C.3 The false position (regula falsi) method
- Page ID
- 89670
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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
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
\[ I_{n+1}=[a_{n+1},b_{n+1}]\quad\text{with}\quad a_{n+1}=a_n,\ b_{n+1}=c_n \nonumber \]