Thread: 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, [LaTeX ERROR: Convert failed] and [LaTeX ERROR: Convert failed] , are sufficiently close to the solution, [LaTeX ERROR: Convert failed] , then

[LaTeX ERROR: Convert failed]

for [LaTeX ERROR: Convert failed] where [LaTeX ERROR: Convert failed] is the solution after [LaTeX ERROR: Convert failed] solutions. This essentially says that the error bound between the actual solution and the '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

[LaTeX ERROR: Convert failed]

, which means that the error bound between the actual solution and the k'th iterate is less than the distance between the 'th iterate and the the '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 [LaTeX ERROR: Convert failed] 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?