數(shù)值分析-曲線擬合

數(shù)學(xué)與計(jì)算科學(xué)學(xué)院《數(shù)值分析》實(shí)驗(yàn)報(bào)告題目：曲線擬合專業(yè)：信息與計(jì)算科學(xué)學(xué)號(hào)：姓名：指導(dǎo)教師：成績(jī)：二零一五年四月二十四日《數(shù)值分析》實(shí)驗(yàn)報(bào)告（2）學(xué)號(hào)：姓名：班級(jí)：成績(jī)：實(shí)驗(yàn)名稱：曲線擬合實(shí)驗(yàn)地點(diǎn)：化工樓317所使用的工具軟件及環(huán)境：Matlab實(shí)驗(yàn)任務(wù)與目的學(xué)習(xí)使用最小二乘法的原理了解法方程的特性理解和分析生成的過(guò)程以及結(jié)果分析二、實(shí)驗(yàn)內(nèi)容或問(wèn)題描述通過(guò)實(shí)驗(yàn)的觀察和測(cè)量給出離散的一些點(diǎn)，再來(lái)求出近似的函數(shù)解析式。最小二乘法是求解這一問(wèn)題的很好的方法，本實(shí)驗(yàn)運(yùn)用這一方法實(shí)現(xiàn)對(duì)給定數(shù)據(jù)的擬合。解題思路與方法，或解題詳細(xì)步驟1.擬合曲線三次多項(xiàng)式nm=3+1,基函數(shù)｛1，x，x^2，x^3｝2.擬合節(jié)點(diǎn)數(shù)：np=10+13.擬合數(shù)據(jù)：xp[np],yp[np],nm矩陣階數(shù)4.基函數(shù)在各擬合節(jié)點(diǎn)上值構(gòu)成向量xe(i)，全部基函數(shù)向量構(gòu)成二維矩陣xe5.正則方程系數(shù)矩陣G,常數(shù)項(xiàng)b6.解方程組，求基函數(shù)的系數(shù)a(1),a(2),a(3),a(4)四、程序設(shè)計(jì)代碼%%擬合曲線【-1，1】，f(x)=1/(1+25x^2）%準(zhǔn)確解x=-1:0.01:1,n=2/0.01+1;y=f(x)x=-1:0.01:1;n=2/0.01+1;fori=1:ny(i)=1/(1+25*x(i)^2);end;%%擬合曲線三次多項(xiàng)式nm=3+1,基函數(shù)｛1，x，x^2，x^3｝，擬合節(jié)點(diǎn)數(shù)：np=10+1%%擬合數(shù)據(jù)：xp[np],yp[np],nm矩陣階數(shù)nm=4;np=10+1;%nm=2;%np=4+1;%xp=1:5;%yp=[4,4.5,6,8,8.5];%w=[2,1,3,1,1];%xp=-1:2/(np-1):1;fori=1:npxp(i)=-1+0.2*(i-1);yp(i)=1/(1+25*xp(i)^2);end;w=ones(1,np);%擬合曲線步驟一，基函數(shù)在各擬合節(jié)點(diǎn)上值構(gòu)成向量xe(i)，全部基函數(shù)向量構(gòu)成二維矩陣xexe=ones(nm,np);t=xe(1,:);fori=2:nmt=t.*xp;xe(i,:)=t;end;xe%擬合曲線步驟二，正則方程系數(shù)矩陣G,常數(shù)項(xiàng)bfori=1:nmforj=1:iG(i,j)=sum(xe(i,:).*xe(j,:).*w);G(j,i)=G(i,j);end;b(i)=sum(xe(i,:).*yp.*w);end;%擬合曲線步驟三，解方程組，求基函數(shù)的系數(shù)a(1),a(2),a(3),a(4)a=b/G;Gbapause%%求擬合曲線上各點(diǎn)x=-1:0.01:1的近似值：s(x),點(diǎn)下標(biāo)k=1:n:近似值s(x(k))fork=1:ns(k)=a(1);xk=1;fori=2:nm;xk=xk*x(k);s(k)=s(k)+a(i)*xk;end;end;%%繪制函數(shù)曲線y=f(x)和擬合曲線s=s(x)figure(1);plot(x,y,'*');holdon;plot(x,s,'-');holdon;plot(xp,yp,'.');%%繪制誤差曲線figure(2);plot(x,y-s,'r');五、運(yùn)行結(jié)果>>quxianyihexe=Columns1through81.00001.00001.00001.00001.00001.00001.00001.0000-1.0000-0.8000-0.6000-0.4000-0.200000.20000.40001.00000.64000.36000.16000.040000.04000.1600-1.0000-0.5120-0.2160-0.0640-0.008000.00800.0640Columns9through111.00001.00001.00000.60000.80001.00000.36000.64001.00000.21600.51201.0000G=11.00000.00004.40000.00000.00004.40000.00003.13284.40000.00003.132800.00003.132802.6259b=2.7946-0.00000.32820.0000a=0.4841-0.0000-0.57520.0000quxianyihexe=Columns1through81.00001.00001.00001.00001.00001.00001.00001.0000-1.0000-0.8000-0.6000-0.4000-0.200000.20000.40001.00000.64000.36000.16000.