The problem is asking you to apply Newton's method to the equation $f(x) = x^3 - x^2 - x + 1 = 0$ and verify the convergence order, initially of first order, and then use the modified Newton's method for second-order convergence. Here's how we can approach this:

Step 1: Applying Newton's Method with $x_0 = 0.8$

Newton's method is defined by the iterative formula:

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

For the function $f(x) = x^3 - x^2 - x + 1$, the derivative is:

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

Now, applying the first few iterations of Newton's method with $x_0 = 0.8$, we will observe the rate of convergence.

Step 2: First-Order Convergence

First-order convergence implies that the error $|x_n - \alpha|$ decreases linearly. We will check if Newton's method converges with first-order to the root $\alpha = 1$, which is verified by calculating the error after each iteration.

Step 3: Root $\alpha = 1$ with Multiplicity 2

A root $\alpha = 1$ has multiplicity 2, meaning the function can be factored as $f(x) = (x - 1)^2 \cdot g(x)$. This changes the convergence behavior because the standard Newton’s method converges more slowly (only linearly) for multiple roots. To achieve quadratic convergence, a modified Newton’s method is used.

Step 4: Modified Newton's Method

The modified Newton's method for multiplicity $m$ is given by:

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

In this case, the multiplicity $m = 2$. Using this modified method, we expect to see second-order convergence, where the error decreases quadratically.

Step 5: Verifying Second-Order Convergence

Second-order convergence implies that the error $|x_n - \alpha|$ decreases quadratically, meaning $|x_{n+1} - \alpha| \propto |x_n - \alpha|^2$. After applying the modified Newton’s method, we will check the error behavior to confirm second-order convergence.

Would you like to see detailed steps of the iterations for both methods?

Here are 5 related questions:

1. How is the derivative of $f(x)$ used in Newton's method?
2. What is the significance of the multiplicity of a root in Newton's method?
3. How do you derive the modified Newton’s method formula for a root with multiplicity?
4. Can you explain the difference between first-order and second-order convergence?
5. What are other numerical methods for solving non-linear equations besides Newton's method?

Tip: In Newton’s method, if the convergence is slow, it's often a sign that the root has higher multiplicity.