函数逼近与曲线拟合 105. 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),(ii yx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),(ii yx x i -2.5 -1.7 -1.1 -0.8 0 0.1 1.5 2.7 3.6 y i -192.9 -85.50 -36.15 -26.52 -9.10 -8.43 -13.12 6.50 68.04 解 ( 1) 在 MATLAB 工作窗口输入程序 >> x=[-2.5 -1.7 -1.1 -0.8 0 0.1 1.5 2.7 3.6]; y=[-192.9 -85.50 -36.15 -26.52 -9.10 -8.43 -13.12 6.50 68.04]; 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)(xf在),(ii yx处的函数值,即输入程序 >> syms a1 a2 a3 a4 x=[-2.5 -1.7 -1.1 -0.8 0 0.1 1.5 2.7 3.6]; 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 -13.12 6.50 68.04]; 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, 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+(5832/1...