Gaussian elimination method & Gauss-Jordan elimination method

Gaussian elimination method & Gauss-Jordan elimination method

18:02:10 | classification: Control Theory
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Gaussian elimination method is a common method for solving linear equations, and it may not be very familiar to users. Next we will first introduce some basic concepts of linear equations and matrices and Gaussian elimination method. Then we will focus on the advantages of Gaussian approximate elimination method over Gaussian elimination method: Simple Program (no back-generation is required ), it is widely used (for example, inverse matrix) and is easy to judge and handle special cases (with infinite solutions ).

This is a linear equations:

Calculate the coefficient matrix in the equations
And result Vectors
To obtain the following Augmented Matrix:

The basic method of Gaussian elimination method is to use addition and subtraction to select each unknown number in sequence.
Use one of the rows in the Augmented Matrix to remove
Finally, the return process is used to obtain the solution of the equations. To reduce the error, select
The row with the largest absolute value of the coefficient.

Gaussian elimination method has some difficult problems to solve:
1. Back-to-generation process: although the formula is not complex, it is still a great obstacle for beginners (such as me) to combine the consumption and addition and subtraction in the programming age.
2. Free variable: if there is a free variable in the process of elimination, it must be assigned immediately; otherwise, it cannot continue. It is difficult to find the functional relationship between other variables and free variables.
3. Swap two rows: during processing, the two rows of the original matrix must be exchanged repeatedly, which makes debugging difficult.
　　
Gaussian Describe method solves these problems. Unlike the Gaussian elimination method
Coefficient
In addition, not only does
, Also removes
. In this way, the back-to-generation process is not required after the elimination of the element. Every row in the equations is
. At the same time, because the Back-to-generation process is not required, the order of the elements for each unknown in the equations is no longer so strict. Therefore, we can change "select row principal component" to "Select column principal component ", that is, select each equation in sequence and select the largest absolute value of the coefficient.
. In this way, the equation can be processed in sequence, and free variables will not appear before the completion of the elimination. If there is "no yuan can be eliminated", there are only two situations:
And
, The former is ignored directly, and the latter is unsolvable directly.