https://r2.erweima.ai/mathbot/image/17189541829662p14rm98.jpg

From the additional image provided, let's dive into the continuation of the explanation:

### Key Points from the Additional Image

1. **Polynomial Selection and Transformation**:
   - The polynomial \( L_3(x) = \frac{1}{2} (5x^3 - 3x) \) is chosen.
   - The change of variable formula \( x = \frac{2\bar{x} - (x_{n+k} + x_n)}{x_{n+k} - x_n} \) is used.

2. **Calculations for Polynomial Values**:
   - \( x = -\frac{1}{3} \) results in \( L_3(x_{n+1}) = \frac{11}{27} \).
   - \( x = \frac{1}{3} \) results in \( L_3(x_{n+2}) = -\frac{11}{27} \).
   - \( x = 1 \) results in \( L_3(x_{n+3}) = 1 \).

3. **Equation Formulation**:
   - Using the properties \( c_0 p_0(x) = 0, c_1 p_1(x) = 0, c_2 p_2(x) = 2c_2 \), and \( c_3 p_3(x) = 6c_3 x_n \), the equations are reduced to:
     \[
     0 + 0 + 2c_2 + 6c_3 x_n = f(x, y, y') + \lambda L_3(x_{n+i}), i = 1, 2, 3
     \]

4. **Matrix Formulation and Gaussian Elimination**:
   - The matrix form of the system of equations is:
     \[
     \begin{pmatrix}
     1 & x_n & x_n^2 & x_n^3 & 0 \\
     1 & x_{n+1} & x_{n+1}^2 & x_{n+1}^3 & 0 \\
     0 & 0 & 2 & 6x_n & -\frac{11}{27} \\
     0 & 0 & 2 & 6x_{n+2} & -\frac{11}{27} \\
     0 & 0 & 2 & 6x_{n+3} & 1
     \end{pmatrix}
     \begin{pmatrix}
     c_0 \\
     c_1 \\
     c_2 \\
     c_3 \\
     \lambda
     \end{pmatrix}
     =
     \begin{pmatrix}
     y_n \\
     y_{n+1} \\
     f_{n+1} \\
     f_{n+2} \\
     f_{n+3}
     \end{pmatrix}
     \]
   - Solving this matrix using Gaussian elimination provides the values for \( c_0, c_1, c_2, c_3, \) and \( \lambda \).

5. **Final Three-Step Method Formula**:
   - The derived formula for the three-step method is:
     \[
     y(t) = \alpha_0(x) y_n + \alpha_1(x) y_{n+1} + h^2 \sum_{i=1}^k \beta_i(x) f_{n+i}
     \]
   - Where \( \alpha_0(t) = -1 - t \) and \( \alpha_1(t) = 2 + t \).

### Questions to Consider:

1. Can you detail the steps involved in solving the matrix equation using Gaussian elimination?
2. How are the values of \( \alpha_0(t) \) and \( \alpha_1(t) \) derived?
3. What is the significance of each term in the final three-step method formula?
4. How does the change of variable impact the interpolation process?
5. Can you provide an example application where this three-step method would be useful?

### Tip:

When dealing with complex polynomial equations and transformations, it's helpful to verify each step by substituting back into the original equations to ensure correctness. This is especially crucial in numerical methods where small errors can propagate.

Would you like further details on any specific part of this method or have any other questions?

Explore the three-step method in numerical methods with detailed explanations and formulas. Learn about polynomial interpolation, Gaussian elimination, and the impact of variable changes on accuracy. Suitable for advanced undergraduate students.

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