SOFTWARE IMPLEMENTATION OF DECOMPOSITION METHODS FOR DETERMINING COMPLEX ONE-PARAMETER GENERALIZED INVERSE MOORE-PENROSE MATRICES (II)

Abstract

Software implementations of analytical and numerical-analytical decomposition 
methods for determining complex one-parameter generalized inverse Moore-Penrose 
matrices are presented. Analytical methods are based on the first two Moore-Penrose 
conditions: in both cases the same solution to the problem is obtained, as it should be. 
Numerical-analytical methods are based on the obtained analytical relations and differential 
Pukhov transformations. Computational methods are implemented using modern 
information technologies, in particular Python programming language, NumPy and SymPy 
libraries. Software implementations consist of programs for entering the dimensions and 
elements of a given matrix, implementation of an analytical method, determination of 
matrix elements derivatives and their values, calculation of matrix discretes and restoration 
of original matrices, as well as determining pseudo-inverse matrix. In addition, the software 
implementation of the numerical-analytical method also contains a custom implemented 
functions for determining the given matrix elements derivatives and for calculating the 
matrix discretes at the approximation center with a given scaling coefficient. Algorithm 
flowcharts and Python source codes for all listed programs are presented. 
The software implementation is tested on a model example with a square matrix, for 
which an analytical solution is obtained using two approaches - the determination of the 
usual inverse one-parameter matrix and the decomposition method for determining the 
same matrix. This matrix is defined in exactly the same way as numerical-analytical 
decomposition methods.