Error-bound on Regula Falsi

**Mollier** · September 5th 2010, 11:05 PM

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.

**CaptainBlack** · September 6th 2010, 12:07 AM

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

**Mollier** · September 6th 2010, 02:41 AM

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,
$<br /> c_2 = b_1 - \frac{f(b_1)}{\frac{f(b_1)-f(c_1)}{b_1-c_1}}$

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

**CaptainBlack** · September 6th 2010, 02:48 AM

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,
$<br /> c_2 = b_1 - \frac{f(b_1)}{\frac{f(b_1)-f(c_1)}{b_1-c_1}}$

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

**Mollier** · September 6th 2010, 03:42 AM

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

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

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