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<TITLE> 一維內插 </TITLE>
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<H1>7.1.1 一維內插</H1>
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線性內插是假設在二個已知數據中的變化為線性關系，因此可由已知二點的座標(<I>a, b</I>)去計算通過這二點的
斜線，公式如下：
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<IMG SRC="img00006-3.gif" tppabs="http://166.111.167.223/computer/cai/matlabjc/img7/img00006.gif">
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其中<FONT FACE="Times New Roman"> <I>a<b<c</I> </FONT>在上式的<FONT FACE="Times New Roman">
<I>b</I> </FONT>點即是代表要內插的點，<I><FONT FACE="Times New Roman">f(b)</FONT></I><FONT FACE="Times New Roman">
</FONT>則是要計算的內插函數值。下圖即是一個以二種內插
法的比較
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\<FONT FACE="Times New Roman">pcxfile[12cm,5cm]{fig9_1.pcx}</FONT>
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\caption{線性式與 spline 函數的曲線契合}<BR>
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線性內插是最簡單的內插方法，但其適用范圍很小；如果原來數據的函數<I>f</I>有極大的變化，假設其數據點之
間為線性變化并不合理。所以我們可以用二次、三次方程式或是另一種稱為spline函數來近似原來數據的函
數。MATLAB的一維內插函數是<FONT COLOR=#FF0000>interp1</FONT>，其語法為<FONT COLOR=#FF0000>interp1(x,y,xi)</FONT>,<FONT COLOR=#FF0000>interp1(x,y,xi,'method')</FONT>；其中的<FONT COLOR=#FF0000>x</FONT>,<FONT COLOR=#FF0000>y</FONT>是原已知的
數據的<I>x,y</I>值，而<FONT COLOR=#FF0000>xi</FONT>則是要內插的數據點，另外<FONT COLOR=#FF0000>method</FONT>可以設定內插方法有
<FONT COLOR=#FF0000>linear</FONT>,<FONT COLOR=#FF0000>cubic</FONT>,<FONT COLOR=#FF0000>spline</FONT>，分別是一次、三
次方程式和spline函數，其中預設方法是<FONT COLOR=#FF0000>linear</FONT>。如果數據的變化較大，以
<FONT COLOR=#FF0000>spline</FONT>函數內插所形成的曲線最平滑
，所以效果最好。而三次方程式所得到的內插曲線平滑度，則介于線性與<FONT COLOR=#FF0000>spline</FONT>函數之間。<BR>
<P>
我們以下面的例子說明。假設有一個汽車引擎在定轉速下，溫度與時間（單位為sec）的三次量測值如下<P>
<TABLE BORDER=1>
<TR><TD WIDTH=66><CENTER>time</CENTER></TD><TD WIDTH=72><CENTER>temp1</CENTER>
</TD><TD WIDTH=80><CENTER>temp2</CENTER></TD><TD WIDTH=72><CENTER>temp3</CENTER>
</TD></TR>
<TR><TD WIDTH=66><CENTER>0</CENTER></TD><TD WIDTH=72><CENTER>0</CENTER>
</TD><TD WIDTH=80><CENTER>0</CENTER></TD><TD WIDTH=72><CENTER>0</CENTER>
</TD></TR>
<TR><TD WIDTH=66><CENTER>1</CENTER></TD><TD WIDTH=72><CENTER>20</CENTER>
</TD><TD WIDTH=80><CENTER>110</CENTER></TD><TD WIDTH=72><CENTER>176</CENTER>
</TD></TR>
<TR><TD WIDTH=66><CENTER>2</CENTER></TD><TD WIDTH=72><CENTER>60</CENTER>
</TD><TD WIDTH=80><CENTER>180</CENTER></TD><TD WIDTH=72><CENTER>220</CENTER>
</TD></TR>
<TR><TD WIDTH=66><CENTER>3</CENTER></TD><TD WIDTH=72><CENTER>68</CENTER>
</TD><TD WIDTH=80><CENTER>240</CENTER></TD><TD WIDTH=72><CENTER>349</CENTER>
</TD></TR>
<TR><TD WIDTH=66><CENTER>4</CENTER></TD><TD WIDTH=72><CENTER>77</CENTER>
</TD><TD WIDTH=80><CENTER>310</CENTER></TD><TD WIDTH=72><CENTER>450</CENTER>
</TD></TR>
<TR><TD WIDTH=66><CENTER>5</CENTER></TD><TD WIDTH=72><CENTER>110</CENTER>
</TD><TD WIDTH=80><CENTER>405</CENTER></TD><TD WIDTH=72><CENTER>503</CENTER>
</TD></TR>
</TABLE>
<P>
其中溫度的數據從<FONT FACE="Times New Roman"> 20<SUP>o</SUP>C</FONT>變化到<FONT FACE="Times New Roman">
503<SUP>o</SUP>C</FONT>，如果要估計在<FONT FACE="Times New Roman">t=2.6,
4.9 sec </FONT>的溫度，可以下列指令計算
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<FONT COLOR=#FF0000 FACE="Times New Roman">>> x=[0 1 2 3
4 5]'; % </FONT><FONT COLOR=#FF0000>鍵入時間</FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman">>> y=[0 20 60
68 77 110]'; % </FONT><FONT COLOR=#FF0000>鍵入第一組時間</FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman">>> y1=interp1(x,y,2.6)
% </FONT><FONT COLOR=#FF0000>要內插的數據點為</FONT><FONT COLOR=#FF0000 FACE="Times New Roman">
