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

TL;DR

This paper introduces two iterative algorithms utilizing Lagrange multipliers and minimization techniques to solve various types of linear algebraic systems, including underdetermined and overdetermined cases, with demonstrated numerical examples.

Contribution

The paper presents novel iterative algorithms that combine Lagrange multipliers with relaxation and conjugate gradient methods for solving general SLAE.

Findings

Algorithms effectively solve diverse SLAE types.

Numerical examples validate the methods.

Approach handles dependent equations.

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.