Jordan-Gauss's method is one of ways of the solution of systems of the linear equations. It is usually used, for finding of variables when other ways are powerless. Its essence consists in use of a triangular matrix or the flowchart for performance of an objective.
Gauss's method
Let's say that it is necessary to solve the system of the linear equations of the following look:
1) X1+X2+X4=0;
2) – X2-X3-5kh4=0;3)-4Х2-Х3-7Х4=0;4) 3Х2-3Х3-2Х4=0; Apparently, all is available four variables which should be found. There are several ways to make it. For a start, it is necessary to write down the system equations in the form of a matrix. In this case it will have three columns and four lines:
X1 X2 of X4-X2 X3 5X4-4X2 of X3 - 7Х4 3Х2 - 3Х3 - 2Х4 First and easiest way of the decision is substitution of a variable from one equation of a system in another. Thus, it is possible to achieve that all variables, except one will be excluded and there will be one equation. For example, it is possible to remove and substitute X2 variable from the second line in the first. This procedure can be performed also for other lines. As a result all variables, except one will be excluded from the first column.
Then Gauss's exception needs to be applied similarly and to the second column. Further by the same method it is possible to arrive also with other lines of a matrix.
Thus, all lines of a matrix take a triangular form as a result of these actions: 0 X1 0 0 X2 0 0 0 0 X3 0 X4
Jordan-Gauss's method
Jordan-Gauss's exception includes an additional step. By means of it all variables, except four are eliminated, and the matrix takes almost ideal diagonal form: X1 0 0 0 X2 0 0 X3 0 0 0 X4 Further it is possible to look for values of these variables. In this case, x1=-1, x2=2 and so on.
Need of reserve replacement is solved for each variable separately, as in Gaussian replacement therefore all unnecessary elements will be eliminated.
Additional operations in Jordan-Gauss's exception play a role of substitution of variables in a matrix of a diagonal form. It triples the number of the calculations, necessary, even in comparison with operations of reserve replacement of Gauss. However it the unknown with a bigger accuracy helps to find values and helps to count deviations better.
Shortcomings
Additional operations of a method of Jordan-Gauss increase the probability of emergence of a mistake and increase time necessary for calculation. A shortcoming both is that they demand the correct algorithm. If the sequence of actions gets off, then the result can be wrong too. For this reason such methods are most often used not for calculations on paper, and for computer programs. It is possible to realize them practically in any way and in all programming languages: from Basic to Page.