Determine-enumerate fixed point iteration functions?

Questions
,
1

Mathematica has some excellent tools for fixed point iteration.

What I am curious about is if there is a way to have Mathematica enumerate the choice(s) for the iteration function?

For example, if we have f(x) = x^3+4x^2-10, we want various test functions x = g(x), and we can calculate various ones from f(x) as:

  • g_1(x) = x - x^3 - 4 x^2 + 10
  • g_2(x) = \sqrt{\dfrac{10}{x} - 4 x}
  • g_3(x) = \dfrac{1}{2} \sqrt{10 - x^3}
  • g_4(x) = \sqrt{\dfrac{10}{4 + x}}
  • g_5(x) = x - \dfrac{x^3 + 4 x^2 - 10}{3 x^2 + 8 x}

It is possible there are more, we are just creating permutations of f(x) by solving x = g_i(x).

Is there some way to coax Mathematica to calculate all these variations?

Regards

2

You get a few this way: replace x with y in the equation, but not everywhere, just in one occurrence, then solve for y:

eq = 0 == x^3 + 4 x^2 - 10;
pos = Position[eq, x];
Flatten@Map[Solve[ReplacePart[eq, # -> y], y] &, pos] /. y -> x /. 
 Rule -> Equal
3

@Gianluca Gorni Interesting idea, I will play around with it some more. Regards!