Good data sites, Chebyshev-Demko points
tau = chbpnt(t,k)
[tau,sp] = chbpnt(...)
tau = chbpnt(t,k) are the extreme sites of the Chebyshev spline of order k with knot sequence t. These are particularly good sites at which to interpolate data by splines of order k with knot sequence t because the resulting interpolant is often quite close to the best uniform approximation from that spline space to the function whose values at tau are being interpolated.
chbpnt(t,k,tol) also specifies the tolerance tol to be used in the iterative process that constructs the Chebyshev spline. This process is terminated when the relative difference between the absolutely largest and the absolutely smallest local extremum of the spline is smaller than tol. The default value for tol is .001.
chbpnt([-ones(1,k),ones(1,k)],k) provides (approximately) the extreme sites on the interval [–1 .. 1] of the Chebyshev polynomial of degree k-1.
If you have decided to approximate the square-root function on the interval [0 .. 1] by cubic splines, with knot sequence t as given by
k = 4; n = 10; t = augknt(((0:n)/n).^8,k);
then a good approximation to the square-root function from that specific spline space is given by
x = chbpnt(t,k); sp = spapi(t,x,sqrt(x));
as is evidenced by the near equi-oscillation of the error.
The Chebyshev spline for the given knot sequence and order is constructed iteratively, using the Remez algorithm, using as initial guess the spline that takes alternately the values 1 and −1 at the sequence aveknt(t,k). The example "Constructing the Chebyshev Spline" gives a detailed discussion of one version of the process as applied to a particular example.