Acceptable knot sequence
knots = aptknt(tau,k)
[knots,k] = aptknt(tau,k)
knots = aptknt(tau,k) returns a knot
sequence suitable for interpolation at the data sites tau by
splines of order k with that knot sequence, provided tau has at
least k entries, is nondecreasing, and satisfies
tau(i)<tau(i+k-1) for all
i. In that case, there is exactly one spline of order
k with knot sequence knots that matches
given values at those sites. This is so because the sequence
knots returned satisfies the Schoenberg-Whitney
conditions
knots(i) < tau(i) < knots(i+k), i=1:length(tau)
with equality only at the extreme knots, each of which occurs with exact
multiplicity k.
If tau has fewer than k entries, then
k is reduced to the value length(tau). An
error results if tau fails to be nondecreasing and/or
tau(i) equals tau(i+k-1) for some
i.
[knots,k] = aptknt(tau,k) also returns the
actual k used (which equals the smaller of the input
k and length(tau)).
If tau is equally spaced, e.g., equal to
linspace(a,b,n) for some n>=4, and
y is a sequence of the same size as tau,
then sp = spapi(aptknt(tau,4),tau,y) gives the cubic spline
interpolant with the not-a-knot end condition. This is the same cubic spline as
produced by the command spline(tau,y), but in B-form rather than
ppform.
If tau is very nonuniform, then use of the resulting knot
sequence for interpolation to data at the sites tau may lead to
unsatisfactory results.
The (k-1)-point averages
sum(tau(i+1:i+k-1))/(k-1) of the sequence
tau, as supplied by aveknt(tau,k), are
augmented by a k-fold tau(1) and a
k-fold tau(end). In other words, the
command gives the same result as
augknt([tau(1),aveknt(tau,k),tau(end)],k), provided
tau has at least k entries and k
is greater than 1.