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The provided text includes detailed mathematical derivations for a three-step method. Here's the extracted content along with some clarification:

1. The formula for \( x \):
   \[
   x = \frac{2\overline{x} - (x_{n+k} + x_n)}{x_{n+k} - x_n}
   \]

2. Systems of equations derived from interpolation and collocation:
   \[
   \sum_{i=0}^{k} c_i p_i(x) = y_{n+s}, \quad s = 0, 1 \quad (2.7)
   \]
   \[
   \sum_{i=0}^{k} c_i p'_i(x) = f_{n+j} + \lambda L_k(x_{n+j}), \quad j = 1(1)k \quad (2.8)
   \]

3. For the three-step method (\(k = 3\)), using the polynomial \( L_3(x) = \frac{1}{2}(5x^3 - 3x) \) and the formula for \( x \), values are determined for \( L_3(x_{n+1}) \), \( L_3(x_{n+2}) \), and \( L_3(x_{n+3}) \):
   \[
   x = -\frac{1}{3} \quad \text{and} \quad L_3(x_{n+1}) = \frac{11}{27}
   \]
   \[
   x = \frac{1}{3} \quad \text{and} \quad L_3(x_{n+2}) = -\frac{11}{27}
   \]
   \[
   x = 1 \quad \text{and} \quad L_3(x_{n+3}) = 1
   \]

4. Additional relations from (2.3):
   \[
   c_0 p'_0(x) = 0, \quad c_1 p'_1(x) = 0, \quad c_2 p'_2(x) = 2c_2, \quad c_3 p'_3(x) = 6c_3 x_n
   \]

5. Reduction of equation (2.8):
   \[
   0 + 0 + 2c_2 + 6c_3 x_n = f(x, y, y') + \lambda L_3(x_{n+i}), \quad i = 1, 2, 3 \quad (2.9)
   \]

6. Matrix form of the system of equations:
   \[
   \begin{pmatrix}
   1 & x_n & x_n^2 & x_n^3 & 0 \\
   1 & x_{n+1} & x_{n+1}^2 & x_{n+1}^3 & 0 \\
   0 & 0 & 2 & 6x_{n+1} & \frac{11}{27} \\
   0 & 0 & 2 & 6x_{n+2} & -\frac{11}{27} \\
   0 & 0 & 2 & 6x_{n+3} & 1
   \end{pmatrix}
   \begin{pmatrix}
   c_0 \\
   c_1 \\
   c_2 \\
   c_3 \\
   \lambda
   \end{pmatrix}
   =
   \begin{pmatrix}
   y_n \\
   y_{n+1} \\
   f_{n+1} \\
   f_{n+2} \\
   f_{n+3}
   \end{pmatrix} \quad (2.10)
   \]

7. Solving the system using Gaussian elimination gives the values of the unknown parameters \(c_i\) and \(\lambda\).

8. Continuous implicit three-step method:
   \[
   y(x) = \alpha_0(x) y_n + \alpha_1(x) y_{n+1} + h^2 \sum_{i=1}^{k} \beta_i(x) f_{n+i} \quad (2.11)
   \]
   Where:
   \[
   \alpha_0(t) = -1 - t, \quad \alpha_1(t) = 2 + t
   \]

Would you like any further details or explanations on these equations?

Here are five related questions you might consider asking:
1. Can you explain the Gaussian elimination method used in this context?
2. How do the polynomials \( L_3(x) \) influence the solution of the system?
3. What are the steps to derive the continuous implicit three-step method?
4. Can you provide a practical example where this method is applied?
5. How does the parameter \(\lambda\) affect the solution?

**Tip:** When solving systems of equations using matrices, always check if the system is consistent and the determinant of the coefficient matrix is non-zero for a unique solution.

Explore detailed mathematical derivations for a three-step method, including interpolation, collocation, Gaussian elimination, and matrix operations. Understand the influence of polynomial L_3(x) and the continuous implicit three-step method (2.11).

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