Good data sites, Chebyshev-Demko points

`tau = chbpnt(t,k) `

chbpnt(t,k,tol)

[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.

`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.

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