interval halving converges linearly
Theorem 1.
The interval halving algorithm converges linearly.
Proof.
To see that interval halving (or bisection) converges linearly we use the alternative definition of linear convergence that says that for some constant .
In the case of interval halving, is the length of the interval we should search for the solution in and has as its midpoint
. We have then that this interval has half the length of the previous interval which means, . Thus and we have exact linear convergence.∎