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Thread: Secant Method - Error bound between kth iteration and root

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    Secant Method - Error bound between kth iteration and root

    Hey guys.

    I know from the proof of the secant method that given the two initial values, $\displaystyle x_{0}$ and $\displaystyle x_{1}$, are sufficiently close to the solution, $\displaystyle x_{\ast}$, then

    $\displaystyle |x_{k+1} - x_{\ast}| < \frac{1}{2}|x_{k} - x_{\ast}|$

    for $\displaystyle k>0$ where $\displaystyle x_{k+1}$ is the solution after $\displaystyle k$ solutions. This essentially says that the error bound between the actual solution and the $\displaystyle k$'th iterate is less than half the distance between the iterate before that and the actualy solution.

    How would i show using this fact (and possible other results) that

    $\displaystyle |x_{k+1} - x_{\ast}| < |x_{k+1} - x_{k}|$

    , which means that the error bound between the actual solution and the k'th iterate is less than the distance between the $\displaystyle k$'th iterate and the the $\displaystyle k-1$'th iterate.

    I'm very stumped on this question so if anyone has an idea i'd be extremely grateful

    Edit:
    I've been told that the Contraction Mapping Method could help me show this. As a reminder,



    It turns out that the error bound for this method is given by

    However, just like the error bound for the secant method that i gave in my question, this is not computable as the value of $\displaystyle x_{\ast}$ is unknown.
    Now for the Contraction Mapping Method, the way they have manipulated the error bound is as shown,



    I can see the similarities between this and what i am trying to show with the Secant Method, but i don't understand exactly what they have done here, especially concerning the part under 'Now' on the above image.
    Can someone explain this to me and how i could use in my original problem?
    Last edited by nugiboy; Dec 2nd 2010 at 04:02 AM.
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