Computes the variable knot B-spline least squares approximation to given data.

The routine BSVLS attempts to find the best placement of knots that will minimize the leastsquares error to given data by a spline of order k = KORDER with N = NCOEF coefficients. The user provides the order k of the spline and the number of coefficients N. For this problem to make sense, it is necessary that N > k. We then attempt to find the minimum of the functional

The user must provide the weights w = WEIGHT, the data xi = XDATA and fi = FDATA, and
M = NDATA. The minimum is taken over all admissible knot sequences t.

The technique employed in BSVLS uses the fact that for a fixed knot sequence t the minimization in a is a linear least-squares problem that can be solved by calling the IMSL routine BSLSQ. Thus, we can think of our objective function F as a function of just t by setting

A Gauss-Seidel (cyclic coordinate) method is then used to reduce the value of the new objective function G. In addition to this local method, there is a global heuristic built into the algorithm that will be useful if the data arise from a smooth function. This heuristic is based on the routine NEWNOT of de Boor (1978, pages 184 and 258− 261).

The user must input an initial guess, tg = XGUESS, for the knot sequence. This guess must be a valid knot sequence for the splines of order k with

with tg nondecreasing, and

The routine BSVLS returns the B-spline representation of the best fit found by the algorithm as well as the square root of the sum of squares error in SSQ. If this answer is unsatisfactory, you may reinitialize BSVLS with the return from BSVLS to see if an improvement will occur. We have found that this option does not usually (substantially) improve the result. In regard to execution speed, this routine can be several orders of magnitude slower than one call to the least-squares routine BSLSQ.

In this example, we try to fit the function ∣x − .33∣ evaluated at 100 equally spaced points on [0, 1]. We first use quadratic splines with 2 interior knots initially at .2 and .8. The eventual error should be zero since the function is a quadratic spline with two knots stacked at .33. As a second example, we try to fit the same data with cubic splines with three interior knots initially located at .1, .2, and, .5. Again, the theoretical error is zero when the three knots are stacked at .33.

We include a graph of the initial least-squares fit using the IMSL routine BSLSQ for the above quadratic spline example with knots at .2 and .8. This graph overlays the graph of the spline computed by BSVLS, which is indistinguishable from the data.