Math Help - Fixed Point Iteration

1. Fixed Point Iteration

I'm having problems finding a good way to simplify F(x)=0 in the form of x=
Its there a Tip or trick to find it faster. I'm posting some examples

$F(x)=x^3-1.33x^2+x-1.33$
Not converges $x= -x^3 +1.33x^2 +1.33$
it converges $x=(1.33*(x^2+1)-x)^1/3$

what about this one $g(x) = e^3 - tan(x)$
How can I Rewrite g(x)=0 as another fixed-point equation, that actually let find a suitiable fixed point.

Cheers,
Alexis

2. Originally Posted by aleba008
I'm having problems finding a good way to simplify F(x)=0 in the form of x=
Its there a Tip or trick to find it faster. I'm posting some examples

$F(x)=x^3-1.33x^2+x-1.33$
Not converges $x= -x^3 +1.33x^2 +1.33$
it converges $x=(1.33*(x^2+1)-x)^1/3$

what about this one $g(x) = e^3 - tan(x)$
How can I Rewrite g(x)=0 as another fixed-point equation, that actually let find a suitable fixed point.
You could write g(x)=0 as $\sin x = e^3\cos x$ or $\cos x = e^{-3}\sin x$, leading to $x = \arcsin(e^3\cos x)$ or $x= \arccos(e^{-3}\sin x)$. The first of those does not look promising, but the second one converges very rapidly from the starting point $x_0 = \pi/2$.