The Computation and Application of the Generalized Inverse via Maple

Abstract

In Karampetakis (1995) an algorithm for computing the generalized inverse of a singular polynomial matrix A(s) ∈ R[s] n×m has been presented. In this paper the algorithm is extended to that of the singular rational matrix, A(s) ∈ R(s) n×m , and the algorithm is subsequently implemented in the symbolic computational package Maple. Several applications of its use are given.

Interpolation algorithm of Leverrier–Faddev type for polynomial matrices

“…The group inverse, denoted by A # , is the unique {1, 2, 5}-inverse of A, and it exists if and only if ind(A) = min{k : rank(A k+1 ) = rank(A k )}=1. A matrix X = A D is said to be the Drazin inverse of A if (1 k ) (for some positive integer k), (2) and (5) are satisfied. By A −1 R and A −1 L we denote a right and a left inverse of A, respectively.…”

“…Corresponding algorithm for two-variable polynomial matrix is presented in [7]. In [5] it is described an implementation of the algorithm for computing the Moore-Penrose inverse of a singular one variable rational matrix in the symbolic computational language MAPLE.…”

Inverses of Multivariable Polynomial Matrices by Discrete Fourier Transforms

“…An extension of this algorithm to the one and two-variable polynomial matrices has been proposed by [10], [12], [13], [14]. A Leverrier-Faddeev algorithm has also been proposed by Grevile [7] for the computation of the Drazin inverse of square constant matrices with extensions to the one-variable polynomial matrices by [11], [18].…”

Computation of generalized inveses

“…Izračunavanje Moore-Penroseovog inverza za polinomijalne i racionalne matrice, baziran na Leverrier-Faddeevom algoritmu proučavan je u [2,17,22,23,24,25]. U literaturi je poznat veći broj aplikacija za izračunavanje generisanje inverza polinomijalnih matrica [22,23,24,25,30,31,32].…”

“…U literaturi je poznat veći broj aplikacija za izračunavanje generisanje inverza polinomijalnih matrica [22,23,24,25,30,31,32]. U [22] opisana je implementacija algoritma za izračunavanje Moore-Penroseovog inverza singularnih racionalnih matrica u programskom jeziku MAPLE. Algoritmi koji koriste reprezentaciju matrice u polinomijalnoj formi, su naročito pogodni za implementaciju u programskim paketima koji ne mogu vršiti simbolička izračunavanja.…”