Least squares cubic splines without B-splines

Readers of the Gazette will be familiar with the use of least squares to find the polynomial that best fits a set of data points. As with all curve fitting problems, there are many situations where one low order polynomial is inadequate and a high order polynomial fit is inappropriate. A least squares cubic spline would provide a better fit. There are thousands of articles published on variants of splines, including least squares cubic splines. One of the first least squares articles was de Boor and Rice [1], and a comprehensive explanatory textbook is Dierckx [2]. Unfortunately, every example in the literature and on the web of a least squares cubic spline makes use of B-splines. While B-splines have a certain elegance, they are sufficiently complex to be beyond the typical undergraduate level, and have the disadvantage of being more expensive to evaluate than traditional cubic splines. For example, Schumacker [4] points out that it is more efficient to convert a cubic B-spline to a traditional cubic spline and then evaluate if you require two or more function evaluations per interval. The only exception to using B-splines is Ferguson and Staley [3], where they find a least squares cubic fit to data that enforces continuity of function and first derivative, but not second derivative. Thus, their formulation does not lead to a cubic spline. My colleague, Basil Benjamin