Fitting polynomials to data

Question

Is there a way, given a set of values (x,f(x)), to find the polynomial of a given degree that best fits the data?

I know polynomial interpolation, whic

Answer 1

Lagrange polynomials (as @j w posted) give you an exact fit at the points you specify, but with polynomials of degree more than say 5 or 6 you can run into numerical instability.

Least squares gives you the "best fit" polynomial with error defined as the sum of squares of the individual errors. (take the distance along the y-axis between the points you have and the function that results, square them, and sum them up) The MATLAB polyfit function does this, and with multiple return arguments, you can have it automatically take care of scaling/offset issues (e.g. if you have 100 points all between x=312.1 and 312.3, and you want a 6th degree polynomial, you're going to want to calculate u = (x-312.2)/0.1 so the u-values are distributed between -1 and +=).

NOTE that the results of least-squares fits are strongly influenced by the distribution of x-axis values. If the x-values are equally spaced, then you'll get larger errors at the ends. If you have a case where you can choose the x values and you care about the maximum deviation from your known function and an interpolating polynomial, then the use of Chebyshev polynomials will give you something that is close to the perfect minimax polynomial (which is very hard to calculate). This is discussed at some length in Numerical Recipes.

Edit: From what I gather, this all works well for functions of one variable. For multivariate functions it is likely to be much more difficult if the degree is more than, say, 2. I did find a reference on Google Books.