多項式插值實驗報告.doc

實驗報告 實驗報告一題目：多項式插值 摘要：熟悉插值多項式構造，通過計算機解決實驗問題；龍格現(xiàn)象的發(fā)生、防止，通過拉格朗日插值、分段線性插值以及三次樣條插值進行插值效果的比較通過分析、推導，掌握數(shù)據(jù)插值的思想方法；通過對插值方法的進一步討論，了解插值的“龍格”現(xiàn)象；熟悉常用的分段線性插值和樣條插值的使用方法；掌握上機編程與調試能力。由于高次多項式插值不收斂，會產(chǎn)生Runge現(xiàn)象，本實驗在給出具體的實例后，采用分段線性插值和三次樣條插值的方法有效的克服了這一現(xiàn)象，而且還取的很好的插值效果。前言：（目的和意義）1. 掌握拉格朗日插值法，分段線性插值法，三次樣條插值法。2. 深刻認識多項式插值的缺點。3. 明確插值的不收斂性怎樣克服。4. 明確精度與節(jié)點和插值方法的關系。數(shù)學原理：在給定n+1個節(jié)點和相應的函數(shù)值以后構造n次的Lagrange插值多項式，實驗結果表明（見后面的圖）這種多項式并不是隨著次數(shù)的升高對函數(shù)的逼近越來越好，這種現(xiàn)象就是Rung現(xiàn)象。解決Rung現(xiàn)象的方法通常有分段線性插值、三次樣條插值等方法。分段線性插值：設在區(qū)間a, b上，給定n+1個插值節(jié)點a=x0x1xn=b和相應的函數(shù)值y0，y1，yn，求作一個插值函數(shù)，具有如下性質：1) ，j=0，1，n。2) 在每個區(qū)間xi, xj上是線性連續(xù)函數(shù)。則插值函數(shù)稱為區(qū)間a, b上對應n個數(shù)據(jù)點的分段線性插值函數(shù)。三次樣條插值：給定區(qū)間a, b一個分劃 ：a=x0x1xN=b 若函數(shù)S(x)滿足下列條件：1) S(x)在每個區(qū)間xi, xj上是不高于3次的多項式。2) S(x)及其2階導數(shù)在a, b上連續(xù)。則稱S(x)使關于分劃的三次樣條函數(shù)。程序設計：本實驗采用Matlab的M文件編寫。其中待插值的方程寫成function的方式，如下function y=f(x);y=1/（1+25*x*x）;寫成如上形式即可，下面給出主程序Lagrange插值源程序：function p=Lag_polyfit(X,Y)%Matlab函數(shù)文件Lag_polyfit.m%拉格朗日插值法多項式擬合 P17%p=Lag_polyfit(X,Y)% X 擬合自變量% Y 擬合函數(shù)值% p 所得的擬合多項式系數(shù)if size(X)=size(Y)error(變量不匹配);end %如果要擬合的函數(shù)值與自變量維數(shù)不一樣,則退出報錯tic %開始記時format long g %設置最合適的數(shù)字格式r=size(Y); n=r(2); %n為要擬合的數(shù)據(jù)長度p=zeros(1,n); %保存所得多項式系數(shù)p0=p; b=0; %工作變量W=poly(X); %W為以X為根的多項式dW=polyder(W); %dW為對多項式W 求導后的多項式系數(shù)z=polyval(dW,X); %z為以dW 為系數(shù)的多項式對X的值A=1 1; r=A; %A,r為長度為2 的(1,2)向量for i=1:n %計算循環(huán)開始A=1,-X(i); %A為一次多項式x-x(i)系數(shù)p0,r=deconv(W,A); %進行多項式除法W/A,p0為商b=Y(i)/z(i);p0=b.*p0; %p0為累加項p=p+p0;end %循環(huán)結束disp(toc) %分段線性插值源程序clearn=input(將區(qū)間分為的等份數(shù)輸入：n);s=-1+2/n*0:n; %給定的定點，Rf為給定的函數(shù)m=0;hh=0.001;for x=-1:hh:1; ff=0; for k=1:n+1; %求插值基函數(shù) switch k case 1 if xs(n); l=(x-s(n)./(s(n+1)-s(n); else l=0; end otherwise if x=s(k-1)&x=s(k)&x=s(k+1); l=(x-s(k+1)./(s(k)-s(k+1); else l=0; end end end ff=ff+Rf(s(k)*l; %求插值函數(shù)值 end m=m+1; f(m)=ff;end %作出曲線x=-1:hh:1;plot(x,f,r);grid onhold on 三次樣條插值源程序：（采用第一邊界條件）clearn=input(將區(qū)間分為的等份數(shù)輸入：n);%插值區(qū)間a=-1;b=1;hh=0.001; %畫圖的步長s=a+(b-a)/n*0:n; %給定的定點，Rf為給定的函數(shù)%第一邊界條件Rf(-1),Rf(1)v=5000*1/(1+25*a*a)3-50/(1+25*a*a)4;for k=1:n; %取出節(jié)點間距 h(k)=s(k+1)-s(k);endfor k=1:n-1; %求出系數(shù)向量lamuda,miu la(k)=h(k+1)/(h(k+1)+h(k); miu(k)=1-la(k);end%賦值系數(shù)矩陣Afor k=1:n-1; for p=1:n-1; switch p case k A(k,p)=2; case k-1 A(k,p)=miu(p+1); case k+1 A(k,p)=la(p-1); otherwise A(k,p)=0; end endend%求出d陣for k=1:n-1; switch k case 1 d(k)=6*f2c(s(k) s(k+1) s(k+2)-miu(k)*v; case n-1 