Two Iterative Algorithms for Solving Systems of Simultaneous Linear Algebraic Equations with Real Matrices of Coefficients

The paper describes two iterative algorithms for solving general systems of M simultaneous linear algebraic equations (SLAE) with real matrices of coefficients. The system can be determined, underdetermined, and overdetermined. Linearly dependent equations are also allowed. Both algorithms use the method of Lagrange multipliers to transform the original SLAE into a positively determined function F of real original variables and Lagrange multipliers λm. Function F is differentiated with respect to variables xi and the obtained relationships are used to express F in terms of Lagrange multipliers λm. The obtained function is minimized with respect to variables λm with the help of one of two the following minimization techniques: (1) relaxation method or (2) conjugate gradient method by Fletcher and Reeves. Numerical examples are given. INTRODUCTION Most of well-known methods for solving systems of simultaneous (consistent) linear algebraic equations that can be found in standard software packages either fail to solve systems with degenerate matrices of coefficients (for the systems containing linearly dependent equations) and the systems in which the number of equations does not coincide with the number of unknowns (in the case of underdetermined or overdetermined systems) or require tedious preliminary manipulations. Therefore, the development of a universal algorithm that is suitable for solving both underdetermined and overdetermined systems and does not require a preliminary analysis of the system type seems to be useful. 1. FORMULATION OF THE PROBLEM Let us we have a system of simultaneous linear equations