函数逼近与曲线拟合1曲线拟合的线性最小二乘法及其MATLAB程序例7.2.1给出一组数据点列入表7–2中，试用线性最小二乘法求拟合曲线，并用（7.2），（7.3）和（7.4）式估计其误差，作出拟合曲线.表7–2例7.2.1的一组数据xi-2.5-1.7-1.1-0.800.11.52.73.6yi-192.9-85.50-36.15-26.52-9.10-8.43-13.126.5068.04解（1）在MATLAB工作窗口输入程序>>x=[-2.5-1.7-1.1-0.800.11.52.73.6];y=[-192.9-85.50-36.15-26.52-9.10-8.43-13.126.5068.04];plot(x,y,'r*'),legend('实验数据(xi,yi)')xlabel('x'),ylabel('y'),title('例7.2.1的数据点(xi,yi)的散点图')运行后屏幕显示数据的散点图（略）.（3）编写下列MATLAB程序计算在处的函数值，即输入程序>>symsa1a2a3a4x=[-2.5-1.7-1.1-0.800.11.52.73.6];fi=a1.*x.^3+a2.*x.^2+a3.*x+a4运行后屏幕显示关于a1,a2,a3和a4的线性方程组fi=[-125/8*a1+25/4*a2-5/2*a3+a4,-4913/1000*a1+289/100*a2-17/10*a3+a4,-1331/1000*a1+121/100*a2-11/10*a3+a4,-64/125*a1+16/25*a2-4/5*a3+a4,a4,1/1000*a1+1/100*a2+1/10*a3+a4,27/8*a1+9/4*a2+3/2*a3+a4,19683/1000*a1+729/100*a2+27/10*a3+a4,5832/125*a1+324/25*a2+18/5*a3+a4]编写构造误差平方和的MATLAB程序>>y=[-192.9-85.50-36.15-26.52-9.10-8.43-13.126.5068.04];fi=[-125/8*a1+25/4*a2-5/2*a3+a4,-4913/1000*a1+289/100*a2-17/10*a3+a4,-1331/1000*a1+121/100*a2-11/10*a3+a4,-64/125*a1+16/25*a2-4/5*a3+a4,a4,1/1000*a1+1/100*a2+1/10*a3+a4,27/8*a1+9/4*a2+3/2*a3+a4,19683/1000*a1+729/100*a2+27/10*a3+a4,105．函数逼近与曲线拟合5832/125*a1+324/25*a2+18/5*a3+a4];fy=fi-y;fy2=fy.^2;J=sum(fy.^2)运行后屏幕显示误差平方和如下J=(-125/8*a1+25/4*a2-5/2*a3+a4+1929/10)^2+(-4913/1000*a1+289/100*a2-17/10*a3+a4+171/2)^2+(-1331/1000*a1+121/100*a2-11/10*a3+a4+723/20)^2+(-64/125*a1+16/25*a2-4/5*a3+a4+663/25)^2+(a4+91/10)^2+(1/1000*a1+1/100*a2+1/10*a3+a4+843/100)^2+(27/8*a1+9/4*a2+3/2*a3+a4+328/25)^2+(19683/1000*a1+729/100*a2+27/10*a3+a4-13/2)^2+(5832/125*a1+324/25*a2+18/5*a3+a4-1701/25)^2为求使达到最小，只需利用极值的必要条件，得到关于的线性方程组，这可以由下面的MATLAB程序完成，即输入程序>>symsa1a2a3a4J=(-125/8*a1+25/4*a2-5/2*a3+a4+1929/10)^2+(-4913/1000*a1+289/100*a2-17/10*a3+a4...+171/2)^2+(-1331/1000*a1+121/100*a2-11/10*a3+a4+723/20)^2+(-64/125*a1+16/25*a2-4/5*a3+a4+663/25)^2+(a4+91/10)^2+(1/1000*a1+1/100*a2+1/10*a3+a4+843/100)^2+(27/8*a1+9/4*a2+3/2*a3+a4+328/25)^2+(19683/1000*a1+729/100*a2+27/10*a3+a4-13/2)^2+(5832/125*a1+324/25*a2+18/5*a3+a4-1701/25)^2;Ja1=diff(J,a1);Ja2=diff(J,a2);Ja3=diff(J,a3);Ja4=diff(J,a4);Ja11=simple(Ja1),Ja21=simple(Ja2),Ja31=simple(Ja3),Ja41=simple(Ja4),运行后屏幕显示J分别对a1,a2,a3,a4的偏导数如下Ja11=56918107/10000*a1+32097579/25000*a2+1377283/2500*a3+23667/250*a4-8442429/625Ja21=32097579/25000*a1+1377283/2500*a2+23667/250*a3+67*a4+767319/625Ja31=1377283/2500*a1+23667/250*a2+67*a3+18/5*a4-232638/125Ja41=23667/250*a1+67*a2+18/5*a3+18*a4+14859/25解线性方程组Ja11=0，Ja21=0，Ja31=0，Ja41=0，输入下列程序>>A=[56918107/10000,32097579/25000,1377283/2500,23667/250;32097579/25000,1377283/2500,23667/250,67;1377283/2500,23667/250,67,18/5;23667/250,67,18/5,18];B=[8442429/625,-767319/625,232638/125,-14859/25];C=B/A,f=poly2sym(C)运行后屏幕显示拟合函数f及其系数C如下106．函数逼近与曲线拟合C=5.0911-14.19056.4102-8.2574f=716503695845759/140737488355328*x^3-7988544102557579/562949953421312*x^2+1804307491277693/281474976710656*x-4648521160813215/562949953421312故所求的拟合曲线为.（4）编写下面的MATLAB程序估计其误差，并作出拟合曲线和数据的图形.输入程序>>xi=[-2.5-1.7-1.1-0.800.11.52.73.6];y=[-192.9-85.50-36.15-26.52...
