📄 多项式与插值1.htm
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<html><head><title>学用MatLab</title><meta http-equiv="Content-Type" content="text/html; charset=gb2312"><style type="text/css"><!--body { font-family: "宋体"; font-size: 9pt; text-decoration: none}h2 { font-family: "楷体_GB2312"; font-size: 18pt; text-decoration: underline; color: #FF9999}h1 { font-family: "隶书"; font-size: 24pt; font-style: italic; font-weight: bolder; color: #CC66CC; text-decoration: blink}.explain { border-color: black black #00FF00; font-weight: bold; color: #333333}.code { font-family: "Arial", "Helvetica", "sans-serif"; font-size: 12pt; background-color: #FFFFFF; line-height: 24pt}h3 { font-size: 12pt; font-style: italic; font-weight: bold; color: #9999FF}--></style></head><body bgcolor="#CCFFCC" text="#666600" link="#009900" alink="#00FF00" vlink="#006600"><h1 align="center">多项式与插值</h1><h2>多项式</h2><h3>多项式的表达</h3><p>MatLab中用按降幂排列的多项式系数组成的行向量表示多项式,如:<br> p(x)=x^3-2x-5被表示为:</p><p class="code">p = [1 0 –2 –5]; </p><h3>多项式的根</h3><p class="code">r = roots(p)<br> r = <br> 2.0946 <br> –1.0473 + 1.1359i <br> –1.0473 – 1.1359i </p><p>根被储存为<span class="explain">列向量</span>．<br> 若要由方程的根构造多项式，则</p><p class="code">p2 = poly(r) <br> p2 = <br> 1 8.8818e-16 –2 –5 </p><h3>多项式估计</h3><p>可以用多项式估计出多项式在某一点的值：</p><p class="code">polyval(p,5) <br> ans = <br> 110</p><p>同样也可以估计矩阵多项式的值p(X) = X^3 – 2X – 5I,</p><p class="code">X = [2 4 5; –1 0 3; 7 1 5]; <br> Y = polyvalm(p,X) <br> Y = <br> 377 179 439 <br> 111 81 136 <br> 490 253 639</p><h3>卷积</h3><p>多项式相乘是一个卷积的过程,conv()</p><p class="code">a = [1 2 3]; <br> b = [4 5 6]; <br> c = conv(a,b) <br> c = <br> 4 13 28 27 18</p><p>多项式相除是其逆过程，用deconv():</p><p class="code">[q,r] = deconv(c,a) <br> q = <br> 4 5 6 <br> r = <br> 0 0 0 0 0</p><h3>多项式曲线逼近</h3><p>polyfit(x,y,n)能用多项式逼近由x,y向量提供的数据,n是其阶数,如:</p><p class="code">x = [1 2 3 4 5]; <br> y = [5.5 43.1 128 290.7 498.4];<br> p = polyfit(x,y,3) <br> p = <br> –0.1917 31.5821 –60.3262 35.3400 </p><p>将图画出</p><p class="code">x2 = 1:.1:5; <br> y2 = polyval(p,x2); <br> plot(x,y,’o’,x2,y2) <br> grid on </p><p><img src="image/poly1.jpg" width="531" height="404"> </p><h3>分式多项式分解</h3><p>residue()可将分式多项式分解如下:</p><p><img src="image/poly2.jpg" width="294" height="62"></p><p>对于下式</p><p><img src="image/poly3.jpg" width="200" height="66"></p><p>分解为:</p><p class="code">b = [–4 8]; <br> a = [1 6 8]; <br> [r,p,k] = residue(b,a) <br> r = <br> –12 8 <br> p = <br> –4 –2 <br> k = <br> []</p><p>重载此函数可以完成分式多项式相加：</p><p class="code">[b2,a2] = residue(r,p,k) <br> b2 = <br> –4 8 <br> a2 = <br> 1 6 8</p><h2>插值</h2><p>插值是在已知的数据列中，估计别点的函数值．</p><h3>一维插值</h3><p>一维插值在MatLab中有两种方法:<br> @ 多项式插值<br> @ 建立在FFT上的插值</p><p><span class="explain">多项式插值</span></p><p>yi = interp1(x,y,xi,method)<br> x是坐标向量,y是数据向量,xi是待估计点向量,method是插值方法,<br> method有四种:<br> 1.nearest 寻找最近数据点,由其得出函数值;<br> 2.linear 线性插值(该函数的默认方法);<br> 3.spline 样条插值,数据点处光滑--左导等于右导;<br> 4.cubic 三次插值</p><p>以上四种方法得出的数据值一个比一个精确,而所需内存及计算时间也一个比一个要大要长.</p><p><span class="explain">建立在FFT上的插值</span></p><p>这种方法利用了快速傅立叶变换</p><p>y = interpft(x,n),其中,x含有周期性的函数值.</p><h3>二维插值</h3><p>ZI = interp2(X,Y,Z,XI,YI,method)<br> method有三种:<br> 1.nearest 寻找最近数据点,由其得出函数值;<br> 2.linear 二维线性插值<br> 3.cubic 二维三次插值</p><p>下面来看看二维插值的例子:</p><p>先创造数据点:</p><p class="code">[x,y] = meshgrid(–3:1:3); <br> z = peaks(x,y); <br> surf(x,y,z)</p><p><img src="image/poly4.jpg" width="559" height="406"></p><p>再比较一下不同的插值</p><p class="code">[xi,yi] = meshgrid(–3:0.25:3);<br> zi1 = interp2(x,y,z,xi,yi,'nearest');<br> zi2 = interp2(x,y,z,xi,yi,'bilinear');<br> zi3 = interp2(x,y,z,xi,yi,'bicubic');</p><p><img src="image/poly51.jpg" width="462" height="396"><br> <img src="image/poly52.jpg" width="466" height="418"><br> <img src="image/poly53.jpg" width="495" height="411"></p><h3>三维及多维插值</h3><p>列出函数,其余从略</p><p>VI = interp3(X,Y,Z,V,XI,YI,ZI,method)<br> VI = interpn(X1,X2,X3...,V,Y1,Y2,Y3,...,method)<br></p><p> </p><p> </p><p> </p><p> </p></body></html>⌨️ 快捷键说明
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