FUNCTION: NewtonInterp - Newton Interpolation formula

CALLING SEQUENCE:

NewtonInterp(f, n)

SP[NewtonInterp](f, n)

PARAMETERS:

f = any expression in the basis of monomials

n = any positive integer denoting the degree of a symmetric group

SYNOPSIS:

• The NewtonInterp function expresses the function f(x1, x2, ...) in terms of XX Schubert polynomials. It uses the Newton interpolation formula involving the permutations in the symmetric group of degree n.

• Don't forget it is an interpolation...

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

• When called with one argument, the second argument is set to the value of the variable _Sn that fixes the degree of the symmetric group.

• The result is not expanded.

• Whenever there is a conflict between the function name NewtonInterp and another name used in the same session, use the long form SP['NewtonInterp'].

EXAMPLES:

> with(SP):
> Flag(-1):
> NewtonInterp(2*x2 + 4*a^2*x1^2, 4);
 
            2   2                                         2
 (2 y2 + 4 a  y1 ) XX[1, 2, 3, 4] + 2 XX[1, 3, 2, 4] + 4 a  XX[3, 1, 2, 4]
 
             2             2
       + (4 a  y2 - 2 + 4 a  y1) XX[2, 1, 3, 4]
 
> expand( Tox( NewtonInterp(2*x2 + 4*a^2*x1^2, 4) ) );
 
                                         2   2
                               2 x2 + 4 a  x1