I want to draw a curve and generate a polynomial that closely fits it. How would I go about this?

Question

I have an arbitrary curve (defined by a set of points) and I would like to generate a polynomial that fits that curve to an arbitrary precision. What is the best way to tackle this problem, or is there already a library or online service that performs this task?

Thanks!

Answer 1

If your "arbitrary curve" is described by a set of points (x_i,y_i) where each x_i is unique, and if you mean by "fits" the calculation of the best least-squares polynomial approximation of degree N, you can simply obtain the coefficients b of the polynomial using

    b = polyfit(X,Y,N)

where X is the vector of x_i values, Y is the vector of Y_i values. In this way you can increase N until you obtain the accuracy you require. Of course you can achieve zero approximation error by calculating the interpolating polynomial. However, data fitting often requires some thought beforehand - you need to give thought to what you want the approximation to achieve. There are a variety of mathematical ways of assessing approximation error (by using different norms), the choice of which will depend on your requirements of the resulting approximation. There are also many potential pitfalls (such as overfitting) that you may come across and blindly attempting to fit curves may result in an approximation that is theoritically sound but utterly useless to you in practical terms. I would suggest doing a little research on approximation theory if the above method does not meet your requirements, as has been suggested in the comments on your question.