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Homework Help: Matrix/Error Proof?

  1. Aug 29, 2010 #1
    1. The problem statement, all variables and given/known data

    Not too sure what this proof would be under, but am pretty sure it has to do with approximation errors, can anyone give me a hint as to how to even start this?


    2. Relevant equations



    3. The attempt at a solution

    For the first one, my guess at the solution would be...

    1)take the norm of both sides of x = Tx + b
    2) then substitute in x^k = Tx^(k-1) + b into the x's
    3)since x^k = Tx^(k-1) + b, we get norm( x^k = T(Tx^(k-1) + b) + b), which then we get infinite multiples of T. Hence T^k?

    That's really wild guess at the solution, any help would be much appreciated
     

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  2. jcsd
  3. Aug 29, 2010 #2
    for the first part you might look at something like
    |xk+1 - x| = |(Txk + b) - (Tx + b)| = |T(xk -x)|
    and continue till k->0.


    for the second, maybe try to establish something like
    |xk - x| <= ||T|| |xk-1 - x| <= ||T|| ( |xk-1 - xk| + |xk - x|).

    Then solve for |xk - x| in terms of |xk-1 - xk|.

    And then try to figure out how to go from
    |xk-1 - xk| to |x0 - x1|.
     
  4. Sep 10, 2010 #3

    Dear qbert

    For the 2nd part of the question. Are we meant to figure out how to go from |xk - xk-1| to |x1 - x0| rather than the given above |xk-1 - xk| to |x0 - x1|

    my attempt at the question is as follows

    ||x(k) - x|| <= ||T|| ||x(k-1) - x(k)|| + ||T|| ||x(k) - x||

    ||x(k) - x|| - ||T|| ||x(k) - x|| <= ||T|| ||x(k-1) - x(k)||

    ||x(k) - x|| (1 - ||T||) <= ||T|| ||x(k-1) - x(k)||

    solving for ||x(k) - x|| <= (||T||/(1 - ||T||)) ||x(k-1) - x(k)||

    hence my query above.

    Kind regards,
    kcp
     
  5. Sep 10, 2010 #4

    Dear qbert

    For the 2nd part of the question. Are we meant to figure out how to go from |xk - xk-1| to |x1 - x0| rather than the given above |xk-1 - xk| to |x0 - x1|

    my attempt at the question is as follows

    ||x(k) - x|| <= ||T|| ||x(k-1) - x(k)|| + ||T|| ||x(k) - x||

    ||x(k) - x|| - ||T|| ||x(k) - x|| <= ||T|| ||x(k-1) - x(k)||

    ||x(k) - x|| (1 - ||T||) <= ||T|| ||x(k-1) - x(k)||

    solving for ||x(k) - x|| <= (||T||/(1 - ||T||)) ||x(k-1) - x(k)||

    hence my query above.

    Kind regards,
    kcp
     
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