$ can be (and is often) less than $n$. A number of techniques for this are used as well: there's the normal equations, and then there are special matrix decompositions that can be used to efficiently solve this problem.

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Nice explanation. –  Mike Spivey Oct 18 '10 at 15:39
+1 Very nice indeed. –  Adrián Barquero Oct 18 '10 at 16:12
+1 Wonderful explanation! I think now I have understood what is polynomial fitting! I'd vote you if I could! –  rubik Oct 18 '10 at 16:14