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Solving a set of nonlinear quadratic equations

  1. Jun 3, 2018 #1
    I would like to solve this system, which is a sets of non linear quadratic equations, the system needed to be solved can be expressed in general as follow:

    ϒϒ'C – ϒα = B

    Where ϒ=(ϒ1,ϒ2,....ϒn)’ is a column vector and ϒ’ its transpose

    C=(c1,c2,…,cn)’ and B=(b1,b2,…bn)’ are a columns vector

    And α is a reel scalar

    I would like to solve for ϒ, with approximatively about 30

    Can someone propose me an algorihm/method to solve this system.
    also a code to do it wil be very useful.
    Bests
     
    Last edited: Jun 3, 2018
  2. jcsd
  3. Jun 3, 2018 #2

    Mark44

    Staff: Mentor

    Y' Y is a scalar, ##\vec Y \cdot \vec Y = |\vec Y|^2##. If you know the value of ##\vec Y \cdot \vec Y## (but don't know the components of ##\vec Y##), you can rewrite the equation above as a system of equations:
    ##\alpha Y_1 = \vec Y \cdot \vec Y - b_1##
    ##\alpha Y_2 = \vec Y \cdot \vec Y - b_2##
    .
    .
    .
    ##\alpha Y_n = \vec Y \cdot \vec Y - b_n##

    Divide both sides by ##\alpha## to get
    ##Y_1 = 1/\alpha (\vec Y \cdot \vec Y - b_1)##
    ##Y_2 = 1/\alpha (\vec Y \cdot \vec Y - b_2)##
    .
    .
    .
    ##Y_n = 1/\alpha (\vec Y \cdot \vec Y - b_n)##
     
  4. Jun 3, 2018 #3
  5. Jun 3, 2018 #4

    StoneTemplePython

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    Science Advisor
    Gold Member

    so we're dealing with real scalars here.
    - - - -

    So your 'equation' is:

    ϒϒ'C – ϒα = B
    or

    ϒϒ'C = ϒα + B

    so you have a real symmetric rank one matrix on the Left Hand side (LHS).

    The issue is that every possible c you can choose on the LHS gets mapped to zero or is an eigenvector (i.e. ϒ) or a linear combination of the two aforementioned things. So lets hope that B is either a scaled version of ϒ or else the zero vector. If your B is the zero vector, it should be pretty easy. Otherwise you have problems.

    More issues: For starters, why write ϒα + B on the Right hand side.... why not just write
    ##\propto ϒ##
     
  6. Jun 3, 2018 #5
    B is nonzero column, there is a way to solve that?
     
  7. Jun 3, 2018 #6

    StoneTemplePython

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    Science Advisor
    Gold Member

    then ##B \propto ϒ## or this is not an equation
     
  8. Jun 3, 2018 #7
    B is a constant.
     
  9. Jun 3, 2018 #8

    StoneTemplePython

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    Science Advisor
    Gold Member

    I don't know what this means. Your original post, and a quick dimensional check say B is a a column vector.

    What I am trying to tell you is your original post is analogous to

    ## 2 = 3##

    or

    ## 2 = 3 +x##
    for real ## x \geq 0##

    this is not an equation. It is just wrong.
     
  10. Jun 3, 2018 #9
    Yes i mean B is a constant column vector. Do you think that is wrong to?
     
  11. Jun 3, 2018 #10

    StoneTemplePython

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    Science Advisor
    Gold Member

    You're not hearing me. It is one of the 3 options

    option a)
    ## B = \mathbf 0##

    option b)
    ##B \propto ϒ##

    option c)
    this is not an equation. It is just wrong.
    - - - - -
    I have nothing more to say on the matter. Good luck.
     
  12. Jun 3, 2018 #11
    Ok, please look at eq (2 19) page 8 on this link, this paper, maybe some thing wrong.
    Portfolio Theory: Origins, Markowitz and CAPM Based Selection - Springer
    PDFhttps://www.springer.com › document
     
  13. Jun 4, 2018 #12

    Mark44

    Staff: Mentor

    Please provide the actual link to the document. The link you show is just to the Springer site.
     
  14. Jun 4, 2018 #13
    Please find enclosed the document. Go to page 7 to see the original problem, the resolution of lagrangian (which may be wrong) lead to equation posted which is (2 19)
     

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