How to find interpolating polynomial

I'm trying to find the interpolating polynomial of degree 20 for the function f(x)=(x^2+1)^-1 using 21 equally spaced nodes on the interval [-5,5]. The syntax matlab says to use is

interpolate(nodes, values, ind, F)

But I only know the amount of nodes, function, and the interval. Thanks for any help!

Answer 1

[Walter Roberson]

interpolate() is not the correct function to extract an interpolating polynomial. See polyfit() and polyval() and linspace().

[Maciej Rzymek]

polyfit won't give you interpolation, but approximation which is way different. He should use interp1.

[John D'Errico]

@Maciej Rzymek - actually, you are incorrect, on several counts. Hard to do in such a short space. :)

1. polyfit CAN produce an interpolating polynomial, if you choose the proper order polynomial. Thus with n data points, set the order to be n-1. This will produce a polynomial with n coefficients to estimate, and therefore, an interpolating polynomial.
2. Since the goal was to produce an interpolating polynomial, interp1 will NOT satisfy that goal.

In fact, this problem is a classical one, chosen to show how poorly an interpolating polynomial can react. That particular function is one that will generate a wildly oscillatory polynomial, with immense swings between the points. You may wish to read about Runge's phenomenon.

http://www.tlu.ee/~tonu/Arvmeet/Runge%27s%20phenomenon.pdf

The one thing I agree with what you said is that if your goal is purely to interpolate such a function, and to do it well, then there are far better tools than a polynomial. For example, interp1 in this case by use of pchip as the interpolant, will produce a well behaved, non-oscillatory result.

But if the goal is to satisfy the requirements of a homework assignment, then you have no choice.