% $Date: 2004/10/04 17:25:12 $ $Author: nthiery $ $Revision: 1.1.1.3 $

\begin{Mhelppage}{NewtonInterp}{S. Veigneau}
  
  \begin{MInfo}
    \eng{Newton interpolation formula}
  \end{MInfo}
      
\begin{MCalls}
 \Mcall{f,n}
\end{MCalls}
%
\begin{MParameters}
  \param{f}
  \eng{any expression in the basis of monomials}
  \param{n}
  \eng{any positive integer denoting the degree of a symmetric group} 
\end{MParameters}
%
\begin{MIntro}
  \eng{
The {\Mthis} function expresses the function \Mex{f(x1, x2, ...)} in terms
of \Mex{XX} Schubert polynomials. It uses the Newton interpolation formula
involving the permutations in the symmetric group of degree \Mex{n}.

Don't forget it is an interpolation...

On the contrary, the function \Mex{x2XX} uses Monk formula instead of divided
differences to decompose (exactly) a polynomial into the \Mex{XX} basis.

When called with one  argument, the second argument is set to the value of
the variable \Mmath{Sn}{S_n} that fixes the degree of the symmetric group.
 
The result is not expanded.
}
\end{MIntro}
%
\begin{MExample}{ex1}
%
\begin{Mexin}
muEC::SP::NewtonInterp(2*x2 + 4*a^2*x1^2, 4);
\end{Mexin}
\begin{Mexout}
    2   2
(4 a  y1  + 2 y2) XX[1, 2, 3, 4] +

       2         2
   (4 a  y1 + 4 a  y2 - 2) XX[2, 1, 3, 4] +

      2
   4 a  XX[3, 1, 2, 4] + 2 XX[1, 3, 2, 4]
\end{Mexout}
% \begin{Mexout}[hide]
         % XX[1, 3, 2, 4]*2 + a^2*XX[3, 1, 2, 4]*4 +
         % XX[1, 2, 3, 4]*(y2*2 + a^2*y1^2*4) +
         % XX[2, 1, 3, 4]*(a^2*y1*4 + a^2*y2*4 + (-2))
% \end{Mexout}
%
\begin{Mexin}
expand( muEC::SP::Tox(%) );
\end{Mexin}
\begin{Mexout}
                           2   2
                        4 a  x1  + 2 x2
\end{Mexout}
% \begin{Mexout}[hide]
                      % x2*2 + a^2*x1^2*4
% \end{Mexout}
\end{MExample}
%
\MRelatedFunctions{x2XX}
\end{Mhelppage}
