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

Abstract

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 X(i) (i=1,...,N) and Lagrange multipliers Lambda(i) (i=1,...,M). Function F is differentiated with respect to variables X(i) and the obtained relationships are used to express F in terms of Lagrange multipliers Lambda(i). The obtained function is minimized with respect to variables Lambda(i) with the help of one of two the following minimization techniques: (1) relaxation method or (2) method of conjugate gradients by Fletcher and Reeves. Numerical examples are given.