Least-squares fitting of data by rational functions: Levy's method (Part 1) [By the Numbers]

We have, in various previous columns 1-5], looked at the "approximation" problem. There are two broad approaches to dealing with this problem: 1. Interpolation, which finds a suitable fitting function that passes through all the data points available, and 2. Fitting, according to some criterion for goodness, a suitable function to a set of data without restricting that function to coincide with the data points. In the second approach, both the criterion for goodness of fit and the class of fitting functions are important to the outcome of the approximation, and both are left to the person working with the data. One popular criterion is the minimization of the sum (or the integral) of the squared error between the chosen fitting function and the data. An alternative to this "least-squares" criterion is to minimize the maximum error between the fitting function and the data. The fitting function itself can be anything that "makes sense"--a polynomial, a trigonometric function, etc. For a wide variety of problems in science and engineering (e.g., the characterization of a linear, time-invariant system from frequency-response data), it might be particularly appropriate to choose a rational function as the fitting function. In this installment, we will provide a brief introduction to least-squares fitting of data by rational functions by presenting Levy's Method 6], a classical approach that is relatively straightforward in concept and easy to program, and which gives respectable results in a broad range of circumstances.