實(shí)驗(yàn)一-函數(shù)插值方法報(bào)告

實(shí)驗(yàn)一函數(shù)插值方法報(bào)告一、問題提出對(duì)于給定的一元函數(shù)y=f(x)的n+1個(gè)節(jié)點(diǎn)值。=fj=0,1,…，〃。試用Lagrange公式求其插值多項(xiàng)式或分段二次Lagrange插值多項(xiàng)式。數(shù)據(jù)如下：xj0.40.550.650.800.951.05yj0.410750.578150.696750.901.001.25382求五次Lagrange多項(xiàng)式L5(x)，和分段三次插值多項(xiàng)式，計(jì)算f(0.596),f(0.99)的值。(提示：結(jié)果為f(°-596)X0.625732,f(0-99)X105423 )xjxj123yj0.3680.1350.05045670.0180.0070.0020.001試構(gòu)造Lagrange多項(xiàng)式L6(x)，計(jì)算的f(18,f(6.15)值。(提示：結(jié)果為f(1.8)X0.164762,f(6.15)X0.001266 )二、要求1、利用Lagrange插值公式l(x)二fn三x-1k=01i=0,i豐k k編寫出插值多項(xiàng)式程序；給出插值多項(xiàng)式或分段三次插值多項(xiàng)式的表達(dá)式；根據(jù)節(jié)點(diǎn)選取原則，對(duì)問題(2)用三點(diǎn)插值或二點(diǎn)插值，其結(jié)果如何；口(x-x)口(x-x)jj=0,j豐k(x)=f(x)+Xf[x,??.，x]?0 0kk=1其中：f[xn，…，七]二苫