040000.04000.1600-1.0000-0.5120-0.2160-0.0640-0.008000.00800.0640Columns9through111.00001.00001.00000.60000.80001.00000.36000.64001.00000.21600.51201.0000G=11.00000.00004.40000.00000.00004.40000.00003.13284.40000.00003.132800.00003.132802.6259b=2.7946-0.00000.32820.0000a=0.4841-0.0000-0.57520.0000quxianyihexe=Columns1through81.00001.00001.00001.00001.00001.00001.00001.0000-1.0000-0.8000-0.6000-0.4000-0.200000.20000.40001.00000.64000.36000.16000.040000.04000.1600-1.0000-0.5120-0.2160-0.0640-0.008000.00800.0640Columns9through111.00001.00001.00000.60000.80001.00000.36000.64001.00000.21600.51201.0000G=11.00000.00004.40000.00000.00004.40000.00003.13284.40000.00003.132800.00003.132802.6259b=2.7946-0.00000.32820.0000a=0.4841-0.0000-0.57520.0000quxianyihexe=Columns1through81.00001.00001.00001.00001.00001.00001.00001.0000-1.0000-0.8000-0.6000-0.4000-0.200000.20000.40001.00000.64000.36000.16000.040000.04000.1600-1.0000-0.5120-0.2160-0.0640-0.008000.00800.0640Columns9through111.00001.00001.00000.60000.80001.00000.36000.64001.00000.21600.51201.0000G=11.00000.00004.40000.00000.00004.40000.00003.13284.40000.00003.132800.00003.132802.6259b=2.7946-0.00000.32820.0000a=0.4841-0.0000-0.57520.0000quxianyihexe=Columns1through81.00001.00001.00001.00001.00001.00001.00001.0000-1.0000-0.8000-0.6000-0.4000-0.200000.20000.40001.00000.64000.36000.16000.040000.04000.1600-1.0000-0.5120-0.2160-0.0640-0.008000.00800.0640Columns9through111.00001.00001.00000.60000.80001.00000.36000.64001.00000.21600.51201.0000G=11.00000.00004.40000.00000.00004.40000.00003.13284.40000.00003.132800.00003.132802.6259b=2.7946-0.00000.32820.0000a=0.4841-0.0000-0.57520.0000quxianyihexe=Columns1through81.00001.00001.00001.00001.00001.00001.00001.0000-1.0000-0.8000-0.6000-0.4000-0.200000.20000.40001.00000.64000.36000.16000.040000.04000.1600-1.0000-0.5120-0.2160-0.0640-0.008000.00800.0640Columns9through111.00001.00001.00000.60000.80001.00000.36000.64001.00000.21600.51201.0000G=11.00000.00004.40000.00000.00004.40000.00003.13284.40000.00003.132800.00003.132802.6259b=2.7946-0.00000.32820.0000a=0.4841-0.0000-0.57520.0000quxianyihexe=Columns1through81.00001.00001.00001.00001.00001.00001.00001.0000-1.0000-0.8000-0.6000-0.4000-0.200000.20000.40001.00000.64000.36000.16000.040000.04000.1600-1.0000-0.5120-0.2160-0.0640-0.008000.00800.0640Columns9through111.00001.00001.00000.60000.80001.00000.36000.64001.00000.21600.51201.0000G=11.00000.00004.40000.00000.00004.40000.00003.13284.40000.0000