Gauss-Jacobi and Gauss-Siedel

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In summary, Gauss-Jacobi and Gauss-Siedel methods are two iterative techniques used to solve systems of linear equations. The main difference between them is the way in which the solution vector elements are updated. The choice between these methods depends on the convergence rate and diagonal dominance of the system. They cannot be used for non-linear systems and have a convergence criteria based on a tolerance value. However, they can be computationally expensive and may not converge for certain systems.
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viciado123
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Solving the system of linear equations using the methods of Gauss-Jacobi and Gauss-Siedel. Using precision of [tex]1x10^{-3}[/tex].

A=[-4 -1 2; 1 -10 6; 1 -3 -6]
B=[x1; x2; x3]
C=[1 -5 7]

A.B = C



Using Gauss-Jacobi I find:
I used x^0 = [0;0;0]. I can ?
With 11 iterations I find
[tex]X_1 = -0,762[/tex]
[tex]X_2 = -0,271[/tex]
[tex]X_3 = -1,159[/tex]
Test Stop
[tex]M_r = \frac{|-1,159 - (-1,158)|}{|-1,159|} = 8,63x10^{-4} < 1x10^{-3}[/tex]
The test is stopped for any X (x1, x2, x3) or for all ?


Using Gauss-Siedel
With 9 iterations I find
Used x^0 = [0;0;0]
[tex]X_1 = -0,758[/tex]
[tex]X_2 = -0,269[/tex]
[tex]X_3 = -1,155[/tex]
Test Stop
[tex]M_r = \frac{|-1,155 - (-1,154)|}{|-1,155|} = 8,65x10^{-4} < 1x10^{-3}[/tex]


Are correct ?
 
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  • #2


I would like to point out that the methods of Gauss-Jacobi and Gauss-Siedel are both valid methods for solving systems of linear equations. However, it is important to note that the number of iterations required to reach a certain level of precision may vary for each method and for each specific system of equations.

In this case, it seems that both methods have provided solutions that meet the desired precision of 1x10^{-3}. However, it is always important to test the solutions obtained to ensure their accuracy.

In order to test the solutions, the relative error (M_r) is calculated using the formula provided in the forum post. The values obtained for M_r in both methods are less than the desired precision, indicating that the solutions obtained are accurate.

In conclusion, both methods seem to have provided correct solutions. However, it is always a good practice to test the solutions obtained and to also consider the number of iterations required for each method.
 

1. What is the difference between Gauss-Jacobi and Gauss-Siedel methods?

Gauss-Jacobi method is an iterative technique used to solve a system of linear equations while Gauss-Siedel method is an improved version of Gauss-Jacobi that converges faster. The main difference between the two methods is that in Gauss-Jacobi, all the elements of the solution vector are updated simultaneously, while in Gauss-Siedel, the elements are updated one at a time, using the most recently computed values.

2. How do I know which method to use for a specific problem?

The choice between Gauss-Jacobi and Gauss-Siedel methods depends on the convergence rate of each method for the given system of equations. Generally, if the system has a high degree of diagonal dominance, Gauss-Siedel method is preferred, while for systems with low diagonal dominance, Gauss-Jacobi method might be a better choice. Additionally, it is recommended to try both methods and compare the number of iterations needed for convergence.

3. Can Gauss-Jacobi and Gauss-Siedel methods be used to solve non-linear systems?

No, these methods can only be used to solve systems of linear equations. Non-linear systems require different techniques, such as Newton's method or the Broyden method.

4. What is the convergence criteria for Gauss-Jacobi and Gauss-Siedel methods?

The convergence criteria for both methods is that the absolute value of the difference between the current iteration and the previous iteration should be less than a given tolerance value, usually set to a small number close to zero. Additionally, for Gauss-Siedel method, the diagonal entries of the coefficient matrix should be non-zero for the method to converge.

5. Are there any disadvantages to using Gauss-Jacobi and Gauss-Siedel methods?

Yes, these methods can be computationally expensive, especially for large systems of equations, as they require multiple iterations to converge. Additionally, these methods may not converge for certain systems of equations, and it can be difficult to predict this beforehand.

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