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Math Help - Accuracy of principal Eigen vector and Eigen value estimates

  1. #1
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    Accuracy of principal Eigen vector and Eigen value estimates

    Hi,

    I am writing a small piece of software that will use the AHP method ("Analytic Hierarchy Process (AHP) is a structured technique for dealing with complex decisions"). This requires calculating principal Eigen vectors and Eigen values.

    Since I don't have a mathematics background, I found this great tutorial:
    Analytic Hierarchy Process AHP Tutorial

    The methods used to calculate the principal Eigen vector and Eigen value in that tutorial are only approximations. I wanted to ask if you guys have any idea as to how accurate these approximations are as the matrix size grows beyond 3x3 (i.e. would using the approximation method be sufficient?).

    Any help would be much appreciated.
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  2. #2
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    Quote Originally Posted by silverback566 View Post
    Hi,

    I am writing a small piece of software that will use the AHP method ("Analytic Hierarchy Process (AHP) is a structured technique for dealing with complex decisions"). This requires calculating principal Eigen vectors and Eigen values.

    Since I don't have a mathematics background, I found this great tutorial:
    Analytic Hierarchy Process AHP Tutorial

    The methods used to calculate the principal Eigen vector and Eigen value in that tutorial are only approximations. I wanted to ask if you guys have any idea as to how accurate these approximations are as the matrix size grows beyond 3x3 (i.e. would using the approximation method be sufficient?).

    Any help would be much appreciated.
    I should not worry too much about the accuracy of the numerical method in use. I would worry more about the reasoning and logic behind the AHP (it is virtually non-existent).

    CB
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    Care to elaborate?
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  4. #4
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    Quote Originally Posted by silverback566 View Post
    Care to elaborate?
    What is the optimality condition associated with the eigen vector?

    (there is a substantial literature critical of the AHP for other reasons as well)

    CB
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  5. #5
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    Quote Originally Posted by CaptainBlack View Post
    What is the optimality condition associated with the eigen vector?

    (there is a substantial literature critical of the AHP for other reasons as well)

    CB
    I'm afraid I can't get deep into the mathematics of AHP, but according to wikipedia:

    The AHP is now included in most operations research and management science textbooks, and is taught in numerous universities; it is used extensively in organizations that have carefully investigated its theoretical underpinnings.[3] While the general consensus is that it is both technically valid and practically useful, the method does have its critics.


    I realise its not the best source, but this is sufficient for me. In any case, my assignment was pretty much to use AHP, so I have little choice in the matter.

    Getting back to my original question, can you perhaps tell me if the estimate is accurate enough?

    Thanks
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  6. #6
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    Quote Originally Posted by silverback566 View Post
    I'm afraid I can't get deep into the mathematics of AHP, but according to wikipedia:



    [/U]I realise its not the best source, but this is sufficient for me. In any case, my assignment was pretty much to use AHP, so I have little choice in the matter.

    Getting back to my original question, can you perhaps tell me if the estimate is accurate enough?

    Thanks
    It seems to be favoured by modern proponents of the technique and so as you are not concerned about the validity of the process I can see no reason for not using it.

    It is about par for the AHP since there are perfectly valid numerical methods/libraries available that will find the principle eigen vector limited only by machine precision. Obviously the level of ambiguity in the AHP is great enough to not have to worry about the accuracy of the method of extracting the e-vector and value.

    I don't recall how we used to extract the principle eigen vector but I do recall that the errors from the pairwise comparisons could be relativity large and so presumably the favoured approximation is good enough (the number of options in a single stage/level of the process should never be very large anyway)

    CB
    Last edited by CaptainBlack; April 5th 2010 at 02:56 AM.
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