The qcheb algorithm is a mystery and is inflexible.

[emsr]

There is actually a good bit online about this. it's one of those ancient algos that gets passed around.

c***purpose  this routine computes the chebyshev series expansion
c            of degrees 12 and 24 of a function using a
c            fast fourier transform method
c            f(x) = sum(k=1,..,13) (cheb12(k)*t(k-1,x)),
c            f(x) = sum(k=1,..,25) (cheb24(k)*t(k-1,x)),
c            where t(k,x) is the chebyshev polynomial of degree k.
c        parameters
c          on entry
c           x      - real
c                    vector of dimension 11 containing the
c                    values cos(k*pi/24), k = 1, ..., 11
c
c           fval   - real
c                    vector of dimension 25 containing the
c                    function values at the points
c                    (b+a+(b-a)*cos(k*pi/24))/2, k = 0, ...,24,
c                    where (a,b) is the approximation interval.
c                    fval(1) and fval(25) are divided by two
c                    (these values are destroyed at output).
c
c          on return
c           cheb12 - real
c                    vector of dimension 13 containing the
c                    chebyshev coefficients for degree 12
c
c           cheb24 - real
c                    vector of dimension 25 containing the
c                    chebyshev coefficients for degree 24

So the polynomial expansion is in terms of T_n(x) and the roots are those of another Chebyshev polynomial U_{n-1}(x).

Let's have the ability to have expansions in T_{2m}(x) (and T_m(x)) at roots of U_{2m-1}(x).

In fact, I want a general Clenshaw thing for arbutrary polynomials with a recurrence relation.