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Multiplication bloards after factorization

  1. Nov 18, 2013 #1
    Let a positive definite matrix A be factorized to P and Q, A=P*Q and let an arbitrary matrix B.
    I am calculating the relative error of the factorization through the norm:

    [itex]\epsilon = \left\| \textbf{A}-\textbf{PQ} \right\| / \left\| \textbf{A} \right\|[/itex]

    which gives

    [itex]\epsilon <1\text{e}-16[/itex]

    so I assume factorization is correct.

    But things go messy when I try to multiply the factorized form of A with B.
    In particular, the relative error, r, of the product

    [itex]r = \left\| \textbf{AB}-\textbf{PQB} \right\| / \left\| \textbf{AB} \right\|[/itex]

    now bloats, i.e. I get
    [itex]r>0.1.[/itex]

    Note that B is arbitrary, in particular I have tried several different types: random, structured, all-ones matrix, even the identity matrix.
    I'm confused. How come factorization is correct and then the multiplication bloats?
    Has anything to do with condition number?

    (Unfortunately I can't disclose the type of factorization but I can tell that P and Q are not triangular)
     
  2. jcsd
  3. Nov 19, 2013 #2
    Modification: I mistakenly added also "identity matrix" to previous post. Please ignore this from the list of matrices I have tried.
     
  4. Nov 19, 2013 #3

    AlephZero

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    Science Advisor
    Homework Helper

    It's hard to know exactly what you are doing since you won't tell us all the facts, but I think the basic issue is the difference between "forward" and "backward" error analysis.

    If you are trying to solve ##Ax = b## numerically, you can estimate the error two different ways.

    Forward error analysis: try to estimate the error in ##x##, i.e assume the exact solution is ##A(x + e) = b ## and find an estimate for the vector ##e##.
    Backward error analysis: consider you have the exact solution to the "wrong" equation, i.e. estimate the size of a matrix ##e## such that ##(A+e)x = b##.

    Backward error analysis (first proposed by Wilkinson in the 1960s) is generally more useful than the more "obvious" forward analysis.
     
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