Gauss-Jordan Theory

ESCUELA DE INGENIERÍA DE PETROLEOS


So named because Carl Friedrich Gauss and Wilhelm Jordan, are linear algebra algorithms to determine the solutions of a system of linear equations, matrices and inverse finding.

GAUSS METHOD

Gaussian Elimination Elimination of Gauss

Gauss-Jordan Elimination


A system of equations is solved by the method of Gauss where solutions are obtained by reducing an equivalent system given in which each equation has one fewer variables than the last.

When applying this process, the resulting matrix is known as "stagger."


This method, which is a variation of Gauss elimination method, can solve up to 15 or 20 simultaneous equations, with 8 or 10 significant digits in the arithmetic of the computer.

This procedure differs from the Gaussian method in which when you delete an unknown, is removed from all remaining equations, ie, the preceding equation as well as pivot to follow.


Also all the rows are normalized when taken as pivot equation. The end result of such disposal creates an identity matrix instead of a triangular Gauss as it does, so do not use the back substitution.


The method is best illustrated with an example. Solve the following set of equations

First we express the coefficients and the vector of independent terms as an augmented matrix.

1 2 33.0 0.1 0.2 7.8500x x x

1 2 30.1 7.0 0.3 19.3x x x

1 2 30.3 0.2 10 71.4000x x x


The first line is normalized by dividing by 3 to obtain:

The term X1 can be removed from the second row by subtracting 0.1 times the first in the second row. In a similar way, subtracting 0.3 times the first in the third line delete the term with the third row X1


Then, the second line is normalized by dividing by 7.00333:

Reducing X2 terms in the first and third equation is obtained:


The third line is normalized dividing by 10 010:

Finally, the terms with X3 be reduced in the first and second equation to get:


Notice is not required substitution n s reverse direction for the solution n. The advantages and disadvantages of n Gaussian Elimination also apply to method of Gauss-Jordan.


Although the methods of Gauss-Jordan and Gauss elimination can look almost identical, the former requires approximately 50% fewer operations. Therefore, the Gaussian elimination method is simple for excellence in obtaining exact solutions to simultaneous linear equations.

One of the main reasons for including the Gauss-Jordan, is to provide a direct method for obtaining the inverse matrix.


http://personal.redestb.es/ztt/tem/t6_matrices.htm http://es.wikipedia.org/wiki/Matriz_(matem%C3%A1tica) http://www.gaussjordan.edu.mx/metodogj.html http://docencia.udea.edu.co/GeometriaVectorial/uni2/seccion21.html http://thales.cica.es/rd/Recursos/rd99/ed99-0289-02/inv02.html

Gauss-Jordan Theory

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