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117 lines
3.4 KiB
Python
117 lines
3.4 KiB
Python
"""
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Gamma function is a very useful tool in math and physics.
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It helps calculating complex integral in a convenient way.
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for more info: https://en.wikipedia.org/wiki/Gamma_function
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In mathematics, the gamma function is one commonly
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used extension of the factorial function to complex numbers.
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The gamma function is defined for all complex numbers except
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the non-positive integers
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Python's Standard Library math.gamma() function overflows around gamma(171.624).
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"""
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import math
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from numpy import inf
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from scipy.integrate import quad
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def gamma_iterative(num: float) -> float:
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"""
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Calculates the value of Gamma function of num
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where num is either an integer (1, 2, 3..) or a half-integer (0.5, 1.5, 2.5 ...).
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>>> gamma_iterative(-1)
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Traceback (most recent call last):
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...
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ValueError: math domain error
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>>> gamma_iterative(0)
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Traceback (most recent call last):
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...
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ValueError: math domain error
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>>> gamma_iterative(9)
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40320.0
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>>> from math import gamma as math_gamma
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>>> all(.99999999 < gamma_iterative(i) / math_gamma(i) <= 1.000000001
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... for i in range(1, 50))
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True
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>>> gamma_iterative(-1)/math_gamma(-1) <= 1.000000001
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Traceback (most recent call last):
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...
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ValueError: math domain error
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>>> gamma_iterative(3.3) - math_gamma(3.3) <= 0.00000001
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True
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"""
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if num <= 0:
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raise ValueError("math domain error")
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return quad(integrand, 0, inf, args=(num))[0]
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def integrand(x: float, z: float) -> float:
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return math.pow(x, z - 1) * math.exp(-x)
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def gamma_recursive(num: float) -> float:
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"""
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Calculates the value of Gamma function of num
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where num is either an integer (1, 2, 3..) or a half-integer (0.5, 1.5, 2.5 ...).
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Implemented using recursion
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Examples:
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>>> from math import isclose, gamma as math_gamma
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>>> gamma_recursive(0.5)
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1.7724538509055159
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>>> gamma_recursive(1)
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1.0
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>>> gamma_recursive(2)
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1.0
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>>> gamma_recursive(3.5)
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3.3233509704478426
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>>> gamma_recursive(171.5)
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9.483367566824795e+307
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>>> all(isclose(gamma_recursive(num), math_gamma(num))
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... for num in (0.5, 2, 3.5, 171.5))
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True
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>>> gamma_recursive(0)
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Traceback (most recent call last):
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...
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ValueError: math domain error
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>>> gamma_recursive(-1.1)
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Traceback (most recent call last):
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...
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ValueError: math domain error
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>>> gamma_recursive(-4)
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Traceback (most recent call last):
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...
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ValueError: math domain error
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>>> gamma_recursive(172)
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Traceback (most recent call last):
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...
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OverflowError: math range error
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>>> gamma_recursive(1.1)
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Traceback (most recent call last):
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...
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NotImplementedError: num must be an integer or a half-integer
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"""
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if num <= 0:
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raise ValueError("math domain error")
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if num > 171.5:
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raise OverflowError("math range error")
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elif num - int(num) not in (0, 0.5):
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raise NotImplementedError("num must be an integer or a half-integer")
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elif num == 0.5:
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return math.sqrt(math.pi)
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else:
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return 1.0 if num == 1 else (num - 1) * gamma_recursive(num - 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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num = 1.0
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while num:
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num = float(input("Gamma of: "))
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print(f"gamma_iterative({num}) = {gamma_iterative(num)}")
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print(f"gamma_recursive({num}) = {gamma_recursive(num)}")
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print("\nEnter 0 to exit...")
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