  Inverse Vandermonde matrix calculation is a long-standing problem to solve
nonsingular linear system $Vc=b$ where the rows of a square matrix $V$ are
constructed by progression of the power polynomials. It has many applications
in scientific computing including interpolation, super-resolution, and
construction of special matrices applied in cryptography. Despite its numerous
applications, the matrix is highly ill-conditioned where specialized treatments
are considered for approximation such as conversion to Cauchy matrix, spectral
decomposition, and algorithmic tailoring of the numerical solutions. In this
paper, we propose a generalized algorithm that takes arbitrary pairwise
(non-repetitive) sample nodes for solving inverse Vandermonde matrix. This is
done in two steps: first, a highly balanced recursive algorithm is introduced
with $\mathcal{O}(N)$ complexity to solve the combinatorics summation of the
elementary symmetric polynomials; and second, a closed-form solution is
tailored for inverse Vandermonde where the matrix' elements utilize this
recursive summation for the inverse calculations. The numerical stability and
accuracy of the proposed inverse method is analyzed through the spectral
decomposition of the Frobenius companion matrix that associates with the
corresponding Vandermonde matrix. The results show significant improvement over
the state-of-the-art solutions using specific nodes such as $N$th roots of
unity defined on the complex plane. A basic application in one dimensional
interpolation problem is considered to demonstrate the utility of the proposed
method for super-resolved signals.
