# chbpnt

Chebyshev-Demko points

## Syntax

`tau = chbpnt(t,k) `

chbpnt(t,k,tol)

[tau,sp] = chbpnt(...)

## Description

`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.

## Examples

## Algorithms

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.