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<title> 多項(xiàng)式的根 </title>
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<font COLOR="#0000FF">
<h1>8.1 多項(xiàng)式的根</h1>
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<p>一個(gè)多項(xiàng)式視其階數(shù)而定，它的根可以有一個(gè)到數(shù)個(gè)，可能為實(shí)數(shù)也可能是復(fù)數(shù)。要求一高階多項(xiàng)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(xiàng)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(duì)應(yīng)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(xiàng)系數(shù)為零須要輸入，否則多項(xiàng)式的階數(shù)就不對(duì)</font>
</p>
<p><font COLOR="#FF0000" FACE="Times New Roman">>> r=roots(p) % </font><font COLOR="#FF0000">有實(shí)數(shù)根及復(fù)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(guān)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(gè)函數(shù)的用途是要驗(yàn)算求解的根展開能求得原多項(xiàng)式。
例如有一個(gè)二次方程式的根為<font FACE="Times New Roman">2, 1</font>，則以下式計(jì)算原多項(xiàng)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(xiàng)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(jì)算時(shí)產(chǎn)生的假虛部系數(shù)，為何會(huì)有此種情形請(qǐng)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(gè)多項(xiàng)式的系數(shù)與原多項(xiàng)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(gè)多項(xiàng)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(jì)算的誤差會(huì)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(xiàng)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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