d(k)=6*f2c(s(k) s(k+1) s(k+2)-la(k)*v; otherwise d(k)=6*f2c(s(k) s(k+1) s(k+2); endend%求解M陣M=Ad;M=v;M;v;%m=0;f=0;for x=a:hh:b; if x=a; p=1; else p=ceil(x-s(1)/(b-a)/n); end ff1=0; ff2=0; ff3=0; ff4=0; m=m+1; ff1=1/h(p)*(s(p+1)-x)3*M(p)/6; ff2=1/h(p)*(x-s(p)3*M(p+1)/6; ff3=(Rf(s(p+1)-Rf(s(p)/h(p)-h(p)*(M(p+1)-M(p)/6)*(x-s(p); ff4=Rf(s(p)-M(p)*h(p)*h(p)/6; f(m)=ff1+ff2+ff3+ff4 ; end %作出插值圖形x=a:hh:b;plot(x,f,k)hold ongrid on 結果分析和討論：本實驗采用函數(shù)進行數(shù)值插值，插值區(qū)間為-1，1，給定節(jié)點為xj=-1+jh,h=0.1，j=0,，n。下面分別給出Lagrange插值，三次樣條插值，線性插值的函數(shù)曲線和數(shù)據(jù)表。圖中只標出Lagrange插值的十次多項式的曲線，其它曲線沒有標出，從數(shù)據(jù)表中可以看出具體的誤差。表中，L10(x)為Lagrange插值的10次多項式，S10(x)，S40(x)分別代表n=10，40的三次樣條插值函數(shù)，X10(x)，X40(x)分別代表n=10，40的線性分段插值函數(shù)。x f(x) L10(x) S10(x) S40(x) X10(x) X40(x) -1.00000000000000 0.03846153846154 0.03846153846154 0.03846153846154 0.03846153846154 0.03846153846154 0.03846153846154 -0.95000000000000 0.04244031830239 1.92363114971920 0.04240833151040 0.04244031830239 0.04355203619910 0.04244031830239 -0.90000000000000 0.04705882352941 1.57872099034926 0.04709697585458 0.04705882352941 0.04864253393665 0.04705882352941 -0.85000000000000 0.05245901639344 0.71945912837982 0.05255839923979 0.05245901639344 0.05373303167421 0.05245901639344 -0.80000000000000 0.05882352941176 0.05882352941176 0.05882352941176 0.05882352941176 0.05882352941176 0.05882352941176 -0.75000000000000 0.06639004149378 -0.23146174989674 0.06603986172744 0.06639004149378 0.06911764705882 0.06639004149378 -0.70000000000000 0.07547169811321 -0.22619628906250 0.07482116198866 0.07547169811321 0.07941176470588 0.07547169811321 -0.65000000000000 0.08648648648649 -0.07260420322418 0.08589776360849 0.08648648648649 0.08970588235294 0.08648648648649 -0.60000000000000 0.10000000000000 0.10000000000000 0.10000000000000 0.10000000000000 0.10000000000000 0.10000000000000 -0.55000000000000 0.11678832116788 0.21559187891257 0.11783833017713 0.11678832116788 0.12500000000000 0.11678832116788 -0.50000000000000 0.13793103448276 0.25375545726103 0.14004371555730 0.13793103448276 0.15000000000000 0.13793103448276 -0.45000000000000 0.16494845360825 0.23496854305267 0.16722724315883 0.16494845360825 0.17500000000000 0.16494845360825 -0.40000000000000 0.20000000000000 0.20000000000000 0.20000000000000 0.20000000000000 0.20000000000000 0.20000000000000 -0.35000000000000 0.24615384615385 0.19058046675376 0.24054799403464 0.24615384615385 0.27500000000000 0.24615384615385 -0.30000000000000 0.30769230769231 0.23534659131080 0.29735691695860 0.30769230769231 0.35000000000000 0.30769230769231 -0.25000000000000 