2 曲线拟合的线性最小二乘法及其 MATLAB 程序
例 2 给出一组数据点(x , y ) 列入表2 中,试用线性最小二乘法求拟合曲线,
-192.9 -85.50 -36.15 -26.52
68.04
>> 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('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 f (x) 在 (x , y ) 处的函数值,即输入程序
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a ,a , a 和a 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,
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,
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+(-4913/100
0*a1+289/100*a2-17/10*a3+a4+171/2)^2+(-1331/1000*a1+121/1
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