Given a set of data points, how should you best fit a curve to those points to get a good result?
This is the most basic method, where a high order polynomial is found which satisfies the data set given. This polynomial is of the same order as the number of data points given to fit, and therefore often results in extrememly large oscillations.
This method involves computing a monotonicity-preserving piecewise cubic Hermite interpolant to the data set. The curve will never go above the highest point, or below the lowest, eliminating the large oscillations found in the high-order polynomial fitting technique.
A better technique is to approximate a cubic spline to the data set. This allows a more natural curve to be formed, while still avoiding the issues caused when fitting high-order polynomials. This technique allows the user to specify a smoothing coefficient.