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<h1 align="center">数值分析</h1>
<p>本节介绍的是关于函数的函数(function functions),这些函数是用来处理函数而非数值的;</p>
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<td width="24%" height="41">类别</td>
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<td width="25%">函数</td>
<td width="75%">描述</td>
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<p>绘图</p>
<p>优化</p>
<p>求解</p>
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<td width="25%" height="27">fplot</td>
<td width="75%" height="27">画出函数</td>
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<td width="25%" height="32">fminbnd</td>
<td width="75%" height="32">由一有范围限制的变量找出函数的最小值</td>
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<td height="34" width="25%">fminsearch</td>
<td height="34" width="75%">由几个变量找出函数的最小值</td>
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<td width="25%">fzero</td>
<td width="75%">找出函数的解(零值)</td>
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<td width="24%" height="107">数值积分</td>
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<td width="25%" height="35">quad </td>
<td width="75%" height="35">低阶数值估计积分</td>
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<td width="25%" height="32">quad8</td>
<td width="75%" height="32">高阶数值估计积分</td>
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<td width="25%">dblquad</td>
<td width="75%">二重积分</td>
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<td width="24%" height="32">数值微分</td>
<td width="76%" height="32">见下一章</td>
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<h2>MatLab中的函数表达</h2>
<p>MatLab中<span class="explain">用M文件来表示函数</span>,设有如下函数:</p>
<p><img src="image/fun1.jpg" width="342" height="63"></p>
<p>他别表示为一称为hump.m的文件中:</p>
<p class="code">function y = humps(x) <br>
y = 1./((x – 0.3).^2 + 0.01) + 1./((x – 0.9).^2 + 0.04) – 6;</p>
<p>这个函数文件可用于数值分析的函数中.<br>
第二种方法就是<span class="explain">创造一个行内对象(inline())</span>,方法如下:</p>
<p class="code">f = inline(‘1./((x–0.3).^2 + 0.01) + 1./((x–0.9).^2 + 0.04)–6’);</p>
<p>用了上面的方法创造了函数文件,我们就可以找出函数在2的值:</p>
<p class="code">f(2.0) <br>
ans = <br>
–4.8552 </p>
<p>用创造<span class="explain">行内对象</span>的方法还可以<span class="explain">创造多参数的函数</span>,如下:</p>
<p class="code">f= inline('y*sin(x)+x*cos(y)','x','y') <br>
f(pi,2*pi) <br>
ans = <br>
3.1416</p>
<h2>把函数画出来</h2>
<p>fplot()可画出在给定范围内的函数值,如下</p>
<p class="code">fplot('humps',[–5 5]) <br>
grid on</p>
<p><img src="image/fun2.jpg" width="511" height="408"></p>
<p>可通过限制y轴来放大图形</p>
<p>fplot('humps',[–5 5 –10 25]) <br>
grid on</p>
<p><img src="image/fun3.jpg" width="506" height="390"></p>
<p>你也可直接<span class="explain">在fplot()中传递表达式</span>,如:</p>
<p class="code">fplot('2*sin(x+3)',[–1 1])</p>
<p>更可在一附图中画多个函数,如下</p>
<p class="code">fplot('[2*sin(x+3), humps(x)]',[–1 1])</p>
<p>式中,[2*sin(x+3), humps(x)]组成了一个矩阵,每一列都是对应于x的函数</p>
<h2>函数的最小值与解</h2>
<h3>找出一变量的函数的极值</h3>
<p class="code">x = fminbnd(’humps’,0.3,1)<br>
x = <br>
0.6370 </p>
<p>你可通过向fminbnd()函数传递一个函数optimset()作为参数来把此过程显示为列表形式:</p>
<p class="code">x = fminbnd(’humps’,0.3,1,optimset(’Display’,’iter’))<br>
Func-count x f(x) Procedure <br>
1 0.567376 12.9098 initial <br>
2 0.732624 13.7746 golden <br>
3 0.465248 25.1714 golden <br>
4 0.644416 11.2693 parabolic <br>
5 0.6413 11.2583 parabolic <br>
6 0.637618 11.2529 parabolic <br>
7 0.636985 11.2528 parabolic <br>
8 0.637019 11.2528 parabolic <br>
9 0.637052 11.2528 parabolic <br>
x = <br>
0.6370 </p>
<h3>多变量函数极值</h3>
<p>先创造一个ｍ文件，three_var.m:</p>
<p class="code">function b = three_var(v) <br>
x = v(1); <br>
y = v(2); <br>
z = v(3); <br>
b = x.^2 + 2.5*sin(y) – z^2*x^2*y^2;</p>
<p>现在,以x = –0.6, y = –1.2,z = 0.135为起始点找出函数的极值:</p>
<p class="code">v = [–0.6 –1.2 0.135]; <br>
a = fminsearch('three_var',v) <br>
a = <br>
0.0000 –1.5708 0.1803</p>
<h3>设置寻找极值的参数</h3>
<p><span class="explain">x = fminbnd(fun,x1,x2,options)</span>或<br>
<span class="explain">x = fminsearch(fun,x0,options) </span></p>
<p>其中，options是优化工具箱中(Optimization Toolbox)中的函数所用的一个结构,可如下设置</p>
<p class="explain">options = optimset('Display','iter');</p>
<p>options.Display用来设置是否显示中间过程,如为:"iter"则显示,为"off"则不显示,为"final"则只显示最后结果;<br>
options.To1X设置结果的误差范围,默认值是:1.e–4.<br>
options.MaxFunEval设置函数运行次数的上限,默认fminbnd()是500次,fminsearch()是200*length(x0)次</p>
<h3>找出函数的解(零点值)</h3>
<p><span class="explain">fzero()</span>找出函数的零点值,你可以给出一个<span class="explain">起始点</span>,函数会从点开始搜索直到找到一个异号的值,最终给出解;<br>
如果你知道函数会于哪两点异号,你可以给出一个<span class="explain">两点的向量,表明起始值和起始搜索步长</span></p>
<p class="code">a = fzero('humps',–0.2) <br>
a = <br>
–0.1316 </p>
<p>验证一下，此函数值的确很接近０，</p>
<p class="code">humps(a) <br>
ans = <br>
8.8818e –16</p>
<p>再看看下面的命令，看看你是否能看懂！</p>
<p class="code">humps(1) <br>
ans = <br>
16 <br>
humps(–1) <br>
ans = <br>
–5.1378<br>
options = optimset('Display','iter'); <br>
a = fzero('humps',[–1 1],options) <br>
Func-count x f(x) Procedure <br>
1 –1 –5.13779 initial <br>
1 1 16 initial <br>
2 –0.513876 –4.02235 interpolation <br>
3 0.243062 71.6382 bisection <br>
4 –0.473635 –3.83767 interpolation <br>
5 –0.115287 0.414441 bisection <br>
6 –0.150214 –0.423446 interpolation <br>
7 –0.132562 –0.0226907 interpolation <br>
8 –0.131666 –0.0011492 interpolation <br>
9 –0.131618 1.88371e–07 interpolation <br>
10 –0.131618 –2.7935e–11 ⌨️ 快捷键说明
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