optknt (Spline Toolbox)

Provide knot sequence optimal for interpolation

Syntax

knots = optknt(tau,k)

Description

t= optknt(tau,k) provides the knot sequence t that is best for interpolation from S_k,t at the point sequence tau. Here, best or optimal is used in the sense of [3] and [2], and this means the following: For any recovery scheme R that provides an interpolant Rg that matches a given g at the points tau(1), ..., tau(n), we may determine the smallest constant constR for which ||g-Rg||

constR||D^kg|| for all smooth functions g.

Here, ||f||:= sup_tau(1)<_x_<tau(n)|f(x)|. Then we may look for the optimal recovery scheme as the scheme R for which constR is as small as possible. Micchelli/Rivlin/Winograd have shown this to be interpolation from Sk_,t, with t uniquely determined by the following conditions:

1.: t(1) = ... = t(k) = tau(1);

2.: t(n+1) = ... = t(n+k) = tau(n);

3.: any absolutely constant function h with sign changes at the points t(k+1), ..., t(n) and nowhere else satisfies

Gaffney/Powell called this interpolation scheme optimal since it provides the center function in the band formed by all interpolants to the given data that, in addition, have their kth derivative between M and -M (for large M).

Examples

See the last part of the demo spapidem for an illustration.

Algorithm

This is the Fortran routine SPLOPT in PGS. It is based on an algorithm described in [1], for the construction of that sign function h mentioned in (3) above. It is essentially Newton's method for the solution of the resulting nonlinear system of equations, with aveknt(tau,k) providing the first guess for t(k+1), ...,t(n).