Iterated Logarithm Function 多重对数函数

来源：互联网发布：乡镇网络舆情自查报告编辑：程序博客网时间：2024/05/12 06:32

在算法导论中 Iterated Logarithm Function是在Functional iteration 之后引出的。

Function Iteration 讲的是函数的迭代

Iterated Logarithm Function 多重对数函数——通俗地讲，就是使得lg * n ≦ 1 的最小 i 值。

We use the notation lg* n (read "log star of n") to denote the iteratedlogarithm, which is

defined as follows. Let lg(i) n be as defined below, with f(n) = lg n. Because the logarithm of a
nonpositive number is undefined, lg(i) n is defined only if lg(i-1)n > 0.Be sure to distinguish
lg(i) n (the logarithm function applieditimes insuccession, starting with argument n) from lgi
n (the logarithm of n raised to the ith power). The iteratedlogarithm function is defined as
lg* n= min {i >= 0: lg(i) n ≤ 1}.

The iterated logarithm is a very slowly growing function:
lg*2 = 1,————lg*2 = 1+lg*(lg2) = 1+ lg*1 = 1+0 = 1（PS. lg是以2为底的对数）

lg*4 = 2,————lg*4 = 1+lg*(lg4) = 1+ lg*2 = 1+1 = 2

lg*16 = 3,———-lg*16 = 1+lg*(lg16) = 1+ lg*4 = 1+2 = 3

lg*65536= 4,—--lg*65536= 1+lg*(lg65536) = 1+ lg*16 = 1+3 = 4

lg*(2^65536) = 5.

星号后面的数位意味着：

Since the number of atoms in the observable universe is estimated to beabout 1080, which is
much less than 265536, we rarely encounter an input size n such that lg* n > 5.

0 0