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<title> 多项式的根 </title>
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<font COLOR="#0000FF">
<h1>8.1 多项式的根</h1>
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<p>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 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。
</p>
<p><font COLOR="#FF0000" FACE="Times New Roman">>> p=[1 3 2];</font> </p>
<p><font COLOR="#FF0000" FACE="Times New Roman">>> r=roots(p)</font> </p>
<p><font COLOR="#FF0000" FACE="Times New Roman">r =</font> </p>
<p><font COLOR="#FF0000" FACE="Times New Roman">-2</font> </p>
<p><font COLOR="#FF0000" FACE="Times New Roman">-1</font> </p>
<p><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>
</p>
<p><font COLOR="#FF0000" FACE="Times New Roman">>> r=roots(p) % </font><font COLOR="#FF0000">有实数根及复数根</font> </p>
<p><font COLOR="#FF0000" FACE="Times New Roman">r =</font> </p>
<p><font COLOR="#FF0000" FACE="Times New Roman">11.7473</font> </p>
<p><font COLOR="#FF0000" FACE="Times New Roman">2.7028</font> </p>
<p><font COLOR="#FF0000" FACE="Times New Roman">-1.2251 + 1.4672i</font> </p>
<p><font COLOR="#FF0000" FACE="Times New Roman">-1.2251 - 1.4672i <br>
</font></p>
<p>与 <font COLOR="#FF0000" FACE="Times New Roman">roots</font><font FACE="Times New Roman"> </font>相关的函数尚有 <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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<p><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 COLOR="#FF0000" FACE="Times New Roman">poly(r)</font>，其中 <font COLOR="#FF0000" FACE="Times New Roman">r</font><font FACE="Times New Roman"> </font>是代表根的阵列。而
<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。
</p>
<p><font COLOR="#FF0000" FACE="Times New Roman">>> r=[-2 1];</font> </p>
<p><font COLOR="#FF0000" FACE="Times New Roman">>> pp=poly(r) %
pp=(x+2)(x-1)=x^2+3x+2</font> </p>
<p><font COLOR="#FF0000" FACE="Times New Roman">pp =</font> </p>
<p><font COLOR="#FF0000" FACE="Times New Roman">1 3 2</font> </p>
<p><font COLOR="#FF0000" FACE="Times New Roman">>> p=[1 -4 6 -4];</font> </p>
<p><font COLOR="#FF0000" FACE="Times New Roman">>> r=roots(p)</font> </p>
<p><font COLOR="#FF0000" FACE="Times New Roman">r =</font> </p>
<p><font COLOR="#FF0000" FACE="Times New Roman">2.0000 1.0000 + 1.0000i 1.0000 - 1.0000i </font></p>
<p><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> </p>
<p><font COLOR="#FF0000" FACE="Times New Roman">pp =</font> </p>
<p><font COLOR="#FF0000" FACE="Times New Roman">1 -4 6 -4</font> </p>
<p><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> </p>
<p><font COLOR="#FF0000" FACE="Times New Roman">>> r=roots(pp)</font> </p>
<p><font COLOR="#FF0000" FACE="Times New Roman">r =</font> </p>
<p><font COLOR="#FF0000" FACE="Times New Roman">-4.9395 </font></p>
<p><font COLOR="#FF0000" FACE="Times New Roman">-1.0303 + 0.8721i</font> </p>
<p><font COLOR="#FF0000" FACE="Times New Roman">-1.0303 - 0.8721i</font> </p>
<p><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> </p>
<p><font COLOR="#FF0000" FACE="Times New Roman">pp =</font> </p>
<p><font COLOR="#FF0000" FACE="Times New Roman">1.0000 7.0000 12.0000 9.0000 + 0.0000i</font>
</p>
<p><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> </p>
<p><font COLOR="#FF0000" FACE="Times New Roman">pp =</font> </p>
<p><font COLOR="#FF0000" FACE="Times New Roman">1.0000 7.0000 12.0000 9.0000<br>
</font></p>
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