Add Fixed Point Iteration and Modified Newton-Raphson Methods

maths/numerical_analysis/fixed_point_iteration.py

@@ -0,0 +1,56 @@
+"""
+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}")

maths/numerical_analysis/modified_newton_raphson.py

@@ -0,0 +1,78 @@
+"""
+Modified Newton-Raphson Method
+
+This method is used to find an approximate solution to a given equation f(x) = 0.
+It is an iterative method that modifies the standard Newton-Raphson method to improve
+convergence in certain cases, mostly when multiplicity is more than 1.
+
+Example:
+--------
+>>> import math
+>>> def f(x):
+...     return x**3 - 2*x - 5
+>>> def f_prime(x):
+...     return 3*x**2 - 2
+>>> modified_newton_raphson(f, f_prime, 2.0)
+2.0945514815423265
+"""
+
+from typing import Callable
+
+
+def modified_newton_raphson(
+    f: Callable[[float], float],
+    f_prime: Callable[[float], float],
+    x0: float,
+    tol: float = 1e-7,
+    max_iter: int = 1000,
+) -> float:
+    """
+    Perform the Modified Newton-Raphson method to find the root of the equation f(x) = 0.
+
+    Parameters:
+    -----------
+    f : function
+        The function for which we want to find the root.
+    f_prime : function
+        The derivative of the function f.
+    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 f(x) = 0.
+
+    Raises:
+    -------
+    ValueError
+        If the method does not converge within the maximum number of iterations.
+    """
+    x = x0
+    for i in range(max_iter):
+        fx = f(x)
+        fpx = f_prime(x)
+        if fpx == 0:
+            raise ValueError("Derivative is zero. No solution found.")
+        x_new = x - fx / fpx
+        if abs(x_new - x) < tol:
+            return x_new
+        x = x_new
+    raise ValueError("Modified Newton-Raphson method did not converge")
+
+
+if __name__ == "__main__":
+    import math
+
+    def f(x):
+        return x**3 - 2 * x - 5
+
+    def f_prime(x):
+        return 3 * x**2 - 2
+
+    root = modified_newton_raphson(f, f_prime, 2.0)
+    print(f"The root is: {root}")