Fixed Point Iteration Convergence of $x^3-x+e^x=0$

Question

I have been trying to solve this problem for a while now, but I am going nowhere with it. The problem states:

Express $x^3 − x + e^x = 0$ as a fixed point problem in two different ways so that its fixed point iteration is locally convergent.

I set $g(x)=x$ and got $g(x)=x^3-2x+e^x$. So $x=-1.468$. However, when I implement the fixed point iteration algorithm on this $g(x)$, it doesn't converge. Can someone please point me in the right direction?

Answer 1

You are allowed to be more free with the transformations of the equation. So other possible fixed-point forms are $$ x=x^3+e^x\\ x=-\sqrt[3]{e^x-x}\\ x=\ln(x-x^3) $$ One would have to examine where these expressions are defined and find at least one interval that gets mapped into itself. Then in addition check for contractivity.

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Additionally, the Newton method for any formulation of the equation as a function is also a fixed-point iteration, the convergence analysis might be a bit more difficult due to the structure of the expression. Functions are trivially $f(x)=e^x+x^3-x$, or $f(x)=e^{-x}(x^3-x)+1$, ... each giving a different iteration.

Answer 2

One easy way is Newton's method: Look for a fixed point of $x \mapsto x-{e^x+x^3-x \over e^x +3x^2 -1}$.

Another way, look for a fixed point of $x \mapsto x - {e^x+x^3-x \over 100}$.