Numer. Math. 48, 627-638 (1986) Numerische Mathematik 9 Springer-Verlag1986 Calculation of Cubic Smoothing Splines for Equally Spaced Data David Culpin CSIRO, Division of Mathematics and Statistics, P.O. Box 218, Lindfield, N.S.W., Australia Summary. A method is described for fitting cubic smoothing splines to samples of equally spaced data. The method is based on the canonical decomposition of the linear transformation from the data to the fitted values. Techniques for estimating the required amount of smoothing, in- cluding generalized cross validation, may easily be integrated into the calculations. For large samples the method is fast and does not require prohibitively large data storage. Subject Classifications: AMS(MOS): 65D07, 65D10; CR: G1.2. 1. Cubic Smoothing Splines for Arbitrarily Spaced Data Let data points be (x t, Yi) for i = 1, 2 ..... n, n > 3. A cubic smoothing spline may be written f(x) = a t + b t (x - xi) + c i (x - xi) 2 + d i (x - xi)3 (1.1) for xt<x<xt+ 1, i= 1..... n-1. It is a continuous function with continuous first and second derivatives. These continuity conditions, together with the con- straints f'(xl)=f"(x,)=O, make the coefficients b~, c t and d t all linear functions of the at's (see Sect. 3 for the formulae). Thus the determination of the smooth- ing spline rests on the determination of the ai's. Note that a i=f(xt). Thus if the spline is required to pass through the data points, then a t= Yt and the spline is determined. A smoothing spline need not pass through the data points, but is de- termined by requiring a certain balance between its smoothness and its close- ness to the data. Let A 3 xn n j, .2 C= T f"(x)2dx and O= F~(y~-~), (1.2) i=l