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

Abstract

Software implementations of analytical and numerical-analytical decomposition methods for determining complex one-parameter generalized inverse Moore-Penrose mat-rices are presented. Analytical methods are based on the first two Moore-Penrose con-ditions: 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 infor-mation 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.