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