Nest not giving desired results

Question

I am trying to use Nest to find result using fixed point iteration method, but when I find $g(x)$ and put it in Nest, it doesn't gives me the desired result.
expr = x^3 + x^2 - 1
gx = generateG[expr]

result = Nest[(gx &, 0.5, 10]
(* value of result after 10 iterations *)

1 - x^2)^(1/3)

What I want is:
result = Nest[(1 - #^2)^(1/3) &, 0.5, 10]
(* value after 10 iterations *)

0.686642

Bob Hanlon · Answer

expr = x^3 + x^2 - 1;

You have not provided the definition of generateG
The value after 10 iterations is not a good estimate of the fixed point.
(data = NestList[(1 - #^2)^(1/3) &, 0.5`15, 50])[[11]] // N

(* 0.686642 *)

The fixed point is
fp = FixedPoint[(1 - #^2)^(1/3) &, 0.5`15, 
   SameTest -> (Abs[#1 - #2] < 1*^-15 &)] // N

(* 0.754878 *)

ListPlot[data,
 Epilog -> {Red, Line[{{0, fp}, {50, fp}}]}]

The fixed point is equal to the positive root
fp == x /. Solve[{expr == 0, x > 0}, x][[1]]

(* True *)

Nest not giving desired results

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