Find the roots of the function: f(x) = x^3-x-1; (1,2). Error: 10^-6 fixed point method

1. Define the function and its fixed point form:
f(x) = x^3 - x - 1
Rewrite as:
x = (x^3 - 1) \to g(x) = x^3 - 1

2. Make an initial guess x_0 within the interval (1,2). Let's choose x_0 = 1.5 .

3. Perform iterations using the fixed-point iteration method:
x_{n+1} = g(x_n) until | x_{n+1} - x_n | < 10^{-6} .

4. Iterations:
- x_1 = 1.5^3 - 1 = 2.375 - 1 = 1.375
- x_2 = 1.375^3 - 1 = 2.59765625 - 1 = 1.375
- Continue iterating until the convergence criterion is met.

5. After multiple iterations (usually more than 20 to ensure the required precision):
- x \approx 1.324717

Thus, the root is approximately 1.324717 with the precision of 10^{-6} .