Talk:Inverse of Vandermonde Matrix/Eisinberg Formula

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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.

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));