函数逼近与曲线(面)拟合的MATLAB程序

运行后屏幕显示拟合函数f及其系数C如下

C = 5.0911 -14.1905 6.4102 -8.2574

f=716503695845759/140737488355328*x^3

-7988544102557579/562949953421312*x^2

+1804307491277693/281474976710656*x

-4648521160813215/562949953421312

故所求的拟合曲线为

f(x) 5.0911x3 14.1905x2 6.4102x 8.2574.

（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 -0.8 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];

n=length(xi);

f=5.0911.*xi.^3-14.1905.*xi.^2+6.4102.*xi -8.2574;

x=-2.5:0.01: 3.6;

F=5.0911.*x.^3-14.1905.*x.^2+6.4102.*x -8.2574;

fy=abs(f-y); fy2=fy.^2; Ew=max(fy),

E1=sum(fy)/n, E2=sqrt((sum(fy2))/n)

plot(xi,y,'r*'), hold on, plot(x,F,'b-'), hold off

legend('数据点(xi,yi)','拟合曲线y=f(x)'),

xlabel('x'), ylabel('y'),

title('例7.2.1的数据点(xi,yi)和拟合曲线y=f(x)的图形')

运行后屏幕显示数据(xi,yi)与拟合函数f的最大误差Ew，平均误差E1和均方根误差E2及其数据点(xi,yi)和拟合曲线y=f(x)的图形（略）.

Ew = E1 = E2 =

3.105 4 0.903 4 1.240 9

7.3 函数rk(x)的选取及其MATLAB程序

例7.3.1 给出一组实验数据点(xi,yi)的横坐标向量为

x=(-8.5,-8.7,-7.1,-6.8,-5.10,-4.5,-3.6,-3.4,-2.6,-2.5, -2.1,-1.5, -2.7,-3.6)，纵横坐标向量为y=(459.26,52.81,198.27,165.60,59.17,41.66,25.92, 22.37,13.47, 12.87, 11.87,6.69,14.87,24.22)，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.

解 （1）在MATLAB工作窗口输入程序

>>x=[-8.5,-8.7,-7.1,-6.8,-5.10,-4.5,-3.6,-3.4,-2.6,-2.5,

-2.1,-1.5, -2.7,-3.6];

y=[459.26,52.81,198.27,165.60,59.17,41.66,25.92,

22.37,13.47, 12.87, 11.87,6.69,14.87,24.22];

plot(x,y,'r*'),legend('实验数据(xi,yi)')

xlabel('x'), ylabel('y'),

title('例7.3.1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f(x)在(xi,yi)处的函数值，即输入程序

>> syms a b

x=[-8.5,-8.7,-7.1,-6.8,-5.10,-4.5,-3.6,-3.4,-2.6,-2.5,-2

.1,-1.5,-2.7,-3.6]; fi=a.*exp(-b.*x)

运行后屏幕显示关于a和b的线性方程组

fi =

[ a*exp(17/2*b), a*exp(87/10*b), a*exp(71/10*b),

a*exp(34/5*b), a*exp(51/10*b), a*exp(9/2*b), a*exp(18/5*b), a*exp(17/5*b), a*exp(13/5*b), a*exp(5/2*b), a*exp(21/10*b),