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Solving equation numerical using lipschitz

  1. Jan 23, 2007 #1
    1. The problem statement, all variables and given/known data
    This problem is about solving an equation system numerical using lipschitz method or whatever the name is.

    x_1 = sqrt(1-x^2)
    x_2 = sqrt((9-5x_1^2)/21)

    and we create
    (x_1,x_2) = x
    g(x) = (x_1,x_2) (fixed point equation)

    lipschitz states:
    ||x^(k+1) - x^*|| <= L*||x^k - x^*||

    where x^* is the fix-point, which means x^* = g(x^*)

    (k is not an exponent, it's indexation)
    2. Relevant equations

    Se above

    3. The attempt at a solution

    The problem now is to really get started and doing the iterations, and thats where I'm kind of stuck. I suppose I'll have to start with a guess on x_1 and x_2, so lets.

    (x_1)^(k =0) = 1
    (x_2)^(k =0) = 1

    I try using the lipschitz for x_1
    ||x_1^(k =2) - sqrt(1-(x_2)^2) || <= L* ||1 - sqrt(1-(x_2)^2) ||
    But this just looks like crap to me. From this I want to solve x_1^(k = 2), but I dont see how that's done. And I'm not even sure this is the right setup with lipschitz. I dont see how the iteration is supposed to be done, and all I know about L is that it's between 0 and 1, what value should L have?. Could someone plz show me the first the of the iteration so I see how it works?
  2. jcsd
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