# False Position Method

Mathematics Asked by Matheus Barreto Alves on August 15, 2020

Is it possible to use False position method to find the root near $$pi/2$$ for this function?

$$f(x)=frac12+frac{x^2}4-xsin(x)-frac{cos(2x)}2$$

As you can see, she is positive so we will never get $$f(a) times f(b)<0$$ It is possible to use a modification of false position for finding minimums of functions to find the root (if it is indeed one).

The idea is that three points $$x_mathrm L are given so that $$f(x_mathrm M).

We then generate a new point $$x_mathrm N$$ by applying false position to the points $$(x_mathrm L,pm f(x_mathrm L))$$ and $$(x_mathrm R,mp f(x_mathrm R))$$, which will effectively flip the sign of the function on one side of the root.

We then update $$(x_mathrm L,x_mathrm M,x_mathrm R)$$ so that $$f(x_mathrm M)$$ is minimal (between $$f(x_mathrm M)$$ and $$f(x_mathrm N)$$) and $$x_mathrm L$$ and $$x_mathrm R$$ are the nearest points found on the left and right sides of $$x_mathrm M$$.

See this graph for a visualization of the process. (There is a slight mistake in the conditions for updating, but the first 3 iterations are still correct.)

See here for code.

Answered by Simply Beautiful Art on August 15, 2020

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