The Jacobi Iterative method question

In summary, the conversation discusses the Jacobi Iterative method and its use in solving a system of equations (Ax = B). It is mentioned that the matrix A is not diagonally dominant, which may affect the convergence of the method. The conversation also mentions the use of initial values and suggests trying different values to improve the accuracy of the solution.
  • #1
Spectre Moncy
2
0

Homework Statement



(Ax = B)

A:

3.1410 -2.7180 1.4140 -1.7321
9.8690 2.7180 -7.3890 0.4280
2.2360 -2.4490 1.0000 -1.4140
31.0060 7.3890 -2.6450 0.1110

B:

3.316
0
3.141
1.414

The question in my Numerical Methods assignment asks to use the Jacobi Iterative method to solve the system.

Homework Equations



The Jacobi Iterative method works ONLY IF a matrix is diagonally dominant. It's not mentioned in my Numerical Analysis textbook. I have only found out about this on wiki.org.

I have found out that the matrix A (See above) is not diagonally dominant. So the method will not work on this system (Ax = B).

Did I miss something? What should I do? I tried the method very carefully. The solution set doesn't make sense.
 
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  • #2
Jacobi method can converge even if the matrix is not diagonally dominant. However, you notice immediately from the iteration formula
[itex] x_i^{n+1} = \frac{b_i - \sum_{j \neq i} a_{ij} x^{n}_j}{a_{ii}} [/itex]
that if the matrix is not diagonally dominant,
[itex] \frac{\sum_{j \neq i} a_{ij}}{a_{ii}} \gt 1 [/itex]
then the convergence depends on the initial value you choose for x. Perhaps you can make a better guess for the initial x, or if that fails, look up the correct value from wolfram alpha and adjust your guess accordingly. :-)
 
  • #3
The initial x^(0) (provided by the assignment question paper) is x^(0) = (3, 0, 3, 1).

I tried this. It converged to some solution set but the problem is that this solution set is extremely inaccurate.

I have no problem finding the right solution set when applying Gaussian Elimination (with partial pivoting) and LU Decomposition method on this system (Ax = b).

I can't say the same for the Jacobi method :(
 

FAQ: The Jacobi Iterative method question

1. What is the Jacobi Iterative method?

The Jacobi Iterative method is a numerical algorithm used to solve systems of linear equations. It is an iterative process that involves repeatedly approximating the solution until it reaches a desired level of accuracy.

2. How does the Jacobi Iterative method work?

The Jacobi Iterative method works by breaking down a system of linear equations into a series of simpler equations, and then repeatedly solving these equations using updated estimates of the variables until the solution converges.

3. What are the advantages of using the Jacobi Iterative method?

The Jacobi Iterative method is relatively easy to implement and can be used to solve large systems of equations. It is also well-suited for solving systems with sparse matrices, where most of the coefficients are zero.

4. What are the limitations of the Jacobi Iterative method?

The Jacobi Iterative method may converge slowly or even fail to converge for certain types of matrices. It also requires an initial guess for the solution, which may affect the accuracy and convergence rate of the method.

5. How does the Jacobi Iterative method compare to other methods for solving systems of linear equations?

The Jacobi Iterative method is a basic algorithm for solving systems of linear equations, and it may not be as efficient as other more advanced methods, such as the Gauss-Seidel method or the conjugate gradient method. However, the Jacobi Iterative method can be a good choice for certain types of systems, and it can also be used as a starting point for more advanced methods.

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