# Thread: Error-bound on Regula Falsi

1. ## Error-bound on Regula Falsi

Hi, in the lack of a Numerical Analysis sub forum, I will post this question here as it is quite "calculus-ish".

Does there exist a error-bound on Regula Falsi (Method of False Position) similar to the one that the Bisection method has? I would like to have an expression for the number of iterations necessary as a function of some prespecified error.

Thanks.

2. Originally Posted by Mollier
Hi, in the lack of a Numerical Analysis sub forum, I will post this question here as it is quite "calculus-ish".

Does there exist a error-bound on Regula Falsi (Method of False Position) similar to the one that the Bisection method has? I would like to have an expression for the number of iterations necessary as a function of some prespecified error.

Thanks.
Probably not, or if it does it is not very widely reported.

CB

3. Say that we have a function $f$ and we know that a root $r$ exists on $[a_1,b_1]$.
If we use the method of false position, our first approximation to this root will be,

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

The next one is,
$
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
$|r-c_n|\geq\frac{c_1}{2^n}$ such that

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

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

4. Originally Posted by Mollier
Say that we have a function $f$ and we know that a root $r$ exists on $[a_1,b_1]$.
If we use the method of false position, our first approximation to this root will be,

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

The next one is,
$
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
$|r-c_n|\geq\frac{c_1}{2^n}$ such that

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

where $\epsilon_{step}=|r-c_n|$ ?
Which depends of the function in question, unlike the bounds for bisection.

It would be interesting to see the error at the n-th step say in terms of bounds on the derivatives etc

(I don't think this impossible, just not worth the candle given false-positions other limitations and the nice properties of bisection - which does have its own problems when you have a double root)

CB

5. You mean we could perhaps bound the secant line from point $(a,f(a)) to (b,f(b))$?
I do not see what sort of bound we could use to do this though, other than

$tan^{-1}\left(\frac{f(b)-f(a)}{b-a}\right)\leq 90deg.$