# Calculating the Moore-Penrose Generalized Inverse on Massively Parallel Systems
## 1 Introduction  
The generalized inverses of a matrix are defined as generalizations to the matrix inverse, so a matrix can have a generalized inverse even if the matrix is not regular or square. For nonsingular matrices, the generalized inverse coincides with its ordinary inverse. In this paper, we consider the Moore-Penrose generalized matrix inverse (or the pseudoinverse), which is the oldest type of generalized matrix inverses [1]. Pseudoinverse exists for arbitrary matrices, and one of its most important applications is finding the least square solution to the overdetermined systems of linear equations [2, 3, 4, 5]. The generalized Moore-Penrose inverse finds multiple applications in graph theory. In [6], the authors proved an equation that relates the Moore-Penrose pseudoinverses of two matrices \(A\), \(B\) such that \(A=N^{-1}BM^{-1}\) and applied it to the Moore-Penrose inverse representation of the normalized Laplacian of a graph. In [7], the authors considered the Moore-Penrose inverse of an oriented graph incidence matrix. In [8], the same authors presented a combinatorial interpretation of the Moore-Penrose inverse problem for complete multipartite and bi-block graphs. In [9], the authors provided a combinatorial interpretation of the Moore-Penrose inverse of the graph incidence matrix, where blocks are either cliques or cycles, and they described the minors of the Moore-Penrose inverse of the incidence matrix, with the rows indexed by cut edges. In [10], the authors considered the Moore-Penrose inverses of the signless Laplacian and signless edge-Laplacian of a graph and proposed combinatorial formulas of the Moore-Penrose inverses for trees and odd unicyclic graphs. In addition to the above, the pseudoinverse matrix finds application in many other areas. In [11], the authors considered two security issues for smart devices such assmarthones, tablets, etc., namely changing a shared key between two users and privacy-preserving auditing in cloud storage. Their solution is a one-way method of linear complexity based on the matrix pseudoinverse. Fast computation enables us to apply this method to devices with limited computational capacity. An interesting approach is the application of pseudoinverse and projective matrices in the areas of studying control, observation, and identification problems [12]. The method of the pseudoinverse perturbation, based on matrix splitting, is extended to projective matrices for application in identification, nonlinear regressive analysis, function approximation, and prediction problems. Many authors [13; 14] proposed various methods for applying the Moore-Penrose inversion to restore digital images. In [14], the authors introduced a method for the restoration of images that have been blurred by uniform and nonuniform linear motion. This method is based on the pseudoinverse solution of the matrix equation modeling the motion blur. The method directly generates blurring matrices based on their special structure, without iterative computations, which leads to a decrease in the CPU time, compared with other methods for the pseudoinverse computation. In [13], the authors presented a new approach to the problem of image reconstruction with the use of the Moore-Penrose generalized inverse, which provides a very high-resolution level of the reconstructed image as well as a considerable decrease in the computational load compared with the classic techniques. The calculation of the pseudoinverse matrix is often necessary when solving forecasting problems. In [15], a flat neural network was considered for the online prediction of engine states using a recursive algorithm based on a pseudoinverse partition matrix. Experimentally, the high accuracy of the developed neural network and its sufficient reliability in real-time forecasting are established, which enables us to use this network for nonlinear systems with multiple inputs and multiple outputs, such as vehicle engines. The problem of prediction was also considered in [16].