Convergence of Bisection Method

[Adyssa]

Homework Statement

Show that the Bisection Method converges linearly with K = 1/2

Homework Equations

Note that x(sub n) converges to the exact root r with an order of convergence p if:

lim(n->oo) (|r - x(n + 1)|) / (|r - x(n)|^p) = lim(n->oo) (|e(n + 1)|) / (|e(n)|^p) = K

The Attempt at a Solution

EDIT: I can rearrange the equation above as follows (K = 1/2):

|e(n+1)| = 1/2|e(n)|^p

and I know that the bisection method is a linear operation, reducing the interval by 1/2 each time and converging on the real root, so p = 1, but I'm not sure how to show this?

 

[Nino MMP]

EDIT: I think I've solved it.By rearranging the equation as above, we can show that p = 1 by taking the log of both sides:log(|e(n+1)|) = log(1/2|e(n)|^p)log(|e(n+1)|) = log(1/2) + plog(|e(n)|)dividing both sides by log(|e(n)|):log(|e(n+1)|)/log(|e(n)|) = log(1/2)/log(|e(n)|) + pSince the left side is always equal to 1, this means that p = 1 and therefore K = 1/2.