On the Closed Form Expression of Elementary Symmetric Polynomials and the Inverse of Vandermonde Matrix
Abstract
Inverse Vandermonde matrix calculation is a longstanding 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, superresolution, and construction of special matrices applied in cryptography. Despite its numerous applications, the matrix is highly illconditioned 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 (nonrepetitive) 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 closedform 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 stateoftheart 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 superresolved signals.
 Publication:

arXiv eprints
 Pub Date:
 September 2019
 arXiv:
 arXiv:1909.08155
 Bibcode:
 2019arXiv190908155H
 Keywords:

 Mathematics  Numerical Analysis;
 05E05;
 11C08;
 11C20;
 15B05;
 15A09;
 65F05
 EPrint:
 Submitted for Journal Publication