Calculation of cubic smoothing splines for equally spaced data (Q1822193)

Let \((x_ i,y_ i)\), \(i=1,...,n\), be given data points, and \(\Delta:=\max_{i}(x_{i+1}-x_ i)\). To each cubic spline f with knots \(x_ i\) associate the numbers \(C:=(\Delta^ 3/6)\int | f''(x)|^ 2dx\) and \(D:=\sum_{i}(y_ i-a_ i)^ 2\) with \(a_ i:=f(x_ i)\). These quantities measure the degree of smoothness and the closeness to the data, resp. The paper deals with the problem of determining the spline which minimizes \(sC+D\), where s is a fixed number. One can write C as a quadratic form a'Qa in the vector \(a':=(a_ 1,...,a_ n)\) with a symmetric, positive semidefinite matrix Q. Then the parameter vector a for the optimal f is determined by the equation (1) \((I+sQ)a=y\). The author gives for the case of equally spaced data \(x_ i\) explicit formulas for the eigenvalues and eigenvectors of the matrix Q. This facilitates the solution of the equation (1) considerably and allows also to evaluate criteria for the determination of a suitable smoothing parameter s.