Results 1 to 7 of 7
Like Tree2Thanks
  • 1 Post By BobP
  • 1 Post By BobP

Math Help - power method

  1. #1
    Senior Member
    Joined
    Feb 2010
    Posts
    456
    Thanks
    34

    power method

    use power method to find largest and smallest eigen value of matrix A= .Hence find all eigen values of matrix A.Also find eigen value that is farthest and nearest from 4 take initial vector as[1,1,1]

    Here how to find the largest eigen value as so many eigen values can be calculated
    also how to find eigen value that is farthest away fom 4 using power method

    please help..
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Newbie
    Joined
    Feb 2014
    From
    florida
    Posts
    2

    Re: power method

    Code:
    |2 -1 0|  |1|        |-1|
    |-1 2 -1|*|1| = 1*|0|
    |0 -1 2|   |1|       |1|
    |2 -1 0|  |-1|       |1|
    |-1 2 -1|*|0| = 1*|0|
    |0 -1 2|   |1|       |1|
    so its occilating between the two.
    the middle term is 0 the top may either be -1 or 1 and the bottom is 1.
    and the eigen value is 1.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Super Member
    Joined
    Jun 2009
    Posts
    660
    Thanks
    133

    Re: power method

    With a start vector of [1, 1, 1]^{T}, constantly pre-multiplying by the given matrix produces the sequence

    [1,0,1]^{T}, [2,-2, 2]^{T}, [6, -8, 6]^{T}, [20, -28, 20]^{T}, [68, -96, 68]^{T}, [232, -328, 232]^{T}

    and so on.

    It's usual to normalize at each step, but in this case it's convenient to leave all of the numbers as integers and you should, (after another two or three steps,) be able to calculate the dominant eigenvalue, (as the converging ratio of the normalizing factors).

    Read your notes relating to the remainder of this question (regarding the use of the inverse matrix and of the technique of shifting), and post again if you need further help, but, you should show that you have made some effort, actually tried to answer this question.
    Thanks from prasum
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Senior Member
    Joined
    Feb 2010
    Posts
    456
    Thanks
    34

    Re: power method

    for eigen value closest to 4 i used the concept if(eigen value-4) is smallest then 1/(eigen value-4) is largest and then i equated this with the largest eigen value of inverse of matrix
    and got the eigen value closest to 4 but how to get eigen value farther away from 4
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Super Member
    Joined
    Jun 2009
    Posts
    660
    Thanks
    133

    Re: power method

    The wording of the question is very strange, in that you are asked to calculate all of the eigenvalues of the matrix and then also to find the eigenvalues furthest from and nearest to 4. Surely that doesn't require further calculation, you already have all of the eigenvalues and all that you need to do is to choose two of the three.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Senior Member
    Joined
    Feb 2010
    Posts
    456
    Thanks
    34

    Re: power method

    but then how will we know whether that eigen value is close or far enough but we stop when we see that our iteration is converging to an eigen value
    Last edited by prasum; February 22nd 2014 at 09:26 PM.
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Super Member
    Joined
    Jun 2009
    Posts
    660
    Thanks
    133

    Re: power method

    Do you have the exact wording of this question, because as it is it does not make clear exactly what you are supposed to do.
    The routine for finding the dominant eigenvalue (using the power method) I indicated earlier. Just how far you go with the sequence of vectors depends on the required accuracy of the eigenvalue. Have you been given a desired accuracy ?

    To find the smallest (in magnitude) eigenvalue (again using the power method), requires you to use the inverse of the matrix. In practice the LU decomposition of A should be used, but that isn't made clear.

    How are you supposed to find the third eigenvalue ? Again it isn't made clear. You could continue with the power method theme, shifting to the midpoint of the two eigenvalues found already and then again using the inverse of A ? Alternatively you might calculate the characteristic polynomial and deduce the third eigenvalue from that ?

    You are now asked to find the eigenvalues nearest to and furthest from 4. Can you not just read those from the three that have just been calculated ? Maybe you are supposed to use a shift of 4 and to repeat the earlier calculations ? If it is to be an exercise in shifting, what's the point, haven't you already (possibly) demonstrated that to find the middle eigenvalue ? I don't know what is required.

    The issue of convergence is not relevant, you should have instructions to calculate the eigenvalues to some degree of accuracy and that is what you should do.
    Thanks from prasum
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Inverse Power Method Iteration - HELP!
    Posted in the Calculus Forum
    Replies: 0
    Last Post: July 26th 2012, 07:08 AM
  2. Power series method problem
    Posted in the Differential Equations Forum
    Replies: 1
    Last Post: May 12th 2011, 06:52 PM
  3. Power method
    Posted in the Advanced Algebra Forum
    Replies: 0
    Last Post: December 31st 2009, 04:43 AM
  4. Replies: 3
    Last Post: November 3rd 2007, 02:43 PM
  5. power series method and frobenius method
    Posted in the Calculus Forum
    Replies: 1
    Last Post: August 26th 2006, 01:25 PM

Search Tags


/mathhelpforum @mathhelpforum