Python程序来计算谐波序列 [英] Python program to calculate harmonic series

问题描述

有人知道如何用Python编写一个程序来计算谐波序列的加法吗?即1 + 1/2 +1/3 +1/4 ...

Does anyone know how to write a program in Python that will calculate the addition of the harmonic series. i.e. 1 + 1/2 +1/3 +1/4...

推荐答案

@ Kiv的答案是正确的，但是如果您不需要无限精度，对于大n来说，它的速度会很慢.在这种情况下，最好使用渐近公式:

@Kiv's answer is correct but it is slow for large n if you don't need an infinite precision. It is better to use an asymptotic formula in this case:

#!/usr/bin/env python
from math import log

def H(n):
    """Returns an approximate value of n-th harmonic number.

       http://en.wikipedia.org/wiki/Harmonic_number
    """
    # Euler-Mascheroni constant
    gamma = 0.57721566490153286060651209008240243104215933593992
    return gamma + log(n) + 0.5/n - 1./(12*n**2) + 1./(120*n**4)

@Kiv对于Python 2.6的答案:

from fractions import Fraction

harmonic_number = lambda n: sum(Fraction(1, d) for d in xrange(1, n+1))

示例:

>>> N = 100
>>> h_exact = harmonic_number(N)
>>> h = H(N)
>>> rel_err = (abs(h - h_exact) / h_exact)
>>> print n, "%r" % h, "%.2g" % rel_err
100 5.1873775176396242 6.8e-16

在N = 100处，相对误差小于1e-15.

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