The Newton-Raphson method is an iterative numerical technique used to find the roots of a real-valued function. The formula for the Newton-Raphson iteration is:

$x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$

Given the equation $f(x) = x^3 - x^2 - x - 1$, we first need to determine its derivative:

$f'(x) = 3x^2 - 2x - 1$

Let's start by selecting an initial guess, $x_0$, within the interval $1.6 < x < 2.1$. For this example, we'll choose $x_0 = 2.0$.

Now, we will apply the Newton-Raphson iteration formula.

1. First Iteration: $x_1 = x_0 - \frac{f(x_0)}{f'(x_0)}$ $x_1 = 2.0 - \frac{2^3 - 2^2 - 2 - 1}{3(2)^2 - 2(2) - 1}$ $x_1 = 2.0 - \frac{8 - 4 - 2 - 1}{12 - 4 - 1}$ $x_1 = 2.0 - \frac{1}{7}$ $x_1 = 2.0 - 0.142857$ $x_1 \approx 1.857143$

2. Second Iteration: $x_2 = x_1 - \frac{f(x_1)}{f'(x_1)}$ $f(1.857143) = (1.857143)^3 - (1.857143)^2 - 1.857143 - 1 \approx 0.482$ $f'(1.857143) = 3(1.857143)^2 - 2(1.857143) - 1 \approx 5.143$ $x_2 = 1.857143 - \frac{0.482}{5.143}$ $x_2 \approx 1.763157$

3. Third Iteration: $x_3 = x_2 - \frac{f(x_2)}{f'(x_2)}$ $f(1.763157) = (1.763157)^3 - (1.763157)^2 - 1.763157 - 1 \approx 0.045$ $f'(1.763157) = 3(1.763157)^2 - 2(1.763157) - 1 \approx 4.088$ $x_3 = 1.763157 - \frac{0.045}{4.088}$ $x_3 \approx 1.752$

4. Fourth Iteration: $x_4 = x_3 - \frac{f(x_3)}{f'(x_3)}$ $f(1.752) = (1.752)^3 - (1.752)^2 - 1.752 - 1 \approx 0.002$ $f'(1.752) = 3(1.752)^2 - 2(1.752) - 1 \approx 4.008$ $x_4 = 1.752 - \frac{0.002}{4.008}$ $x_4 \approx 1.7514$

5. Fifth Iteration: $x_5 = x_4 - \frac{f(x_4)}{f'(x_4)}$ $f(1.7514) = (1.7514)^3 - (1.7514)^2 - 1.7514 - 1 \approx 0$ $f'(1.7514) = 3(1.7514)^2 - 2(1.7514) - 1 \approx 4.004$ $x_5 = 1.7514 - \frac{0}{4.004}$ $x_5 \approx 1.7514$

After five iterations, the root of the equation $x^3 - x^2 - x - 1 = 0$ is approximately $x \approx 1.7514$.

Would you like further details or have any questions?

Further Questions:

1. How does the Newton-Raphson method compare to other root-finding methods like the bisection method?
2. Can you explain the convergence criteria for the Newton-Raphson method?
3. What happens if the initial guess is far from the actual root?
4. Can the Newton-Raphson method fail? If so, under what conditions?
5. How can we improve the accuracy of the Newton-Raphson method?
6. How is the derivative calculated if the function is not given in a simple algebraic form?
7. Can the Newton-Raphson method be used for complex roots? If so, how?
8. How does the choice of the initial guess affect the efficiency of the Newton-Raphson method?

Tip:

Always check the derivative $f'(x)$ at your initial guess to ensure it is not zero, as this would lead to division by zero and make the method inapplicable at that point.