added type hints and doctests to arithmetic_analysis/bisection.py (#2241)

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

continuing in line with #2128

* modified arithmetic_analysis/bisection.py

Put back print statement at the end, replaced algorithm's print statement with an exception.

* modified arithmetic_analysis/bisection.py

Removed unnecessary type import "Optional"

* modified arithmetic_analysis/bisection.py

Replaced generic Exception with ValueError.

* modified arithmetic_analysis/bisection.py

fixed doctests

Author: spamegg1

Diff for: arithmetic_analysis/bisection.py

@@ -1,25 +1,38 @@
-import math
+from typing import Callable
 
 
-def bisection(
-    function, a, b
-):  # finds where the function becomes 0 in [a,b] using bolzano
-
-    start = a
-    end = b
+def bisection(function: Callable[[float], float], a: float, b: float) -> float:
+    """
+    finds where function becomes 0 in [a,b] using bolzano
+    >>> bisection(lambda x: x ** 3 - 1, -5, 5)
+    1.0000000149011612
+    >>> bisection(lambda x: x ** 3 - 1, 2, 1000)
+    Traceback (most recent call last):
+    ...
+    ValueError: could not find root in given interval.
+    >>> bisection(lambda x: x ** 2 - 4 * x + 3, 0, 2)
+    1.0
+    >>> bisection(lambda x: x ** 2 - 4 * x + 3, 2, 4)
+    3.0
+    >>> bisection(lambda x: x ** 2 - 4 * x + 3, 4, 1000)
+    Traceback (most recent call last):
+    ...
+    ValueError: could not find root in given interval.
+    """
+    start: float = a
+    end: float = b
     if function(a) == 0:  # one of the a or b is a root for the function
         return a
     elif function(b) == 0:
         return b
     elif (
         function(a) * function(b) > 0
     ):  # if none of these are root and they are both positive or negative,
-        # then his algorithm can't find the root
-        print("couldn't find root in [a,b]")
-        return
+        # then this algorithm can't find the root
+        raise ValueError("could not find root in given interval.")
     else:
-        mid = start + (end - start) / 2.0
-        while abs(start - mid) > 10 ** -7:  # until we achieve precise equals to 10^-7
+        mid: float = start + (end - start) / 2.0
+        while abs(start - mid) > 10 ** -7:  # until precisely equals to 10^-7
             if function(mid) == 0:
                 return mid
             elif function(mid) * function(start) < 0:
@@ -30,9 +43,13 @@ def bisection(
         return mid
 
 
-def f(x):
-    return math.pow(x, 3) - 2 * x - 5
+def f(x: float) -> float:
+    return x ** 3 - 2 * x - 5
 
 
 if __name__ == "__main__":
     print(bisection(f, 1, 1000))
+
+    import doctest
+
+    doctest.testmod()