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Math Help - Cubic Convergence

  1. #1
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    Question Cubic Convergence

    Question :
    Consider an iteration function of the form F(x) = x + f(x)g(x), where f(r) = 0 and f'(r) != 0. Find the precise conditions on the function g so that the method of functional iteration will converge cubically to r if started near r.

    Any help with this problem would be greatly appreciated! Thanks!
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by theamazingjenny View Post
    Question :
    Consider an iteration function of the form F(x) = x + f(x)g(x), where f(r) = 0 and f'(r) != 0. Find the precise conditions on the function g so that the method of functional iteration will converge cubically to r if started near r.

    Any help with this problem would be greatly appreciated! Thanks!
    I'm a bit confused here. Function iteration produces a sequence of functions:

    f_{n+1}(x) = f(f_{n}(x))

    for all x in some domain D.

    But as far as I can see you are interested in the iteration:

    x_{n+1} = x_{n} + f(x_{n})g(x_{n})

    RonL
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