Implement Gauss-Seidel method for solving linear systems. (#13535)

* Add Gauss-Seidel method implementation

Implement Gauss-Seidel method for solving linear systems.

* Update gauss-seidel-method.py

* Rename gauss-seidel-method.py to gauss_seidel_method.py

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* Rename parameters in Gauss-Seidel method

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* Enhance gauss_seidel function documentation and code style

Refactor gauss_seidel method for improved readability and clarity.

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* Enhance Gauss-Seidel method with result rounding

Added rounding to results for stable test comparison.

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* Clean up comments in Gauss-Seidel method

Removed comments related to convergence check and result rounding.

* updating DIRECTORY.md

---------

Co-authored-by: pre-commit-ci[bot] <66853113+pre-commit-ci[bot]@users.noreply.github.com>
Co-authored-by: Christian Clauss <cclauss@me.com>
Co-authored-by: cclauss <cclauss@users.noreply.github.com>

Author: 3 people

DIRECTORY.md

@@ -822,6 +822,7 @@
     * [Bisection](maths/numerical_analysis/bisection.py)
     * [Bisection 2](maths/numerical_analysis/bisection_2.py)
     * [Brent Method](maths/numerical_analysis/brent_method.py)
+    * [Gauss Seidel Method](maths/numerical_analysis/gauss_seidel_method.py)
     * [Integration By Simpson Approx](maths/numerical_analysis/integration_by_simpson_approx.py)
     * [Intersection](maths/numerical_analysis/intersection.py)
     * [Nevilles Method](maths/numerical_analysis/nevilles_method.py)

maths/numerical_analysis/gauss_seidel_method.py

@@ -0,0 +1,43 @@
+def gauss_seidel(
+    coefficients: list[list[float]],
+    rhs: list[float],
+    tol: float = 1e-10,
+    max_iter: int = 1000,
+) -> list[float]:
+    """
+    Solve the linear system Ax = b using the Gauss-Seidel iterative method.
+
+    Args:
+        coefficients (list[list[float]]): Coefficient matrix (n x n)
+        rhs (list[float]): Right-hand side vector (n)
+        tol (float): Convergence tolerance
+        max_iter (int): Maximum number of iterations
+
+    Returns:
+        list[float]: Approximate solution vector
+
+    Example:
+        >>> A = [[4, 1, 2], [3, 5, 1], [1, 1, 3]]
+        >>> b = [4, 7, 3]
+        >>> gauss_seidel(A, b)
+        [0.5, 1.0, 0.5]
+
+    Wikipedia:
+    https://en.wikipedia.org/wiki/Gauss%E2%80%93Seidel_method
+    """
+    n = len(coefficients)
+    x = [0.0 for _ in range(n)]
+
+    for _ in range(max_iter):
+        x_new = x.copy()
+        for i in range(n):
+            sum_before = sum(coefficients[i][j] * x_new[j] for j in range(i))
+            sum_after = sum(coefficients[i][j] * x[j] for j in range(i + 1, n))
+            x_new[i] = (rhs[i] - sum_before - sum_after) / coefficients[i][i]
+
+        if all(abs(x_new[i] - x[i]) < tol for i in range(n)):
+            return [round(val, 10) for val in x_new]
+
+        x = x_new
+
+    return [round(val, 10) for val in x]