adding doctests to maths/trapezoidal_rule.py (#12193)

* adding doctests to trapezoidal_rule.py

* adding algorithm delta-star transformation

* updating DIRECTORY.md

* delete file star_delta_transform.py

* updating DIRECTORY.md

* modified:   ../DIRECTORY.md

---------

Co-authored-by: MRJPEREZR <MRJPEREZR@users.noreply.github.com>

maths/trapezoidal_rule.py

@@ -5,13 +5,25 @@ This method is the classical approach of suming 'Equally Spaced Abscissas'
 
 method 1:
 "extended trapezoidal rule"
+int(f) = dx/2 * (f1 + 2f2 + ... + fn)
 
 """
 
 
 def method_1(boundary, steps):
-    # "extended trapezoidal rule"
+    """
-    # int(f) = dx/2 * (f1 + 2f2 + ... + fn)
+    Apply the extended trapezoidal rule to approximate the integral of function f(x)
+    over the interval defined by 'boundary' with the number of 'steps'.
+
+    Args:
+        boundary (list of floats): A list containing the start and end values [a, b].
+        steps (int): The number of steps or subintervals.
+    Returns:
+        float: Approximation of the integral of f(x) over [a, b].
+    Examples:
+    >>> method_1([0, 1], 10)
+    0.3349999999999999
+    """
     h = (boundary[1] - boundary[0]) / steps
     a = boundary[0]
     b = boundary[1]
@@ -26,13 +38,40 @@ def method_1(boundary, steps):
 
 
 def make_points(a, b, h):
+    """
+    Generates points between 'a' and 'b' with step size 'h', excluding the end points.
+    Args:
+        a (float): Start value
+        b (float): End value
+        h (float): Step size
+    Examples:
+    >>> list(make_points(0, 10, 2.5))
+    [2.5, 5.0, 7.5]
+
+    >>> list(make_points(0, 10, 2))
+    [2, 4, 6, 8]
+
+    >>> list(make_points(1, 21, 5))
+    [6, 11, 16]
+
+    >>> list(make_points(1, 5, 2))
+    [3]
+
+    >>> list(make_points(1, 4, 3))
+    []
+    """
     x = a + h
-    while x < (b - h):
+    while x <= (b - h):
         yield x
         x = x + h
 
 
 def f(x):  # enter your function here
+    """
+    Example:
+    >>> f(2)
+    4
+    """
     y = (x - 0) * (x - 0)
     return y
 
@@ -47,4 +86,7 @@ def main():
 
 
 if __name__ == "__main__":
+    import doctest
+
+    doctest.testmod()
     main()