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Number of Arithmetic Operations to solve (x^T)(A^-1)x

  1. Feb 21, 2009 #1
    1. The problem statement, all variables and given/known data
    Let A be a nxn real symmetric positive definite matrix and x not equal to 0 a real nx1 vector. Show how to computre xTA-1x in n3/3 + O(n2) arithmetic operations.


    2. Relevant equations



    3. The attempt at a solution
    Some things I think I do know:
    If A is real spd, so is A-1.
    Therefore, A-1=LLT where L is a lower triangular matrix with positive diagonal elements.
    I also know in class it was stated that solving Ax = b by Cholesky Factorization requires n3/3 + O(n2) arithmetic operations.
    Also, I might be wrong, but I think we are looking for there to be (n-1)2 + O(n) operations, then you would say that is equivalent to [tex]\sum[/tex]j=1n-1j2 and then [tex]\sum[/tex]j=1kj2=k(k+1)(2k+1)/6 = 2k3/6 + O(n2) = k3/3 + O(k2).
    I'm not sure how to piece these all together or if I even need this information, and I'm missing something else.

    Does anyone have any ideas? Thank you so much.
     
    Last edited: Feb 21, 2009
  2. jcsd
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