数值分析实验作业matlab编程课题八.docx

曲线拟合的最小二乘法
1、
%采用二次多项式拟合
%a的输出为多项式各项系数
%b为拟合曲线各点函数值
%phi为输出的曲线拟合函数
x=0:5:55;
y=[0 1.27 2.16 2.86 3.44 3.87 4.15 4.37 4.51 4.58 4.02 4.64];
a=polyfit(x,y,2)
b=polyval(a,x)
syms t
phi=a(1)*t^2+a(2)*t+a(3)

运行结果：
>> leastway

a =

-0.0024 0.2037 0.2305

b =

Columns 1 through 5

0.2305 1.1894 2.0293 2.7502 3.3521

Columns 6 through 10

3.8349 4.1987 4.4435 4.5693 4.5760

Columns 11 through 12

4.4637 4.2324

phi =

2
-0.00238051948051948162 t + 0.203690809190809258 t + 0.230467032967031749

2、
%采用三次多项式拟合
x=0:5:55;
y=[0 1.27 2.16 2.86 3.44 3.87 4.15 4.37 4.51 4.58 4.02 4.64];
a=polyfit(x,y,3)
b=polyval(a,x)
syms t
phi=a(1)*t^3+a(2)*t^2+a(3)*t+a(4)

运行结果：
>> leastway

a =

0.0000 -0.0052 0.2634 0.0178

b =

Columns 1 through 5

0.0178 1.2087 2.1646 2.9113 3.4745

Columns 6 through 10

3.8800 4.1536 4.3211 4.4082 4.4407

Columns 11 through 12

4.4444 4.4450

phi =

3 2
0.0000343641543641541613 t - 0.00521556221556219567 t

+ 0.263398527398526872 t + 0.0178388278388323038

3、
%delta为拟合函数值与原函数值的误差
clc
clear
x=0:5:55;
y=[0 1.27 2.16 2.86 3.44 3.87 4.15 4.37 4.51 4.58 4.02 4.64];
a=polyfit(x,y,3);
b=polyval(a,x);
for j=1:12
delta=b(j)-y(j)
end
syms t
phi=a(1)*t^3+a(2)*t^2+a(3)*t+a(4)

运行结果：
delta =

0.0178

delta =

-0.0613

delta =

0.0046

delta =

0.0513

delta =

0.0345

delta =

0.0100

delta =

0.0036

delta =

-0.0489

delta =

-0.1018

delta =

-0.1393

delta =

0.4244

delta =

-0.1950

phi =

3 2
0.0000343641543641541613 t - 0.00521556221556219567 t

+ 0.263398527398526872 t + 0.0178388278388323038

4、
%采用四次多项式拟合与三次多项式拟合进行比较
%输出其与原函数值的误差再与之前输出的三次与原函误差进行比较
%delta4为四次与原函的误差
clc
clear
x=0:5:55;
y=[0 1.27 2.16 2.86 3.44 3.87 4.15 4.37 4.51 4.58 4.02 4.64];
a=polyfit(x,y,4);
b=polyval(a,x)
for j=1:12
delta4=b(j)-y(j)
end
syms t
phi=a(1)*t^4+a(2)*t^3+a(3)*t^2+a(4)*t+a(5)

运行结果：
b =

Columns 1 through 5

0.0604 1.1739 2.1220 2.8945 3.4900

Columns 6 through 10

3.9162 4.1898 4.3366 4.3914 4.3981

Columns 11 through 12

4.4095 4.4876

delta4 =

0.0604

delta4 =

-0.0961

delta4 =

-0.0380

delta4 =

0.0345

delta4 =

0.0500

delta4 =

0.0462

delta4 =

0.0398

delta4 =

-0.0334

delta4 =

-0.1186

delta4 =

-0.1819

delta4 =

0.3895

delta4 =

-0.1524

phi =

-6 4 3
0.602564102564107288 10 t - 0.0000319178969178975504t

2
- 0.00293227466977464190 t +0.238069314944314520t

+ 0.0604487179487194770

5、
%分别采用二次、四次、六次、九次多项式拟合并画出其曲线进行比较
clc
clear
x=0:5:55;
y=[0 1.27 2.16 2.86 3.44 3.87 4.15 4.37 4.51 4.58 4.02 4.64];
a2=polyfit(x,y,2);
b2=polyval(a2,x);
a4=polyfit(x,y,4);
b4=polyval(a4,x);
a6=polyfit(x,y,6);
b6=polyval(a6,x);
a9=polyfit(x,y,9);
b9=polyval(a9,x);
plot(x,b2,'r-^');
hold on
plot(x,b4,'g->');
hold on
plot(x,b6,'b-*');
hold on
plot(x,b9,'y-+');
hold on
plot(x,y,'ko');
title('含碳量-时间曲线');
xlabel('时间t（分）');
ylabel('含碳量y（10^-^4）');
legend('二次拟合曲线', '四次拟合曲线', '六次拟合曲线', '九次拟合曲线', '原函数点',);

运行结果：
警告: 多项式未正确设置条件。请添加具有不同 X 值的点，减少多项式的阶数，或者尝试按照 HELP
POLYFIT 中所述进行居中和缩放。
> In polyfit at 75
In leastway at 9
警告: 多项式未正确设置条件。请添加具有不同 X 值的点，减少多项式的阶数，或者尝试按照 HELP
POLYFIT 中所述进行居中和缩放。
> In polyfit at 75
In leastway at 11

局部放大图