2 research outputs found
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 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.Comment: 5 pages; 0 figures; 4 reference
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 and Lagrange multipliers 位m. Function F is differentiated with respect to variables xi and the obtained relationships are used to express F in terms of Lagrange multipliers 位m. The obtained function is minimized with respect to variables 位m 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