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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.

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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.

Apply the Newton’s method with x0 = 0.8 to the equation f(x) = x^3 - x^2 - x + 1 = 0, and verify that the convergence is only of first-order. Further show that root α = 1 has multiplicity 2 and then apply the modified Newton’s method with m = 2 and verify that the convergence is of second-order.

Learn how to apply Newton’s method with x0 = 0.8 to solve the equation f(x) = x^3 - x^2 - x + 1 = 0. Verify first-order convergence and explore the modified Newton’s method for a root with multiplicity 2, confirming second-order convergence. Suitable for university students studying numerical methods.

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