0k，二0f⑷

i

口(x-x)ijj=0,j豐i三、目的和意義學(xué)會(huì)常用的插值方法，求函數(shù)的近似表達(dá)式，以解決其它實(shí)際問題；明確插值多項(xiàng)式和分段插值多項(xiàng)式各自的優(yōu)缺點(diǎn)；熟悉插值方法的程序編制；如果繪出插值函數(shù)的曲線，觀察其光滑性。四、實(shí)驗(yàn)學(xué)時(shí)：2學(xué)時(shí)五、實(shí)驗(yàn)步驟：1.進(jìn)入C或matlab開發(fā)環(huán)境；2．根據(jù)實(shí)驗(yàn)內(nèi)容和要求編寫程序；3．調(diào)試程序；4．運(yùn)行程序；5．撰寫報(bào)告,討論分析實(shí)驗(yàn)結(jié)果.解：、編寫插值函數(shù)結(jié)構(gòu)程序Lagrange插值多項(xiàng)式M文件：lagrangel.mfunction[A1,LN,L1,B1]=lagrange1(X,Y)m=length(X);LN=ones(m,m);fork=1:mx1=1;fori=1:mifk~=ix1=conv(x1,poly(X(i)))/(X(k)-X(i));endendL1(k,:)=x1;B1(k,:)=poly2sym(x1)endA1=Y*L1;LN=Y*B1分段三次艾爾米特插值多項(xiàng)式的M文件：Hermite3.mfunction[f,ff]=Hermite3(x,y,y1)symst;f=0.0;if(length(x)==length(y))if(length(y)==length(y1))n=length(x);elsedisp('y和y的導(dǎo)數(shù)的維數(shù)不相等’)；return;endelsedisp('x和y的維數(shù)不相等！’);return；endfori=1:nh=1.0;a=0.0;forj=1:nif(j~=i)h=h*(t-x(j))八2/((x(i)-x(j))八2);a=a+1/(x(i)-x(j));endendf=f+h*((x(i)-t)*(2*a*y(i)-y1(i))+y(i));endff=subs(f,'t');(1)、求五次Lagrange多項(xiàng)式L5(")，和分段三次插值多項(xiàng)式。在主顯示區(qū)，輸入五次Lagrange多項(xiàng)式L5(x)程序：>X=[0.40.550.650.800.951.05];>Y=[0.410750.578150.696750.901.001.25382];>[A1,LN,L1,B1]=lagrange1(X,Y)>plot(X,A1);>F=poly2sym(A1)運(yùn)行后，輸出五次Lagrange多項(xiàng)式L5(x)的結(jié)果：A1=121.6264-422.7503572.5667-377.2549121.9718-15.0845F=拉格朗日插值多項(xiàng)式L5(x)的圖如下：在主顯示區(qū)，輸入分段三次艾爾米特插值多項(xiàng)式L5(x)的程序:>x=[0.40.550.650.800.951.05];>y=[0.410750.578150.696750.901.001.25382];>y1=[2.34400.90321.43290.99030.91705.1439];>[f,ff]=Hermite3(x,y,y1);>ff運(yùn)行后，分段三次艾爾米特插值多項(xiàng)式L5(x)的輸出結(jié)果：ff=分段三次艾爾米特插值多項(xiàng)式L5(x)的圖如下：(2)、試構(gòu)造Lagrange多項(xiàng)式L6(x)。結(jié)果為f(1.8)氏0.164762f(6.15)氏0.001266在主顯示區(qū)，輸入程序：>X=[1234567];>Y=[0.3680.1350.0500.0180.0070.0020.001];>[A1,LN,L1,B1]=lagrange1(X,Y)>plot(X,A1);>>F=poly2sym(A1)運(yùn)行后，輸出結(jié)果的Lagrange多項(xiàng)式4(x)的結(jié)果：A1=0.0001 -0.0016 0.0186 -0.1175 0.4419 -0.9683 0.9950F=Lagrange多項(xiàng)式L6(x)的圖如下：二、計(jì)算函數(shù)值計(jì)算函數(shù)值的主程序：lagrangezhi.mfunction[y,R]=lagrangezhi(X,Y,x,M)n=length(X);m=length(x);fori=1:mz=x(i);s=0.0;fork=1:np=1.0;q1=1.0;c1=1.0;forj=1:nifj~=kp=p*(z-X(j))/(X(k)-X(j));endq1=abs(q1*(z-X(j)));c1=c1*j;ends=p*Y(k)+s;endy(i)=s;endR=M*q1/c1;(1)、計(jì)算fQ596)、f(0.99)的值。在主顯示區(qū)，輸入程序：>x=0.596;M=1;X=[0.4,0.55,0.65,0.80,0.95,1.05];>Y=[0.41075,0.57815,0.69675,0.90,1.00,1.25382];>[y,R]=lagrangezhi(X,Y,x,M)運(yùn)行結(jié)果：y=0.6257R=2.2170e-008在主顯示區(qū)，輸入程序：>>x=0.99;M=1;X=[0.4,0.55,0.65,0.80,0.95,1.05];>>Y=[0.41075,0.57815,0.69675,0.90,1.00,1.25382];>>[y,R]=lagrangezhi(X,Y,x,M)運(yùn)行結(jié)果：y=1.0542R=5.5901e-008(2)、計(jì)算f(18、f(6』5)的值在主顯示區(qū)，輸入程序：>>x=1.8;M=1;X=[1,2,3,4,5,6,7];>>Y=[0.368,0.135,0.050,0.018,0.007,0.002,0.001];>>[y,R]=lagrangezhi(X,Y,x,M)運(yùn)行結(jié)果：y=0.1648R=0.0059在主顯示區(qū)，輸入程序：>>x=6.15;M=1;X=[1,2,3,4,5,6,7];>>Y=[0.368,0.135,0.050,0.018,0.007,0.002,0.001];>>[y,R]=lagrangezhi(X,Y,x,M)運(yùn)行結(jié)果：y=0.0013R=0.0042三、Newton插值多項(xiàng)式Newton插值多項(xiàng)式主程序M文件：Newton.mfunction[A,C,L,wcgs,Cw]=Newton(X,Y)n=length(X);A=zeros(n,n);A(:,1)=Y';s=0.0;p=1.0;q=1.0;c1=1.0;forj=2:nfori=j:nA(i,j)=(A(i,j-1)-A(i-1,j-1))/(X(i)-X(i-j+1));endb=poly(X(j-1));q1=conv(q,b);c1=c1*j;q=q1;endC=A(n,n);b=poly(X(n));q1=conv(q1,b);fork=(n-1):-1:1C=conv(C,poly(X(k)));d=length(C);C(d)=C(d)+A(k,k);endL(k,:)=poly2sym(C);Q=poly2sym(q1);symsMwcgs=M*Q/c1;Cw=q1/c1;在主顯示區(qū)，輸入L5a)的程序：>x=[0.40.550.650.800.951.05];>y=[0.410750.578150.696750.901.001.25382];>[A,C,L,wcgs,Cw]=Newton(x,y)>symsx;>ezplot(L,[01.1]);運(yùn)行結(jié)果如下，得到L5a)A=0.4108000000.57821.116000000.69671.18600.28000000.90001.35500.67600.9900001.00000.6667-2.2944-7.4261-15.302001.25382.53827.486124.451463.7551121.6264121.6264-422.7503572.5667-377.2549121.9718-15.0845wcgs=Cw=0.0014 -0.0061 0.0110 -0.0103 0.0054 -0.0014 0.0002牛頓插值多項(xiàng)式L5a)的圖如下：在主顯示區(qū)，輸入L6(x)的程序：> x=[1234567];> y=[0.3680.1350.0500.0180.0070.0020.001];>[A,C,L,wcgs,Cw]=Newton(x,y)>symsx;>ezplot(L,[08]);運(yùn)行結(jié)果如下，得到L6(x)：A=0.36800000000.1350-0.2330000000.0500-0.08500.074000000.0180-0.03200.0265-0.01580000.0070-0.01100.0105-0.00530.0026000.0020-0.00500.0030-0.00250.0007-0.000400.0010-0.00100.0020-0.00030.0005-0.00000.0001C=0.0001 -0.0016 0.0186 -0.1175 0.4419 -0.9683 0.9950wcgs=(M*(xA7-28*xA6+322*xA5-1960*xA4+6769*xA3-13132*xA2+13068*x-5040))/5040Cw=0.0002 -0.0056 0.0639 -0.3889 1.3431 -2.6056 2.5929-1.0000牛頓插值多項(xiàng)式L6(*)的圖如下四、結(jié)果討論和分析通過上