Merge d8c8acece9 into e3bd7721c8

maths/trapezoidal_rule.py

@@ -1,17 +1,32 @@
 """
 Numerical integration or quadrature for a smooth function f with known values at x_i
 
-This method is the classical approach of suming 'Equally Spaced Abscissas'
-
-method 1:
-"extended trapezoidal rule"
+The trapezoidal rule is the classical approach of summing 'Equally Spaced Abscissas'
 
 """
 
 
-def method_1(boundary, steps):
-    # "extended trapezoidal rule"
-    # int(f) = dx/2 * (f1 + 2f2 + ... + fn)
+def trapezoidal_rule(boundary, steps):
+    """
+    This function implements the extended trapezoidal rule for numerical integration.
+    The function f(x) is provided below.
+
+    :param boundary: List containing the lower and upper bounds of integration [a, b]
+    :param steps: The number of steps (intervals) used in the approximation
+    :return: The numerical approximation of the integral
+
+    >>> abs(trapezoidal_rule([0, 1], 10) - 0.33333) < 0.01
+    True
+
+    >>> abs(trapezoidal_rule([0, 1], 100) - 0.33333) < 0.01
+    True
+
+    >>> abs(trapezoidal_rule([0, 2], 1000) - 2.66667) < 0.01
+    True
+
+    >>> abs(trapezoidal_rule([1, 2], 1000) - 2.33333) < 0.01
+    True
+    """
     h = (boundary[1] - boundary[0]) / steps
     a = boundary[0]
     b = boundary[1]
@@ -19,30 +34,67 @@ def method_1(boundary, steps):
     y = 0.0
     y += (h / 2.0) * f(a)
     for i in x_i:
-        # print(i)
         y += h * f(i)
     y += (h / 2.0) * f(b)
     return y
 
 
 def make_points(a, b, h):
+    """
+    Generates the points between a and b with spacing h for trapezoidal integration.
+
+    :param a: The lower bound of integration
+    :param b: The upper bound of integration
+    :param h: The step size
+    :yield: The next x-value in the range (a, b)
+
+    >>> list(make_points(0, 1, 0.1))
+    [0.1, 0.2, 0.30000000000000004, \
+0.4, 0.5, 0.6, 0.7, \
+0.7999999999999999, \
+0.8999999999999999]
+    """
     x = a + h
     while x < (b - h):
         yield x
         x = x + h
 
 
-def f(x):  # enter your function here
-    y = (x - 0) * (x - 0)
-    return y
+def f(x):
+    """
+    This is the function to integrate, f(x) = (x - 0)^2 = x^2.
+
+    :param x: The input value
+    :return: The value of f(x)
+
+    >>> f(0)
+    0
+
+    >>> f(1)
+    1
+
+    >>> f(0.5)
+    0.25
+    """
+    return x**2
 
 
 def main():
-    a = 0.0  # Lower bound of integration
-    b = 1.0  # Upper bound of integration
-    steps = 10.0  # define number of steps or resolution
-    boundary = [a, b]  # define boundary of integration
-    y = method_1(boundary, steps)
+    """
+    Main function to test the trapezoidal rule.
+    :a: Lower bound of integration
+    :b: Upper bound of integration
+    :steps: define number of steps or resolution
+    :boundary: define boundary of integration
+
+    >>> main()
+    y = 0.3349999999999999
+    """
+    a = 0.0
+    b = 1.0
+    steps = 10.0
+    boundary = [a, b]
+    y = trapezoidal_rule(boundary, steps)
     print(f"y = {y}")