Newton's Method

Author: JHay0112

jmath/approximation/__init__.py

@@ -3,15 +3,20 @@
 
     Tools for approximating mathematical equations.
 
-    Optional Packages
+    Default Packages
     -----------------
 
     jmath.approximation.euler_method
-            The euler method for approximating the integrals of differential equations.
+        The Euler Method for approximating the integrals of differential equations.
+    jmath.approximation.newton_method
+        The Newton Method for approximating the roots of an equation.
 '''
 
 # - Namespace
 
 __path__ = __import__('pkgutil').extend_path(__path__, __name__)
 
-# No Defaults
+# - Defaults
+
+from .euler_method import euler_step, euler_step_interval, iterate_euler_step
+from .newton_method import newton_method

jmath/approximation/newton_method.py

@@ -0,0 +1,58 @@
+"""
+    Newton's method for approximating the roots of equations.
+"""
+
+# - Imports
+
+from ..uncertainties import Uncertainty
+from typing import Callable
+from warnings import warn
+
+# - Functions
+
+def newton_method(f: Callable[[float], float], f_prime: Callable[[float], float], x_0: float, threshold: float = 1e-6, max_iter: int = 100) -> float:
+    """
+        Newton's Method for approximation of the root of an equation.
+
+        Parameters
+        ----------
+
+        f
+            The function to determine the root of.
+        f_prime
+            The derivative of the function to determine the root of.
+        x_0
+            The starting point for determining the root from.
+        threshold (optional)
+            The threshold error to approximate the root to.
+        max_iter (optional)
+            The maximum iterations to apply.
+    """
+
+    # Instantiate x_n from x_0
+    x_n = x_0
+    # Create a placeholder new x such that the threshold is not triggered
+    x_new = x_n + 2*threshold 
+    # Iterator counter
+    i = 0
+
+    # While the distance between points is greater than the threshold
+    while abs(f(x_new) - f(x_n)) >= threshold:
+
+        x_new = x_n
+        # Compute new x_n
+        x_n = x_n - f(x_n)/f_prime(x_n)
+
+        # If iterator is too high
+        if i >= max_iter:
+            # Warn user
+            warn(f"Newton Method reached max iterations of {max_iter}.")
+            # Return value of x_n
+            return(Uncertainty(x_n, threshold))
+
+        # Increase iterator
+        i += 1
+
+    # Error under threshold
+    # Return value
+    return(Uncertainty(x_n, threshold))

tests/test_newton_method.py

@@ -0,0 +1,15 @@
+"""
+    Tests that Newton's Method behaves as expected.
+"""
+
+# - Imports
+
+from ..jmath.approximation.newton_method import *
+
+# - Tests
+
+def test_root_2():
+    """Tests that root 2 comes from the Newton Method as expected."""
+    f = lambda x : x**2 - 2
+    f_prime = lambda x : 2*x
+    assert round(newton_method(f, f_prime, 1).value, 6) == 1.414214