# Newton's Method, Analytical Formula

Currently I am learning about Netwon's Method. Given the function f: $\frac{1}{5} x^5 - \frac{2}{3}x^3 + x$ and $x^{(0)} = \sqrt{\frac{ 25+2\sqrt{55} }{27} }$, I want to analytically determine the sequence $x^{(n+1)} = \Psi(x^{(n)})$ with $\Psi(x) := x - \frac{f(x)}{f'(x)}$. Can someone show me how this is done in this particular case?

• Interessting starting point, why did you choose that one ? – Dominic Michaelis Mar 5 '13 at 11:00
• butterfly effect. – TestGuest Mar 5 '13 at 11:42

As this is a numerical method, why do you want to do it analytical? If you want the analytical solutions use $$\frac{1}{5} x^5 - \frac{2}{3} x^3 + x= x( \frac{1}{5} x^4 - \frac{2}{3} x^2 +1)$$ so $0$ is a solution and the others you get by using pq formal with $z=x^2$
• the values are something like $0.233$ than $-0.0184$ – Dominic Michaelis Mar 5 '13 at 18:25
• ..so I get $\pm 1.2146085632857550969148471364800765701899398426479169$ – TestGuest Mar 5 '13 at 18:26