# Math Help - Secant Method - Error bound between kth iteration and root

1. ## 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 $k$'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

[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 $k$'th iterate and the the $k-1$'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 [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?