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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.

[tau,sp] = chbpnt(...) also returns, in sp, the Chebyshev spline.


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Create a knot sequence of ten zeros and ten ones using the augknt function.

k = 10;
t = augknt([0,1],k);

Create a vector of interpolation sites by calculating the Chebyshev-Demko points for the Chebyshev spline of order 10 with knot sequence t.

sites = chbpnt(t,k);

Create a vector of interpolation points by evaluating the function sqrt at the Chebyshev-Demko points. Use the function spapi to interpolate through the evaluated points with a tenth-order spline from the knot sequence t. The knots in t indicate that the spline fit consists of a single polynomial with boundaries at 0 and 1.

interpolationPoints = sqrt(sites);
sp = spapi(t,sites,interpolationPoints)
sp = struct with fields:
      form: 'B-'
     knots: [0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1]
     coefs: [-1.7088e-16 0.8842 -0.7101 2.4484 -1.1869 1.9678 0.3331 0.9940 0.9349 1]
    number: 10
     order: 10
       dim: 1

The structure sp contains the properties of the spline. The coefs field of sp is a vector of coefficients for the polynomial that approximates the square root function in the interval [0,1].

Compare sp and the square root function by displaying them in the same plot.

x = linspace(0,1,1000);

hold on
legend(["square root function" "spline"],Location="best")

The plot shows that the spline closely follows the square root function.


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 Construct Chebyshev Spline gives a detailed discussion of one version of the process as applied to a particular example.

See Also