Newton's Method

[Vader07]

Hey guys,


I have an issue with the following problem

Apply Newtons method to find a root of
x^5 - cos(Pi(x) in the interval. If your intial guess is x0 = 1/2 then x1= ?

The main thing is what do I do with the cos (Pi(x)?

is it - sin x first or something different?


Also another problem I have
is this:

Derive a newton transform, F, whose iterates tend to converge to solutions of the equation

x^4 + 3 = 2/x

Then F(1)

is this how I am supposed to start?


4x^3 +0 = 2x^-1/2

I hope I can some feedback from this, I just need a slight start or hint to get this problem started

Any help would be greatly appreciated? Thanks in advance.

 

[BigGlenntheHeavy]

\(\displaystyle f(x) \ = \ x^{5}-cos(\pi x), \ x_0 \ = \ 1/2, \ find \ x_1\)

\(\displaystyle f \ ' \ (x) \ = \ 5x^{4}+\pi sin(\pi x)\)

\(\displaystyle Newton's \ Method: \ x_{n+1} \ = \ x_n-\frac{f(x_n)}{f'(x_n)}\)

\(\displaystyle x_{0+1} \ = \ x_0-\frac{f(x_0)}{f'(x_0)} \ = \ 1/2-\frac{f(1/2)}{f'(1/2)}\)

\(\displaystyle Hence, \ x_1 \ = \ 1/2-\frac{(1/2)^{5}-cos(\pi/2)}{5(1/2)^4+\pi sin(\pi/2)}\)

\(\displaystyle x_1 \ = \ 1/2-\frac{1/32}{(5/16)+\pi} \ = \ 1/2-\frac{1}{2(5+16\pi)} \ = \ \frac{2+8\pi}{5+16\pi} \ = \ .490952761511\)

\(\displaystyle Actually \ f(x) \ = \ x^{5}-cos(\pi x) \ = \ 0, \ x \ = \ .490922357935,\)

\(\displaystyle therefore \ with \ only \ one \ iteration, \ we \ are \ off\ only \ .000030403576, \ thank \ you \ Sir \ Isaac.\)