數(shù)值分析-曲線擬合

數(shù)學與計算科學學院《數(shù)值分析》實驗報告題目：曲線擬合專業(yè)：信息與計算科學學號：姓名：指導教師：成績：二零一五年四月二十四日《數(shù)值分析》實驗報告（2）學號：姓名：班級：成績：實驗名稱：曲線擬合實驗地點：化工樓317所使用的工具軟件及環(huán)境：Matlab實驗任務與目的學習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程以及結果分析二、實驗內容或問題描述通過實驗的觀察和測量給出離散的一些點，再來求出近似的函數(shù)解析式。最小二乘法是求解這一問題的很好的方法，本實驗運用這一方法實現(xiàn)對給定數(shù)據(jù)的擬合。解題思路與方法，或解題詳細步驟1.擬合曲線三次多項式nm=3+1,基函數(shù)｛1，x，x^2，x^3｝2.擬合節(jié)點數(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)點上值構成向量xe(i)，全部基函數(shù)向量構成二維矩陣xe5.正則方程系數(shù)矩陣G,常數(shù)項b6.解方程組，求基函數(shù)的系數(shù)a(1),a(2),a(3),a(4)四、程序設計代碼%%擬合曲線【-1，1】，f(x)=1/(1+25x^2）%準確解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;%%擬合曲線三次多項式nm=3+1,基函數(shù)｛1，x，x^2，x^3｝，擬合節(jié)點數(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)點上值構成向量xe(i)，全部基函數(shù)向量構成二維矩陣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x=-1:0.01:1的近似值：s(x),點下標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、運行結果>>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