weddle's integration rule

maths/weddles_rule.py

@@ -0,0 +1,135 @@
+import numpy as np
+from sympy import lambdify, symbols, sympify
+
+
+def get_inputs():
+    """
+    Get user input for the function, lower limit, and upper limit.
+
+    Returns:
+        tuple: A tuple containing the function as a string, the lower limit (a),
+               and the upper limit (b) as floats.
+
+    Example:
+        >>> from unittest.mock import patch
+        >>> inputs = ['1/(1+x**2)', 1.0, -1.0]
+        >>> with patch('builtins.input', side_effect=inputs):
+        ...     get_inputs()
+        ('1/(1+x**2)', 1.0, -1.0)
+    """
+    func = input("Enter function with variable as x: ")
+    a = float(input("Enter lower limit: "))
+    b = float(input("Enter upper limit: "))
+    return func, a, b
+
+
+def safe_function_eval(func_str):
+    """
+    Safely evaluates the function by substituting x value using sympy.
+
+    Args:
+        func_str (str): Function expression as a string.
+
+    Returns:
+        float: The evaluated function result.
+    """
+    x = symbols('x')
+    func_expr = sympify(func_str)
+
+    # Convert the function to a callable lambda function
+    lambda_func = lambdify(x, func_expr, modules=["numpy"])
+    return lambda_func
+
+
+def compute_table(func, a, b, acc):
+    """
+    Compute the table of function values based on the limits and accuracy.
+
+    Args:
+        func (str): The mathematical function with the variable 'x' as a string.
+        a (float): The lower limit of the integral.
+        b (float): The upper limit of the integral.
+        acc (int): The number of subdivisions for accuracy.
+
+    Returns:
+        tuple: A tuple containing the table of values and the step size (h).
+
+    Example:
+        >>> compute_table(
+        ...     safe_function_eval('1/(1+x**2)'), 1, -1, 1
+        ... )
+        (array([0.5       , 0.69230769, 0.9       , 1.        , 0.9       ,
+               0.69230769, 0.5       ]), -0.3333333333333333)
+    """
+    # Weddle's rule requires number of intervals as a multiple of 6 for accuracy
+    n_points = acc * 6 + 1
+    h = (b - a) / (n_points - 1)
+    x_vals = np.linspace(a, b, n_points)
+
+    # Evaluate function values at all points
+    table = func(x_vals)
+    return table, h
+
+
+def apply_weights(table):
+    """
+    Apply Simpson's rule weights to the values in the table.
+
+    Args:
+        table (list): A list of computed function values.
+
+    Returns:
+        list: A list of weighted values.
+
+    Example:
+        >>> apply_weights([0.0, 0.866, 1.0, 0.866, 0.0, -0.866, -1.0])
+        [4.33, 1.0, 5.196, 0.0, -4.33]
+    """
+    add = []
+    for i in range(1, len(table) - 1):
+        if i % 2 == 0 and i % 3 != 0:
+            add.append(table[i])
+        if i % 2 != 0 and i % 3 != 0:
+            add.append(5 * table[i])
+        elif i % 6 == 0:
+            add.append(2 * table[i])
+        elif i % 3 == 0 and i % 2 != 0:
+            add.append(6 * table[i])
+    return add
+
+
+def compute_solution(add, table, h):
+    """
+    Compute the final solution using the weighted values and table.
+
+    Args:
+        add (list): A list of weighted values from apply_weights.
+        table (list): A list of function values.
+        h (float): The step size (h) calculated from the limits and accuracy.
+
+    Returns:
+        float: The final computed integral solution.
+
+    Example:
+        >>> compute_solution([4.33, 6.0, 0.0, -4.33], [0.0, 0.866, 1.0, 0.866, 0.0,
+        ... -0.866, -1.0], 0.5235983333333333)
+        0.7853975
+    """
+    return 0.3 * h * (sum(add) + table[0] + table[-1])
+
+
+if __name__ == "__main__":
+    from doctest import testmod
+    testmod()
+
+    func, a, b = get_inputs()
+    acc = 1
+    solution = None
+
+    while acc <= 100_000:
+        table, h = compute_table(func, a, b, acc)
+        add = apply_weights(table)
+        solution = compute_solution(add, table, h)
+        acc *= 10
+
+    print(f'Solution: {solution}')