Root
Usage
• Root[f, k] 表示多项式方程 f[x] 0 的第 k个根.
Notes
• f 必须是一个诸如(#^5 - 2 # + 1)& 的Function对象. • Root[f, k] 自动进行化简以使得 f 具有可能的最小次数和最小整系数. • Root 把实根放在复根之前,而把一对复共轭根放在相邻位置. • 多项式f[x]的系数可以包含符号参数. • 对线性的和二次多项式 f[x],Root[f, k]自动化简为显式有理形式或根形式. • N 给出Root对象的近似数值. • 诸如 Abs, Re, Round 和 Less的运算可以使用到Root对象上. • 如果 f 不包含符号参数, 则Root[f, k]被视作一个数值量. • 缺省情形下,Root使用近似数值方法对多项式的根进行分离. 该方法何种情况下失败还不知道,然而 SetOptions[Root, ExactRootIsolation->True]将使用慢得多的但完全严格的方法产生 Root.
Further Examples
The polynomial that appears in Root is automatically reduced to a minimal polynomial in the form of a pure function.
In[1]:=
|
Out[1]=
|
In[2]:=
|
Out[2]=
|
It can be cast into the form of a polynomial in one variable.
In[3]:=
|
Out[3]=
|
In[4]:=
|
In the quadratic case the Root object is automatically expressed in terms of radicals.
In[5]:=
|
Out[5]=
|
For degree three or higher, Root objects are not automatically expressed in terms of radicals.
In[6]:=
|
Out[6]=
|
ToRadicals attempts to express all Root objects in terms of radicals.
In[7]:=
|
Out[7]=
|
To get a Root object from performing algebraic operations on algebraic numbers, use RootReduce.
In[8]:=
|
Out[8]=
|
In[9]:=
|
Out[9]=
|
Here is a more complicated expression involving algebraic numbers.
In[10]:=
|
Out[10]=
|
Sometimes RootReduce gives a simple result. This is the product of the roots of the polynomial.
In[11]:=
|
Out[11]=
|
|