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Math Help - Rate of Convergence for iterative method.

  1. #1
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    Rate of Convergence for iterative method.

    Let an iterative solution of a system of linear equations be defined by...

    x^{(k+1)} = Gx^{(k)} + c, k = 1, 2, ...

    G = \left[ \begin{array}{cccc} 0.08 & 0 & 0.05 \\ 0.04 & 0.1 & 0.02 \\ 0 & 0 & 0.08 \end{array} \right]

    Determine the rate of convergence for the iterative method and explain your work.

    Once again, finding it hard to find information that explains this sort of thing clearly, so any help greatly appreciated.

    Thanks.
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  2. #2
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    Have you tried Varga, R.S., Matrix Iterative Analysis ?
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  3. #3
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    Okay, I think I have the answer but perhaps someone might check for me...

    G = \left[ \begin{array}{cccc} 0.08 & 0 & 0.05 \\ 0.04 & 0.1 & 0.02 \\ 0 & 0 & 0.08 \end{array} \right]

    H = \left[ \begin{array}{cccc} 0.08 - \lambda & 0 & 0.05 \\ 0.04 & 0.1 - \lambda & 0.02 \\ 0 & 0 & 0.08 - \lambda \end{array} \right]

    det(H) = (0.08 - \lambda)(0.1 - \lambda)(0.08 - \lambda)

    + (0)(0.02)(0) + (0.05)(0.04)(0)

    - (0.08 - \lambda)(0.02)(0) - (0)(0.04)(0.08 - \lambda) - (0.05)(0.1 - \lambda)(0)

    = (0.08 - \lambda)(0.1 - \lambda)(0.08 - \lambda)

    max(|\lambda|) = 0.1 = p(G)

    r(G) = -log_{10}p(G) = -log_{10}(0.1) = 1
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