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?