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<TITLE> 多项式的根 </TITLE>
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<H1>8.1 多项式的根</H1>
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一个多项式视其阶数而定，它的根可以有一个到数个，可能为实数也可能是复数。要求一高阶多项式的根往
往须借助数值方法，所幸<FONT FACE="Times New Roman">MATLAB</FONT>已将这些数值方法写成一函数<FONT COLOR=#FF0000 FACE="Times New Roman">roots(p)</FONT>，我们只要输入多项式的各阶系数（
以<FONT FACE="Times New Roman">
</FONT><FONT COLOR=#FF0000 FACE="Times New Roman">p</FONT><FONT FACE="Times New Roman">
</FONT>代表）即可求解到对应的根。
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<FONT COLOR=#FF0000 FACE="Times New Roman">>> p=[1 3 2];</FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman">>> r=roots(p)</FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman">r =</FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman"> -2</FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman"> -1</FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman">>> p=[1 -12 0
25 116]; % </FONT><FONT COLOR=#FF0000>注意二阶项系数为零须要输入，否则多项式的阶数就不对</FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman">>> r=roots(p)
% </FONT><FONT COLOR=#FF0000>有实数根及复数根</FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman">r =</FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman"> 11.7473</FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman"> 2.7028</FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman"> -1.2251 + 1.4672i</FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman"> -1.2251 - 1.4672i
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与<FONT FACE="Times New Roman"> </FONT><FONT COLOR=#FF0000 FACE="Times New Roman">roots</FONT><FONT FACE="Times New Roman">
</FONT>相关的函数尚有<FONT FACE="Times New Roman"> </FONT><FONT COLOR=#FF0000 FACE="Times New Roman">poly</FONT><TT><FONT FACE="Courier New">,
</FONT></TT><FONT COLOR=#FF0000 FACE="Times New Roman">real</FONT>，这二个函数的用途是要验算求解的根展开能求得原多项式。
例如有一个二次方程式的根为<FONT FACE="Times New Roman">2, 1</FONT>，则以下式计算原多项式
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<FONT COLOR=#FF0000 FACE="Times New Roman">poly</FONT><FONT FACE="Times New Roman">
</FONT>函数就是在求出多项式的各阶系数，其语法为<FONT FACE="Times New Roman">
</FONT><FONT COLOR=#FF0000 FACE="Times New Roman">poly(r)</FONT>，其中<FONT FACE="Times New Roman">
</FONT><FONT COLOR=#FF0000 FACE="Times New Roman">r</FONT><FONT FACE="Times New Roman">
</FONT>是代表根的阵列。而<FONT FACE="Times New Roman"> </FONT><FONT COLOR=#FF0000 FACE="Times New Roman">real</FONT><FONT FACE="Times New Roman">
</FONT>则是用来去除因计算时产生的假虚部系数，为何会有此种情形请参考以下的例子。
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<FONT COLOR=#FF0000 FACE="Times New Roman">>> r=[-2 1];</FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman">>> pp=poly(r)
% pp=(x+2)(x-1)=x^2+3x+2</FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman">pp =</FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman"> 1 3 2</FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman">>> p=[1 -4 6
-4];</FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman">>> r=roots(p)</FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman">r =</FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman"> 2.0000 1.0000
+ 1.0000i 1.0000 - 1.0000i </FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman">>> pp=poly(r)
% </FONT><FONT COLOR=#FF0000>这个多项式的系数与原多项式</FONT><FONT COLOR=#FF0000 FACE="Times New Roman">
p </FONT><FONT COLOR=#FF0000>相同</FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman">pp =</FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman"> 1 -4 6
-4</FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman">>> pp=[1 7 12
9]; % </FONT><FONT COLOR=#FF0000>再看另一个多项式</FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman">>> r=roots(pp)</FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman">r =</FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman"> -4.9395
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<FONT COLOR=#FF0000 FACE="Times New Roman"> -1.0303 + 0.8721i</FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman"> -1.0303 - 0.8721i</FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman">>> pp=poly(r)
% </FONT><FONT COLOR=#FF0000>注意因计算的误差会有假虚部产生</FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman">pp =</FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman"> 1.0000 7.0000
12.0000 9.0000 + 0.0000i</FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman">>> pp=real(pp)
% </FONT><FONT COLOR=#FF0000>可以</FONT><FONT COLOR=#FF0000 FACE="Times New Roman">real</FONT><FONT COLOR=#FF0000>将假虚部去除，将原多项式还原</FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman">pp =</FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman"> 1.0000 7.0000
12.0000 9.0000<BR>
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