0.39024390243902 0.34264123439789 0.38048738140327 0.39024390243902 0.42500000000000 0.39024390243902 -0.20000000000000 0.50000000000000 0.50000000000000 0.50000000000000 0.50000000000000 0.50000000000000 0.50000000000000 -0.15000000000000 0.64000000000000 0.67898957729340 0.65746969368431 0.64000000000000 0.62500000000000 0.64000000000000 -0.10000000000000 0.80000000000000 0.84340742982890 0.82052861660828 0.80000000000000 0.75000000000000 0.80000000000000 -0.05000000000000 0.94117647058824 0.95862704866073 0.94832323122810 0.94117647058824 0.87500000000000 0.94117647058824 0 1.00000000000000 1.00000000000000 1.00000000000000 1.00000000000000 1.00000000000000 1.00000000000000 0.05000000000000 0.94117647058824 0.95862704866073 0.94832323122810 0.94117647058824 0.87500000000000 0.94117647058824 0.10000000000000 0.80000000000000 0.84340742982890 0.82052861660828 0.80000000000000 0.75000000000000 0.80000000000000 0.15000000000000 0.64000000000000 0.67898957729340 0.65746969368431 0.64000000000000 0.62500000000000 0.64000000000000 0.20000000000000 0.50000000000000 0.50000000000000 0.50000000000000 0.50000000000000 0.50000000000000 0.50000000000000 0.25000000000000 0.39024390243902 0.34264123439789 0.38048738140327 0.39024390243902 0.42500000000000 0.39024390243902 0.30000000000000 0.30769230769231 0.23534659131080 0.29735691695860 0.30769230769231 0.35000000000000 0.30769230769231 0.35000000000000 0.24615384615385 0.19058046675376 0.24054799403464 0.24615384615385 0.27500000000000 0.24615384615385 0.40000000000000 0.20000000000000 0.20000000000000 0.20000000000000 0.20000000000000 0.20000000000000 0.20000000000000 0.45000000000000 0.16494845360825 0.23496854305267 0.16722724315883 0.16494845360825 0.17500000000000 0.16494845360825 0.50000000000000 0.13793103448276 0.25375545726103 0.14004371555730 0.13793103448276 0.15000000000000 0.13793103448276 0.55000000000000 0.11678832116788 0.21559187891257 0.11783833017713 0.11678832116788 0.12500000000000 0.11678832116788 0.60000000000000 0.10000000000000 0.10000000000000 0.10000000000000 0.10000000000000 0.10000000000000 0.10000000000000 0.65000000000000 0.08648648648649 -0.07260420322418 0.08589776360849 0.08648648648649 0.08970588235294 0.08648648648649 0.70000000000000 0.07547169811321 -0.22619628906250 0.07482116198866 0.07547169811321 0.07941176470588 0.07547169811321 0.75000000000000 0.06639004149378 -0.23146174989674 0.06603986172744 0.06639004149378 0.06911764705882 0.06639004149378 0.80000000000000 0.05882352941176 0.05882352941176 0.05882352941176 0.058823529411