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Math Help - pwer method

  1. #1
    emn
    emn is offline
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    How can I solve a problem
    find by power method larger egien value of the matrix
    [ 4 1 -1 ]
    2 3 -1
    -2 1 5

    Help me as soon as possible4
    Last edited by CaptainBlack; June 29th 2008 at 10:13 PM.
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by emn View Post
    How can I solve a problem
    find by power method larger egien value of the matrix
    [ 4 1 -1 ]
    2 3 -1
    -2 1 5

    Help me as soon as possible4
    Let \bold{A} be a matrix with an eigen value of absolute value greater than that of its other eigen values, then the ittereation:

    \bold{b}_{k+1}=\frac{\bold{Ab}_{k}}{|\bold{Ab}_k|}  \ \ \ \ \ ...(1)

    converges to the eigen vector corresponding to the eigen vector of largest absolute value.

    If \bold{b} is this limit the largest eigen value can be found from any component of the equation:

    \bold{Ab}=\lambda \bold{b} \ \ \ \ \ \ \ ...(2)

    So you start with a random vector \bold{b}_0 and apply the itteration in (1) to it untill a satisfactory lefel of convergence is observed, then use any non-zero component of \bold{b} and equation (2) to find the eigen value.

    Now this requires a lot of computation and I expect you are supposed to use some sort of tool to perform the itteration.

    RonL
    Last edited by CaptainBlack; June 29th 2008 at 10:50 PM.
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  3. #3
    Grand Panjandrum
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    Quote Originally Posted by CaptainBlack View Post
    Let \bold{A} be a matrix with an eigen value of absolute value greater than that of its other eigen values, then the ittereation:

    \bold{b}_{k+1}=\frac{\bold{Ab}_{k}}{|\bold{Ab}_k|}  \ \ \ \ \ ...(1)

    converges to the eigen vector corresponding to the eigen vector of largest absolute value.

    If \bold{b} is this limit the largest eigen value can be found from any component of the equation:

    \bold{Ab}=\lambda \bold{b} \ \ \ \ \ \ \ ...(2)

    So you start with a random vector \bold{b}_0 and apply the itteration in (1) to it untill a satisfactory lefel of convergence is observed, then use any non-zero component of \bold{b} and equation (2) to find the eigen value.

    Now this requires a lot of computation and I expect you are supposed to use some sort of tool to perform the itteration.

    RonL
    This can be mechanised as in the following:

    Code:
    >A=[4,1,-1;2,3,-1;-2,1,5]
                4             1            -1 
                2             3            -1 
               -2             1             5 
    >
    >function itterate(A)
    $  bm=random(3,1);
    $  repeat
    $
    $    bp=A.bm;
    $    bp=bp/sqrt(sum((bp^2)'));
    $    l=A.bp/bp;
    $    lambda=l(1);
    $    if totalmax(abs(bm-bp))<1e-6
    $       break
    $    endif
    $    bm=bp;
    $  end
    $  return {lambda,bp}
    $endfunction
    >
    >{lambda,b}=itterate(A);lambda," ",b
                6 
     
         0.577351 
         0.577351 
        -0.577349 
    >
    RonL
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