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* adding doctests to trapezoidal_rule.py * adding algorithm delta-star transformation * updating DIRECTORY.md * delete file star_delta_transform.py * updating DIRECTORY.md * modified: ../DIRECTORY.md --------- Co-authored-by: MRJPEREZR <MRJPEREZR@users.noreply.github.com>
93 lines
2.0 KiB
Python
93 lines
2.0 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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This method is the classical approach of suming 'Equally Spaced Abscissas'
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method 1:
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"extended trapezoidal rule"
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int(f) = dx/2 * (f1 + 2f2 + ... + fn)
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"""
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def method_1(boundary, steps):
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"""
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Apply the extended trapezoidal rule to approximate the integral of function f(x)
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over the interval defined by 'boundary' with the number of 'steps'.
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Args:
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boundary (list of floats): A list containing the start and end values [a, b].
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steps (int): The number of steps or subintervals.
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Returns:
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float: Approximation of the integral of f(x) over [a, b].
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Examples:
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>>> method_1([0, 1], 10)
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0.3349999999999999
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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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# print(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', excluding the end points.
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Args:
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a (float): Start value
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b (float): End value
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h (float): Step size
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Examples:
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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 = x + h
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def f(x): # enter your function here
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"""
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Example:
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>>> f(2)
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4
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"""
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y = (x - 0) * (x - 0)
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return y
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def main():
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a = 0.0 # Lower bound of integration
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b = 1.0 # Upper bound of integration
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steps = 10.0 # define number of steps or resolution
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boundary = [a, b] # define boundary of integration
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y = method_1(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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