Rate of convergence of Bisection and false position method

Question

Rate of convergence for both Bisection and false position method is linear(one) but when we solve nonlinear equation $f(x)=0$ with both methods we see that false position method is converges rapidly than Bisection method although both methods have same rate of convergence.what is the reason behind this fact?

Answer 1

Indeed, both method are linear with the error satisfying $\epsilon_{n+1} = C \epsilon_n$.

For the Bisection method, $C$ is roughly equal to $1/2$ while for the Regula-Falsi method $C = \frac{1}{2}\frac{f^{\prime\prime}(\xi)}{f^{\prime}(\xi)}$ for a twice differentiable map $f$, where $\xi$ is the root to be found.

So when you say that the false position method converge faster than the bisection method, this is not true in general. It depends on the position of the two initial points and on the value of $\frac{f^{\prime\prime}(\xi)}{f^{\prime}(\xi)}$. In particular if $\vert f^\prime(\xi) \vert$ is small, the convergence of the false position method can be slow.