# C.3 The false position (regula falsi) method

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Let $$f(x)$$ be a continuous function and let $$a_1\lt b_1$$ with $$f(a_1)$$ and $$f(b_1)$$ being of opposite sign.

As we have seen, the bisection method generates a sequence of intervals $$I_n=[a_n,b_n]\text{,}$$ $$n=1,2,3,\cdots$$ with, for each $$n\text{,}$$ $$f(a_n)$$ and $$f(b_n)$$ having opposite sign (so that, by continuity, $$f$$ has a root in $$I_n$$). Once we have $$I_n\text{,}$$ we choose $$I_{n+1}$$ based on the sign of $$f$$ at the midpoint, $$\frac{a_n+b_n}{2}\text{,}$$ of $$I_n\text{.}$$ Since we always test the midpoint, the possible error decreases by a factor of 2 each step.

The false position method tries to make the whole procedure more efficient by testing the sign of $$f$$ at a point that is closer to the end of $$I_n$$ where the magnitude of $$f$$ is smaller. To be precise, we approximate $$y=f(x)$$ by the equation of the straight line through $$\big(a_n,f(a_n)\big)$$ and $$\big(b_n,f(b_n)\big)\text{.}$$

The equation of that straight line is

$y = F(x) = f(a_n) + \frac{f(b_n)-f(a_n)}{b_n-a_n}(x-a_n) \nonumber$

Then the false position method tests the sign of $$f(x)$$ at the value of $$x$$ where $$F(x)=0\text{.}$$

\begin{align*} & F(x) = f(a_n) + \frac{f(b_n)-f(a_n)}{b_n-a_n}(x-a_n) =0 \\ & \iff x= a_n - \frac{b_n-a_n}{f(b_n)-f(a_n)} f(a_n) = \frac{a_n f(b_n) - b_n f(a_n) }{f(b_n)-f(a_n)} \end{align*}

So once we have the interval $$I_n\text{,}$$ the false position method generates the interval $$I_{n+1}$$ by the following rule.1

##### Equation C.3.1 fale position method.

Set $$c_n=\frac{a_n f(b_n) - b_n f(a_n) }{f(b_n)-f(a_n)}\text{.}$$ If $$f(c_n)$$ has the same sign as $$f(a_n)\text{,}$$ then

$I_{n+1}=[a_{n+1},b_{n+1}]\quad\text{with}\quad a_{n+1}=c_n,\ b_{n+1}=b_n \nonumber$

and if $$f(c_n)$$ and $$f(a_n)$$ have opposite signs, then

$I_{n+1}=[a_{n+1},b_{n+1}]\quad\text{with}\quad a_{n+1}=a_n,\ b_{n+1}=c_n \nonumber$

This page titled C.3 The false position (regula falsi) method is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Joel Feldman, Andrew Rechnitzer and Elyse Yeager via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.