2.6</FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman">y1 = % </FONT><FONT COLOR=#FF0000>對應</FONT><FONT COLOR=#FF0000 FACE="Times New Roman">
2.6 </FONT><FONT COLOR=#FF0000>的函數值為</FONT><FONT COLOR=#FF0000 FACE="Times New Roman">
64.8</FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman"> 64.8</FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman">>> y1=interp1(x,y,[2.6
4.9]) % </FONT><FONT COLOR=#FF0000>內插數據點為</FONT><FONT COLOR=#FF0000 FACE="Times New Roman">
2.6, 4.9</FONT><FONT COLOR=#FF0000>，注意用</FONT><FONT COLOR=#FF0000 FACE="Times New Roman">[
]</FONT><FONT COLOR=#FF0000>將多個內插點放在其中</FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman">y1 = </FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman"> 64.8</FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman"> 106.7 </FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman">>> y1=interp1(x,y,2.6,'cubic')
% </FONT><FONT COLOR=#FF0000>以三次方程式對數據點</FONT><FONT COLOR=#FF0000 FACE="Times New Roman">
2.6 </FONT><FONT COLOR=#FF0000>作內插</FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman">y1 = % </FONT><FONT COLOR=#FF0000>對應</FONT><FONT COLOR=#FF0000 FACE="Times New Roman">
2.6 </FONT><FONT COLOR=#FF0000>的函數值為</FONT><FONT COLOR=#FF0000 FACE="Times New Roman">
66.264</FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman"> 66.264</FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman">>> y1=interp1(x,y,2.6,'spline')
% </FONT><FONT COLOR=#FF0000>以</FONT><FONT COLOR=#FF0000 FACE="Times New Roman">spline</FONT><FONT COLOR=#FF0000>函數對數據點</FONT><FONT COLOR=#FF0000 FACE="Times New Roman">
2.6 </FONT><FONT COLOR=#FF0000>作內插</FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman">y1 = % </FONT><FONT COLOR=#FF0000>對應</FONT><FONT COLOR=#FF0000 FACE="Times New Roman">
2.6 </FONT><FONT COLOR=#FF0000>的函數值為</FONT><FONT COLOR=#FF0000 FACE="Times New Roman">
66.368</FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman"> 66.368<BR>
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以下的例子還配合繪圖功能，用以比較不同內插方法的差異。
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<FONT COLOR=#FF0000 FACE="Times New Roman">>> h=1:12;</FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman">>> temp=[5 8
9 15 25 29 31 30 22 25 27 24]; % </FONT><FONT COLOR=#FF0000>這組溫度數據變化較大</FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman">>> plot(h,temp,'--',h,temp,'+')
% </FONT><FONT COLOR=#FF0000>將線性內插結果繪圖</FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman">>> h_3=1:0.1:12
% </FONT><FONT COLOR=#FF0000>要每</FONT><FONT COLOR=#FF0000 FACE="Times New Roman">0.1</FONT><FONT COLOR=#FF0000>小時估計一次溫度值</FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman">>> t_3=interp1(h,temp,h_3,'cubic')
% </FONT><FONT COLOR=#FF0000>以三次方程式做內插</FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman">>> t_s=interp1(h,temp,h_3,'spline')
% </FONT><FONT COLOR=#FF0000>以</FONT><FONT COLOR=#FF0000 FACE="Times New Roman">spline</FONT><FONT COLOR=#FF0000>函數做內插</FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman">>> hold on</FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman">>> subplot(1,2,1)</FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman">>> plot(h,temp,'--',h,temp,'+',h_3,t_3)
% </FONT><FONT COLOR=#FF0000>將線性及三次方程式內插繪圖</FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman">>> subplot(1,2,2)</FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman">>> plot(h,temp,'--',h,temp,'+',h_3,t_s)
% </FONT><FONT COLOR=#FF0000>將線性方程式及</FONT><FONT COLOR=#FF0000 FACE="Times New Roman">spline</FONT><FONT COLOR=#FF0000>內插繪圖</FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman">>> hold off<BR>
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