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Gaussian Elimination 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.
EXAMPLE Resoverthe following system of equations. 3X1 -0.1X2 - 0.2 X 3 = 7.85 0.1X1 +7X2 - 0.3 X 3 = -19.5 0.3X1 -0.2X2 +10 X 3 = 7.85 First we express the coefficients and the vector of independent terms as an augmented matrix.
The first line is normalized by dividing by 3 for 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 X1F2-0.1F1F3-0.3F1
Then, the second line is normalized by dividing by 7.00333: Reducing X2 terms in the first and third equation is obtained:F1+ 0.033 F2F3+0.0190 F2
The third line is normalized dividing by 10 010: Finally, the terms with X3 be reduced in the first and second equation to get:F1+ 0.068 F3F3+0.0425 F2 X1=3 X2=-2.499 X3=7.001
ADVANTAGES OF THE METHOD OF GAUSS-JORDAN Note that back substitution is required for the solution.Although the methods of Gauss-Jordan and Gauss elimination can look almost identical, the former requires approximately 50% fewer operations.One of the main reasons for including the Gauss-Jordan, is to provide a direct method for obtaining the inverse matrix.