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**Mollier** Say that we have a function $\displaystyle f$ and we know that a root $\displaystyle r$ exists on $\displaystyle [a_1,b_1]$.

If we use the method of false position, our first approximation to this root will be,

$\displaystyle c_1 = b_1 - \frac{f(b_1)}{\frac{f(b_1)-f(a_1)}{b_1-a_1}}.$

The next one is,

$\displaystyle

c_2 = b_1 - \frac{f(b_1)}{\frac{f(b_1)-f(c_1)}{b_1-c_1}}$

and so on..

What if I use a similar bound to the one in the bisection method, and say that

$\displaystyle |r-c_n|\geq\frac{c_1}{2^n}$ such that

$\displaystyle n \geq log_2(\frac{c_1}{\epsilon_{step}})$

where $\displaystyle \epsilon_{step}=|r-c_n|$ ?