python的fractions.limit_denominator是如何實現的？

Question

limit_denominator(max_denominator=1000000) 
Finds and returns the closest Fraction to self that has denominator at most max_denominator. This method is useful for finding rational approximations to a given floating-point number: 

>>> 
>>> from fractions import Fraction 
>>> Fraction('3.1415926535897932').limit_denominator(1000) 
Fraction(355, 113) 

它不應該像嘗試/ 999，b/998，c/997 ..並找到最佳近似值。

Answer 1

fractions模塊是用Python編寫的，你可以看看源代碼。它包含以下評論。

# Algorithm notes: For any real number x, define a *best upper 
    # approximation* to x to be a rational number p/q such that: 
    # 
    # (1) p/q >= x, and 
    # (2) if p/q > r/s >= x then s > q, for any rational r/s. 
    # 
    # Define *best lower approximation* similarly. Then it can be 
    # proved that a rational number is a best upper or lower 
    # approximation to x if, and only if, it is a convergent or 
    # semiconvergent of the (unique shortest) continued fraction 
    # associated to x. 
    # 
    # To find a best rational approximation with denominator <= M, 
    # we find the best upper and lower approximations with 
    # denominator <= M and take whichever of these is closer to x. 
    # In the event of a tie, the bound with smaller denominator is 
    # chosen. If both denominators are equal (which can happen 
    # only when max_denominator == 1 and self is midway between 
    # two integers) the lower bound---i.e., the floor of self, is 
    # taken.