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False Position Method

TOP/Numerical Analysis/Solution method
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Finds x such that f(x)=0 using the false position method. With f(a) and f(b) of opposite signs, it repeatedly finds the x-axis intersection of the line through the 2 points (a,f(a)) and (b,f(b)).
The false position method generally converges faster than bisection. It can be used to find initial values for fast-converging methods such as Newton’s method and Halley’s method.
f(x)
a
,
bChoose a and b such that
f(a)f(b)≦0
n
Maximum number of iterations
\(\normalsize False\ position\ method\\ (1)\ initial\ value a_0,\ b_0\hspace{20px}f(a_0)f(b_0)\le0\\ (2)\ x_n={\large\frac{a_nf(b_n)-b_nf(a_n)}{f(b_n)-f(a_n)}}\\ \hspace{25px}f(x_n)\le\varepsilon\ \Rightarrow\ answer=x_n\\ \hspace{25px} a_{n+1}=x_n,\ b_{n+1}=b_n\hspace{20px}f(a_n)f(x_n)\ge0\\ \hspace{25px} a_{n+1}=a_n,\ b_{n+1}=x_n\hspace{20px}f(b_n)f(x_n)\ge0\\ \)

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