Add Gauss-Seidel method implementation

Implement Gauss-Seidel method for solving linear systems.

Author: rajsriv

maths/numerical_analysis/gauss-seidel-method.py

@@ -0,0 +1,31 @@
+def gauss_seidel(a: list[list[float]], b: 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:
+        a (list[list[float]]): Coefficient matrix (n x n)
+        b (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]
+    """
+    n = len(a)
+    x = [0.0] * n
+    for _ in range(max_iter):
+        x_new = x.copy()
+        for i in range(n):
+            s1 = sum(a[i][j] * x_new[j] for j in range(i))
+            s2 = sum(a[i][j] * x[j] for j in range(i + 1, n))
+            x_new[i] = (b[i] - s1 - s2) / a[i][i]
+        if all(abs(x_new[i] - x[i]) < tol for i in range(n)):
+            return x_new
+        x = x_new
+    return x