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## `SP::NewtonInterp` -- Newton interpolation formula

#### Call(s)

`SP::NewtonInterp(f,n)`

#### Parameters

`f` | - | any expression in the basis of monomials
`n` | - | any positive integer denoting the degree of a symmetric group

#### Introduction

The `SP::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.

#### Example 1

    >> muEC::SP::NewtonInterp(2*x2 + 4*a^2*x1^2, 4);

                     2   2
          (2 y2 + 4 a  y1 ) 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]

    >> expand( muEC::SP::Tox(%) );

                                            2   2
                                  2 x2 + 4 a  x1

#### Related Functions

`x2XX`

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MuPAD Combinat, an open source algebraic combinatorics package