weddle's integration rule

maths/weddles_rule.py

@@ -0,0 +1,146 @@
+from collections.abc import Callable
+
+import numpy as np
+from sympy import lambdify, symbols, sympify
+
+
+def get_inputs() -> tuple[str, float, float]:
+    """
+    Get user input for the function, lower limit, and upper limit.
+
+    Returns:
+        Tuple[str, float, float]: 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: ")
+    lower_limit = float(input("Enter lower limit: "))
+    upper_limit = float(input("Enter upper limit: "))
+    return func, lower_limit, upper_limit
+
+
+def safe_function_eval(func_str: str) -> Callable:
+    """
+    Safely evaluates the function by substituting x value using sympy.
+
+    Args:
+        func_str (str): Function expression as a string.
+
+    Returns:
+        Callable: A callable lambda function for numerical evaluation.
+
+    Examples:
+        >>> f = safe_function_eval('x**2')
+        >>> f(3)
+        9
+        >>> f = safe_function_eval('x + x**2')
+        >>> f(2)
+        6
+    """
+    x = symbols("x")
+    func_expr = sympify(func_str)
+    lambda_func = lambdify(x, func_expr, modules=["numpy"])
+    return lambda_func
+
+
+def compute_table(
+    func: Callable, lower_limit: float, upper_limit: float, acc: int
+) -> tuple[np.ndarray, float]:
+    """
+    Compute the table of function values based on the limits and accuracy.
+
+    Args:
+        func (Callable): The mathematical function as a callable.
+        lower_limit (float): The lower limit of the integral.
+        upper_limit (float): The upper limit of the integral.
+        acc (int): The number of subdivisions for accuracy.
+
+    Returns:
+        Tuple[np.ndarray, float]: 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 = (upper_limit - lower_limit) / (n_points - 1)
+    x_vals = np.linspace(lower_limit, upper_limit, n_points)
+    table = func(x_vals)  # Evaluate function values at all points
+    return table, h
+
+
+def apply_weights(table: list[float]) -> list[float]:
+    """
+    Apply Weddle's rule weights to the values in the table.
+
+    Args:
+        table (List[float]): A list of computed function values.
+
+    Returns:
+        List[float]: 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])
+        elif 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: list[float], table: list[float], step_size: float) -> float:
+    """
+    Compute the final solution using the weighted values and table.
+
+    Args:
+        add (List[float]): A list of weighted values from apply_weights.
+        table (List[float]): A list of function values.
+        step_size (float): The step size 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 * step_size * (sum(add) + table[0] + table[-1])
+
+
+if __name__ == "__main__":
+    from doctest import testmod
+
+    testmod()
+
+    # func_str, a, b = get_inputs()
+    # acc = 1
+    # solution = None
+
+    # func = safe_function_eval(func_str)
+    # 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}")