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The Jacobi Iterative method question

  1. Mar 19, 2012 #1
    1. The problem statement, all variables and given/known data

    (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.

    2. Relevant 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.
     
  2. jcsd
  3. Mar 19, 2012 #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. :-)
     
  4. Mar 19, 2012 #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 :(
     
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