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<TITLE> 一维内插 </TITLE>
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<H1>7.1.1 一维内插</H1>
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线性内插是假设在二个已知数据中的变化为线性关系，因此可由已知二点的座标(<I>a, b</I>)去计算通过这二点的
斜线，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>点即是代表要内插的点，<I><FONT FACE="Times New Roman">f(b)</FONT></I><FONT FACE="Times New Roman">
</FONT>则是要计算的内插函数值。下图即是一个以二种内插
法的比较
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\<FONT FACE="Times New Roman">pcxfile[12cm,5cm]{fig9_1.pcx}</FONT>
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\caption{线性式与 spline 函数的曲线契合}<BR>
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线性内插是最简单的内插方法，但其适用范围很小；如果原来数据的函数<I>f</I>有极大的变化，假设其数据点之
间为线性变化并不合理。所以我们可以用二次、三次方程式或是另一种称为spline函数来近似原来数据的函
数。MATLAB的一维内插函数是<FONT COLOR=#FF0000>interp1</FONT>，其语法为<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
数据的<I>x,y</I>值，而<FONT COLOR=#FF0000>xi</FONT>则是要内插的数据点，另外<FONT COLOR=#FF0000>method</FONT>可以设定内插方法有
<FONT COLOR=#FF0000>linear</FONT>,<FONT COLOR=#FF0000>cubic</FONT>,<FONT COLOR=#FF0000>spline</FONT>，分别是一次、三
次方程式和spline函数，其中预设方法是<FONT COLOR=#FF0000>linear</FONT>。如果数据的变化较大，以
<FONT COLOR=#FF0000>spline</FONT>函数内插所形成的曲线最平滑
，所以效果最好。而三次方程式所得到的内插曲线平滑度，则介于线性与<FONT COLOR=#FF0000>spline</FONT>函数之间。<BR>
<P>
我们以下面的例子说明。假设有一个汽车引擎在定转速下，温度与时间（单位为sec）的三次量测值如下<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>
其中温度的数据从<FONT FACE="Times New Roman"> 20<SUP>o</SUP>C</FONT>变化到<FONT FACE="Times New Roman">
503<SUP>o</SUP>C</FONT>，如果要估计在<FONT FACE="Times New Roman">t=2.6,
4.9 sec </FONT>的温度，可以下列指令计算
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<FONT COLOR=#FF0000 FACE="Times New Roman">>> x=[0 1 2 3
4 5]'; % </FONT><FONT COLOR=#FF0000>键入时间</FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman">>> y=[0 20 60
68 77 110]'; % </FONT><FONT COLOR=#FF0000>键入第一组时间</FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman">>> y1=interp1(x,y,2.6)
% </FONT><FONT COLOR=#FF0000>要内插的数据点为</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>对应</FONT><FONT COLOR=#FF0000 FACE="Times New Roman">
2.6 </FONT><FONT COLOR=#FF0000>的函数值为</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>内插数据点为</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>将多个内插点放在其中</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>以三次方程式对数据点</FONT><FONT COLOR=#FF0000 FACE="Times New Roman">
2.6 </FONT><FONT COLOR=#FF0000>作内插</FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman">y1 = % </FONT><FONT COLOR=#FF0000>对应</FONT><FONT COLOR=#FF0000 FACE="Times New Roman">
2.6 </FONT><FONT COLOR=#FF0000>的函数值为</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>函数对数据点</FONT><FONT COLOR=#FF0000 FACE="Times New Roman">
2.6 </FONT><FONT COLOR=#FF0000>作内插</FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman">y1 = % </FONT><FONT COLOR=#FF0000>对应</FONT><FONT COLOR=#FF0000 FACE="Times New Roman">
2.6 </FONT><FONT COLOR=#FF0000>的函数值为</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>
</FONT>
<P>
以下的例子还配合绘图功能，用以比较不同内插方法的差异。
<P>
<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>这组温度数据变化较大</FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman">>> plot(h,temp,'--',h,temp,'+')
% </FONT><FONT COLOR=#FF0000>将线性内插结果绘图</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>小时估计一次温度值</FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman">>> t_3=interp1(h,temp,h_3,'cubic')
% </FONT><FONT COLOR=#FF0000>以三次方程式做内插</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>函数做内插</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>将线性及三次方程式内插绘图</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>将线性方程式及</FONT><FONT COLOR=#FF0000 FACE="Times New Roman">spline</FONT><FONT COLOR=#FF0000>内插绘图</FONT>
<P>
<FONT COLOR=#FF0000 FACE="Times New Roman">>> hold off<BR>
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