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ContinuedFraction

Usage

ContinuedFraction[x, n] 产生一个在x的连续分数表示中的前 n项的列表。
ContinuedFraction[x, n] 产生一个给定x的精度可得到的所有项的列表。


Notes

• 连续分数表示   ,  ,  , ...  相应于表达式a1+1/(a2+1/(a3+...))
x 或者是一个精确数或者是一个不精确数。
• 例如: ContinuedFraction[Pi, 4]LongRightArrow .
• 对精确数,如果 x 是有理数或二次无理数,可以使用ContinuedFraction[x]
• 对二次无理数,ContinuedFraction[x]返回   ,  , ... ,   ,  , ...   形式的结果,相应于无限项序列,从  开始循环重复
• 由于一个有理数的连续分数表示仅有有限项,在这种情况下ContinuedFraction[x, n]可能产生一个少于 n 个元素的列表。
• 对中断连续函数,  ... , k 总是与  ... , k-1, 1 相等; ContinuedFraction 返回这些形式中的第一个。
FromContinuedFraction[list]ContinuedFraction的结果中重新构造一个数。
• 参见Mathematica全书: 3.2.4节.
• 实现注释: 参见节 A.9.4.
Further Examples

The result of ContinuedFraction is given as a list.

In[1]:=  

Out[1]=

The continued fraction simplifies to the original fraction.

In[2]:=  

Out[2]=

This gives the continued fraction representation of a quadratic number. The sublist represents the repeated part.

In[3]:=  

Out[3]=

You can go back to the number with FromContinuedFraction.

In[4]:=  

Out[4]=

This gives the first few terms of the continued fraction representation of a transcendental number.

In[5]:=  

Out[5]=

Here is a special case of Pell's equation. In a Diophantine equation like this the parameter m and the variables x and y are assumed to be integers.

In[6]:=  

PellSolve gives the least positive solution for x and y when m is not a perfect square.

In[7]:=  

Here is the solution when m is  .

In[8]:=  

Out[8]=

In[9]:=  

Out[9]=

In[10]:=  

同时参见 FromContinuedFraction.