Talk:Inverse of Vandermonde Matrix/Eisinberg Formula

29 Nov 2019: The Knuth Vandermonde inverse formula requires the computation of $n^2$ symmetric functions $\map {e_m} { \set {x_1,\ldots,x_n} \setminus \set {x_j} }$, $1 \le m \le n$, $j=1,\ldots,n$. Eisinberg and Picardi (1981) provide a Vandermonde inverse formula which is perhaps the first to use just $n$ elementary symmetric functions.




 * : $\S 1.2.3$: Sums and Products: Exercise $40$

The 1981 Italian formula can be tested by this brief Maple code:--Gbgustafson (talk) 06:25, 29 November 2019 (EST)

n:=4;V:=VandermondeMatrix[LinearAlgebra]([seq(x[i],i=1..n)]); p:=unapply( product((y-x[ii]),ii=1..n),y); for j from 1 to n do q[j]:=unapply(simplify(p(y)/(y-x[j])),y); od: A:=[seq(expand(coeff(p(y),y,m)),m=0..n)]; B:=(i,j)->(sum(A[i+k+1]*x[j]^(k),k=0..n-i))/q[j](x[j]); # maple indices start at 1 VI:=Matrix(n,n,(i,j)->simplify(B(i,j))); simplify(VI-(1/V));