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I have doubt in derivation. Pls clarify

Let points be $(a, f(a)), (c,0), (b,f(b))$

$y-f(a) = m (x-a) = \frac{f(b)-f(a)}{b-a}(x-a)$

Substituting y=0, x=c,

$0-f(a) = \frac{f(b)-f(a)}{b-a}(c-a) \implies \fbox {c=a-f(a) ( $\frac{b-a}{f(b)-f(a)}$)}$ ---->(1)

But if i use geometry in uploaded figure

$c = a+ f(a) cot \theta$

where $\cot \theta = (1/slope ) = (1/tan \theta) = \frac{(f(b)-f(a))}{(b-a)}$

here $\theta = \angle ACB =$ slope of line AC = $ \tan^{-1}\frac{(f(b)-f(a))}{(b-a)}$

this means $c=a+f(a)/\tan \theta = \fbox {$a+f(a) \frac{b-a}{f(b)-f(a)}$}$ ---> (2)

If we compare (1) and (2) equations, i am getting sign change i..e, a+() and a-()

But in textbook regular falsi method final equation is given by (1) and not (2).

Pls point me to where i went wrong.enter image description here

Magneto
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1 Answers1

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You did not consider that $f(a)$ is negative. Thus your formula should be $$ c=a+(-f(a))\cot θ $$ which gives back the first variant of the midpoint formula.

This is not only the regula falsi formula, this secant root formula is also used in the secant method, Dekker's, Brent's method etc.

Lutz Lehmann
  • 126,666
  • Even in equation (1) also i did not take f(a) negative. – Magneto Jun 10 '18 at 14:16
  • Equation 1) was derived analytically in Cartesian coordinates, the formulas are independent of the signs of the values. However in the geometrical argument you need to consider the sign if you derive a result from distances and their proportions. What you want is $c=a+|f(a)|\cotθ$ and $|f(a)|=-f(a)$ here. – Lutz Lehmann Jun 10 '18 at 15:29
  • I understood that $c = a+|f(a)| \cot \theta$. But i am just not able to get why we need to consider -f(a).

    Geometry is independent of coordinate axis. Even if i remvoe coordinate axis it is magnitude that we need to consider right... Kindly enlighten me ...

     – Magneto Jun 10 '18 at 18:27
  • Computation with angles also requires to consider orientation, which can be confusing sometimes. But anyway, $|x|=-x$ for $x<0$. – Lutz Lehmann Jun 10 '18 at 19:25