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<title> 多項式的根 </title>
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<font COLOR="#0000FF">
<h1>8.1 多項式的根</h1>
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<p>一個多項式視其階數(shù)而定，它的根可以有一個到數(shù)個，可能為實數(shù)也可能是復數(shù)。要求一高階多項式的根往
往須借助數(shù)值方法，所幸<font FACE="Times New Roman">MATLAB</font>已將這些數(shù)值方法寫成一函數(shù)<font COLOR="#FF0000" FACE="Times New Roman">roots(p)</font>，我們只要輸入多項式的各階系數(shù)（
以 <font COLOR="#FF0000" FACE="Times New Roman">p</font><font FACE="Times New Roman"> </font>代表）即可求解到對應的根。
</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">注意二階項系數(shù)為零須要輸入，否則多項式的階數(shù)就不對</font>
</p>
<p><font COLOR="#FF0000" FACE="Times New Roman">>> r=roots(p) % </font><font COLOR="#FF0000">有實數(shù)根及復數(shù)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>與 <font COLOR="#FF0000" FACE="Times New Roman">roots</font><font FACE="Times New Roman"> </font>相關的函數(shù)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>，這二個函數(shù)的用途是要驗算求解的根展開能求得原多項式。
例如有一個二次方程式的根為<font FACE="Times New Roman">2, 1</font>，則以下式計算原多項式
</p>
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<p><font COLOR="#FF0000" FACE="Times New Roman">poly</font><font FACE="Times New Roman"> </font>函數(shù)就是在求出多項式的各階系數(shù)，其語法為
<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>是代表根的陣列。而
<font COLOR="#FF0000" FACE="Times New Roman">real</font><font FACE="Times New Roman"> </font>則是用來去除因計算時產(chǎn)生的假虛部系數(shù)，為何會有此種情形請參考以下的例子。
</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">這個多項式的系數(shù)與原多項式 </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">再看另一個多項式</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">注意因計算的誤差會有假虛部產(chǎn)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">將假虛部去除，將原多項式還原</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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