Added Fixed Point Iteration Method in numerical_analysis

Author: akshat3144

maths/numerical_analysis/fixed_point_iteration.py

@@ -0,0 +1,53 @@
+"""
+Fixed Point Iteration Method
+
+This method is used to find an approximate solution to a given equation f(x) = 0.
+The function g(x) is derived from f(x) such that x = g(x).
+
+Example:
+--------
+>>> def g(x):
+...     return (x**2 + 2) / 3
+>>> fixed_point_iteration(g, 1.0)
+1.618033988749895
+"""
+
+def fixed_point_iteration(g, x0, tol=1e-7, max_iter=1000):
+    """
+    Perform Fixed Point Iteration to find the root of the equation x = g(x).
+
+    Parameters:
+    -----------
+    g : function
+        The function g(x) derived from f(x) such that x = g(x).
+    x0 : float
+        Initial guess for the root.
+    tol : float, optional
+        Tolerance for the convergence of the method. Default is 1e-7.
+    max_iter : int, optional
+        Maximum number of iterations. Default is 1000.
+
+    Returns:
+    --------
+    float
+        The approximate root of the equation x = g(x).
+
+    Raises:
+    -------
+    ValueError
+        If the method does not converge within the maximum number of iterations.
+    """
+    x = x0
+    for i in range(max_iter):
+        x_new = g(x)
+        if abs(x_new - x) < tol:
+            return x_new
+        x = x_new
+    raise ValueError("Fixed Point Iteration did not converge")
+
+if __name__ == "__main__":
+    def g(x):
+        return (x**2 + 2) / 3
+
+    root = fixed_point_iteration(g, 1.0)
+    print(f"The root is: {root}")