Computes the least-squares spline approximation, and return the B-spline coefficients.

The routine BSLSQ is based on the routine L2APPR by de Boor (1978, page 255). The IMSL routine BSLSQ computes a weighted discrete L2 approximation from a spline subspace to a given data set (xi, fi) for i = 1, …, N (where N = NDATA). In other words, it finds B-spline coefficients,
a = BSCOEF, such that

is a minimum, where m = NCOEF and Bj denotes the j-th B-spline for the given order, KORDER, and knot sequence, XKNOT. This linear least squares problem is solved by computing and solving the normal equations. While the normal equations can sometimes cause numerical difficulties, their use here should not cause a problem because the B-spline basis generally leads to well-conditioned banded matrices.

The choice of weights depends on the problem. In some cases, there is a natural choice for the weights based on the relative importance of the data points. To approximate a continuous function (if the location of the data points can be chosen), then the use of Gauss quadrature weights and points is reasonable. This follows because BSLSQ is minimizing an approximation to the integral

The Gauss quadrature weights and points can be obtained using the IMSL routine GQRUL (see Chapter 4, “Integration and Differentiation”).

In this example, we try to recover a quadratic polynomial using a quadratic spline with one interior knot from two different data sets. The first data set is generated by evaluating the quadratic at 50 equally spaced points in the interval (0, 1) and then adding uniformly distributed noise to the data. The second data set includes the first data set, and, additionally, the values at 0 and at 1 with no noise added. Since the first and last data points are uncontaminated by noise, we have chosen weights equal to 105 for these two points in this second problem. The quadratic, the first approximation, and the second approximation are then evaluated at 11 equally spaced points. This example illustrates the use of the weights to enforce interpolation at certain of the data points.