added type hints and doctests to arithmetic_analysis/newton_method.py (#2259)

* added type hints and doctests to arithmetic_analysis/newton_method.py

Continuing #2128
Also changed some variable names, made them more descriptive.

* Added type hints and doctests to arithmetic_analysis/newton_method.py

added a type alias for Callable[[float], float] and cleaned up the exception handling

* added type hints and doctests to arithmetic_analysis/newton_method.py

improved exception handling

* Update newton_method.py

Co-authored-by: Christian Clauss <cclauss@me.com>

arithmetic_analysis/newton_method.py

@@ -1,23 +1,48 @@
 """Newton's Method."""
 
 # Newton's Method - https://en.wikipedia.org/wiki/Newton%27s_method
+from typing import Callable
+
+RealFunc = Callable[[float], float]  # type alias for a real -> real function
 
 
-# function is the f(x) and function1 is the f'(x)
+# function is the f(x) and derivative is the f'(x)
-def newton(function, function1, startingInt):
+def newton(function: RealFunc, derivative: RealFunc, starting_int: int,) -> float:
-    x_n = startingInt
+    """
+    >>> newton(lambda x: x ** 3 - 2 * x - 5, lambda x: 3 * x ** 2 - 2, 3)
+    2.0945514815423474
+    >>> newton(lambda x: x ** 3 - 1, lambda x: 3 * x ** 2, -2)
+    1.0
+    >>> newton(lambda x: x ** 3 - 1, lambda x: 3 * x ** 2, -4)
+    1.0000000000000102
+    >>> import math
+    >>> newton(math.sin, math.cos, 1)
+    0.0
+    >>> newton(math.sin, math.cos, 2)
+    3.141592653589793
+    >>> newton(math.cos, lambda x: -math.sin(x), 2)
+    1.5707963267948966
+    >>> newton(math.cos, lambda x: -math.sin(x), 0)
+    Traceback (most recent call last):
+    ...
+    ZeroDivisionError: Could not find root
+    """
+    prev_guess float(starting_int)
     while True:
-        x_n1 = x_n - function(x_n) / function1(x_n)
+        try:
-        if abs(x_n - x_n1) < 10 ** -5:
+            next_guess = prev_guess - function(prev_guess) / derivative(prev_guess)
-            return x_n1
+        except ZeroDivisionError:
-        x_n = x_n1
+            raise ZeroDivisionError("Could not find root")
+        if abs(prev_guess - next_guess) < 10 ** -5:
+            return next_guess
+        prev_guess = next_guess
 
 
-def f(x):
+def f(x: float) -> float:
     return (x ** 3) - (2 * x) - 5
 
 
-def f1(x):
+def f1(x: float) -> float:
     return 3 * (x ** 2) - 2