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* chore: improve comments and add tests to trapezoidal rule * fix: too much characters in line * Update maths/trapezoidal_rule.py Co-authored-by: Tianyi Zheng <tianyizheng02@gmail.com> * Update maths/trapezoidal_rule.py Co-authored-by: Tianyi Zheng <tianyizheng02@gmail.com> * Update maths/trapezoidal_rule.py Co-authored-by: Tianyi Zheng <tianyizheng02@gmail.com> * Update maths/trapezoidal_rule.py Co-authored-by: Tianyi Zheng <tianyizheng02@gmail.com> * fix: change function name in calls * modify tests, changes numbers to remove coma * updating DIRECTORY.md * Fix doctest whitespace * Try to fix line length in doctest --------- Co-authored-by: Tianyi Zheng <tianyizheng02@gmail.com> Co-authored-by: tianyizheng02 <tianyizheng02@users.noreply.github.com>
106 lines
2.4 KiB
Python
106 lines
2.4 KiB
Python
"""
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Numerical integration or quadrature for a smooth function f with known values at x_i
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"""
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def trapezoidal_rule(boundary, steps):
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"""
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Implements the extended trapezoidal rule for numerical integration.
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The function f(x) is provided below.
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:param boundary: List containing the lower and upper bounds of integration [a, b]
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:param steps: The number of steps (intervals) used in the approximation
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:return: The numerical approximation of the integral
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>>> abs(trapezoidal_rule([0, 1], 10) - 0.33333) < 0.01
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True
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>>> abs(trapezoidal_rule([0, 1], 100) - 0.33333) < 0.01
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True
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>>> abs(trapezoidal_rule([0, 2], 1000) - 2.66667) < 0.01
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True
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>>> abs(trapezoidal_rule([1, 2], 1000) - 2.33333) < 0.01
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True
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"""
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h = (boundary[1] - boundary[0]) / steps
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a = boundary[0]
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b = boundary[1]
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x_i = make_points(a, b, h)
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y = 0.0
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y += (h / 2.0) * f(a)
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for i in x_i:
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y += h * f(i)
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y += (h / 2.0) * f(b)
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return y
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def make_points(a, b, h):
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"""
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Generates points between a and b with step size h for trapezoidal integration.
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:param a: The lower bound of integration
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:param b: The upper bound of integration
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:param h: The step size
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:yield: The next x-value in the range (a, b)
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>>> list(make_points(0, 1, 0.1)) # doctest: +NORMALIZE_WHITESPACE
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[0.1, 0.2, 0.30000000000000004, 0.4, 0.5, 0.6, 0.7, 0.7999999999999999, \
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0.8999999999999999]
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>>> list(make_points(0, 10, 2.5))
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[2.5, 5.0, 7.5]
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>>> list(make_points(0, 10, 2))
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[2, 4, 6, 8]
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>>> list(make_points(1, 21, 5))
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[6, 11, 16]
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>>> list(make_points(1, 5, 2))
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[3]
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>>> list(make_points(1, 4, 3))
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[]
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"""
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x = a + h
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while x <= (b - h):
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yield x
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x += h
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def f(x):
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"""
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This is the function to integrate, f(x) = (x - 0)^2 = x^2.
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:param x: The input value
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:return: The value of f(x)
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>>> f(0)
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0
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>>> f(1)
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1
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>>> f(0.5)
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0.25
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"""
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return x**2
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def main():
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"""
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Main function to test the trapezoidal rule.
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:a: Lower bound of integration
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:b: Upper bound of integration
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:steps: define number of steps or resolution
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:boundary: define boundary of integration
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>>> main()
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y = 0.3349999999999999
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"""
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a = 0.0
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b = 1.0
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steps = 10.0
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boundary = [a, b]
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y = trapezoidal_rule(boundary, steps)
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print(f"y = {y}")
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if __name__ == "__main__":
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import doctest
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doctest.testmod()
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main()
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