Inverse Of Vandermonde Matrix Research Articles

Series and Connections Among Central Factorial Numbers, Stirling Numbers, Inverse of Vandermonde Matrix, and Normalized Remainders of Maclaurin Series Expansions †

This paper presents an extensive investigation into several interrelated topics in mathematical analysis and number theory. The author revisits and builds upon known results regarding the Maclaurin power series expansions for a variety of functions and their normalized remainders, explores connections among central factorial numbers, the Stirling numbers, and specific matrix inverses, and derives several closed-form formulas and inequalities. Additionally, this paper reveals new insights into the properties of these mathematical objects, including logarithmic convexity, explicit expressions for certain quantities, and identities involving the Bell polynomials of the second kind.

A recursive algorithm for computing the inverse of the Vandermonde matrix

The inverse of a Vandermonde matrix has been used for signal processing, polynomial interpolation, curve fitting, wireless communication, and system identification. In this paper, we propose a novel fast recursive algorithm to compute the inverse of a Vandermonde matrix. The algorithm computes the inverse of a higher order Vandermonde matrix using the available lower order inverse matrix with a computational cost of O(n2). The proposed algorithm is given in a matrix form, which makes it appropriate for hardware implementation. The running time of the proposed algorithm to find the inverse of a Vandermonde matrix using a lower order Vandermonde matrix is compared with the running time of the matrix inversion function implemented in MATLAB.

FAST RESOLUTION OF INTEGER VANDERMONDE SYSTEMS

The resolution of polynomial interpolation problems with integer coefficients directly involves the open issue of the integer inversion of a general Vandermonde matrix defined over the field ℤ/p ℤ, for p prime number. The purpose of this paper is to demonstrate the possibility to invert a Vandermonde matrix with integer mod p coefficients and exactly compute the integer inverse matrix in the ring Mat(ℤ/p ℤ) of square matrices over ℤ/p ℤ through the new fast algorithm InVanderMOD. The explicit formula derived for the integer inversion of Vandermonde matrices entirely develops inside the field of the integers mod p , with due consideration to the operation of integer division. The inversion procedure InVanderMOD is valid for any prime number p and competitive in terms of computational effort, since its computational cost is less than O(n 3 ) .

On the Inversion of Vandermonde Matrix

The inversion of Vandermonde matrix has received much attention for its role in the solution of some problems of numerical analysis and con-trol theory. This work deals with the problem of getting an explicit formula for the generic element of the inverse. It has been shown that this generic element turns out to be a linear combination of the elementary symmetric functions associated with the n values defining Vandermonde matrix.