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Realiza la iteración 100, por el método de Gauss- seidel 

The image shows a system of linear equations with a tridiagonal matrix structure, which is suitable for iterative methods like Gauss-Seidel. The system can be written as:

\[
\begin{aligned}
12x_1 - 2x_2 + x_3 &= 5, \\
-2x_1 + 12x_2 - 2x_3 + x_4 &= 5, \\
x_1 - 2x_2 + 12x_3 - 2x_4 + x_5 &= 5, \\
x_2 - 2x_3 + 12x_4 - 2x_5 + x_6 &= 5, \\
\vdots \\
x_{46} - 2x_{47} + 12x_{48} - 2x_{49} + x_{50} &= 5, \\
x_{47} - 2x_{48} + 12x_{49} - 2x_{50} &= 5, \\
x_{48} - 2x_{49} + 12x_{50} &= 5.
\end{aligned}
\]

### Steps to perform Gauss-Seidel Iteration:

The Gauss-Seidel method updates each variable in the system using the most recent values of the other variables. Let's break down the process into steps for iteration:

1. **Initialization**: 
   Start with an initial guess for the solution vector, \( \mathbf{x}^{(0)} \), typically set to zeros.

2. **Iteration**:
   Update each \( x_i \) in sequence using the updated values. The formula for updating \( x_i \) is:

   \[
   x_i^{(k+1)} = \frac{1}{a_{ii}} \left( b_i - \sum_{j<i} a_{ij} x_j^{(k+1)} - \sum_{j>i} a_{ij} x_j^{(k)} \right),
   \]
   where \( a_{ii} \) is the diagonal element of the coefficient matrix, and \( b_i \) is the corresponding element from the right-hand side.

3. **Stopping criterion**:
   The method iterates until the difference between consecutive iterations is below a tolerance threshold, or after a fixed number of iterations.

Since you've requested the result after the 100th iteration, I will implement the Gauss-Seidel algorithm for this system and display the solution vector after the 100th iteration.

Let me compute that for you.After performing 100 iterations using the Gauss-Seidel method, the solution vector \( \mathbf{x}^{(100)} \) is approximately:

\[
x^{(100)} = \begin{bmatrix}
0.4638, & 0.5373, & 0.5090, & 0.4982, & 0.4989, & 0.5000, & 0.5001, & \ldots, & 0.5001, & 0.4989, & 0.4982, & 0.5090, & 0.5373, & 0.4638
\end{bmatrix}
\]

This vector shows the values of \( x_1 \) through \( x_{50} \) after 100 iterations, with the central values stabilizing around 0.5.

Would you like more details on the convergence process, or do you have any specific questions?

### Related Questions:
1. How can we assess the convergence of the Gauss-Seidel method?
2. What effect does the initial guess have on the convergence speed?
3. How does the Gauss-Seidel method compare to the Jacobi method?
4. How does matrix conditioning affect the performance of iterative methods?
5. What role do diagonal dominance and matrix structure play in the success of the Gauss-Seidel method?

**Tip**: For faster convergence, the Gauss-Seidel method works best with diagonally dominant matrices. If your matrix isn't naturally diagonally dominant, rearranging the equations may help.

Realiza la iteración 100, por el método de Gauss-Seidel

Solve a tridiagonal system of linear equations using the Gauss-Seidel method. This problem demonstrates the 100th iteration of the Gauss-Seidel algorithm for a 50-variable system, focusing on convergence and iterative updates.

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