Problem with expand in lagrange interpolation

On 6/14/07, Daniel Lakeland <dlakelan at street-artists.org> wrote:
>
> I spent a little time last night rewriting the lagrange routine's
> computation of the polynomial to use the newton divided differences
> form.
>

I'm not sure I understand.  The interpolation polynomial (of degree n-1) on
n points is unique. There is a variety of ways of rewriting that polynomial
for various purposes and I don't see why you'd want the lagrange routine to
output anything other than the lagrange form.

On the other hand, you might want routines for converting a polynomial (any
polynomial) to the various forms, and we have some of them:

   taylor(...) useful in the neighborhood of a particular point.
   rat(...) tells you about its behavior for very large values.
   The Bernstein form is useful for evaluating in a given range (see code
below)

Etc.

                      -s

/* convert any polynomial in var to the Bernstein form for the range [a,b]
*/
/* result is a formula, not a program, and not very efficient, but
   demonstrates the form */
to_bernstein(poly,var,a,b):=
   block([degree,denom,nvar],
         poly: subst((var-a)/(b-a),var,poly),
         poly: rat(poly,var),
         denom: ratdenom(poly),
         if not constantp(denom)
           then error("polynomials only"),
         degree: hipow(poly,var),
         nvar: var*(b-a)+a,
         sum( binomial(degree,i)*nvar^i*(1-nvar)^(degree-i)*
             sum( binomial(i,j)/binomial(degree,j)
                  *ratdisrep(ratcoeff(poly,var,j)),j,0,i),
              i,0,degree))$