直接解法

Ax=l，求x。利用逆矩阵、矩阵除法、x=inv(A'A)*A'l

lind.m文件如下，

%求解线性方程组

% x1 + 2x2 + 3x3 + 9x4 = 5

%2x1 + 2x2 + 5x3 + 4x4 = 2

%3x1 + 5x2 + x3 + 5x4 = 3

%7x1 + 4x2 + 2x3 -10x4 = 8

A=[1 2 3 9;2 2 5 4;3 5 1 5;7 4 2 -10];

b=[5 2 3 8]';

x=A\b

c=inv(A)*b

d=inv(A'*A)*A'*b

输出

>> lind.m

x =

4.9227

-3.2373

-1.1680

1.1173

c =

4.9227

-3.2373

-1.1680

1.1173

d =

4.9227

-3.2373

-1.1680

1.1173

Warning: Direct access of structure fields returned by a function call (e.g.,

call to lind) is not allowed. See MATLAB 7.10 Release Notes, "Subscripting Into Function Return Values" for details.

??? Attempt to reference field of non-structure array.

求线性方程组通解

lind.m内容如下

%求解线性方程组通解

% X1 +5 *X2 -1 *X3 -1 *X4 = -1

% X1 -2 *X2 +1 *X3 +3 *X4 = 3

%3*X1 +8 *X2 -1 *X3 +1 *X4 = 1

% X1 -9 *X2 +3 *X3 +7 *X4 = 7

%系数矩阵和增广矩阵的秩都为2，说明有多组解，且基础解系4-2=2

%求基础解系用Z=null(A,'r')，可获得标准化的基础解系。特解采用广义逆函数pinv(A)，

%可求得A的广义逆矩阵，采用x0=pinv(A)*b求得特解，通解为x=k1*x1+k2*x2+...+kn*xn+x0

A=[1 5 -1 -1 ;

1 -2 1 3;

3 8 -1 1;

1 -9 3 7];

b=[-1 3 1 7]';

r1=rank(A)%系数矩阵的秩

r2=rank(A,b)%增广矩阵的秩

Z=null(A,'r')%求基础解系

x0=pinv(A)*b%求得特解

%验证该解

k1=rand(1);

k2=rand(1);

x=k1*Z(:,1)+k2*Z(:,2)+x0

err=norm(A*x-b)

>> lind.m

r1 =

2

r2 =

2

Z =

-0.4286 -1.8571

0.2857 0.5714

1.0000 0

0 1.0000

x0 =

0.3785

-0.0876

0.1873

0.7530

x =

-0.1805

0.0994

0.2848

1.0315

err =

1.8971e-015

矛盾方程组最小二乘解

%矛盾方程组的最小二乘解

%原始数据表

% x 1 1 2 2 2 3 3 4 4 4 5 6

% y 14.8 15.9 20.2 20.0 18.55 22.2 20.9 21 18.3 20.7 16.1 14.75

% 分别一次

A=[ 1 1 2 2 2 3 3 4 4 4 5 6]';

t=ones(size(A));

A1=[t,A];

y= [14.8 15.9 20.2 20.0 18.55 22.2 20.9 21 18.3 20.7 16.1 14.75]';

%x=inv(A'*A)*A'*y

x1=pinv(A1)*y%线性回归方程系数

n=length(A);

rmse1=sqrt(sum((y-A1*x1).^2)/n)%均方误差

xsq=A.^2;

A2=[t,A,xsq]

x2=pinv(A2)*y%线性回归方程系数

rmse2=sqrt(sum((y-A2*x2).^2)/n)%均方误差

xmin=min(A);

xmax=max(A);

nx1=xmin:0.2:xmax;

ny1=x1(1)+x1(2)*nx1;

nx2=nx1;

ny2=x2(1)+x2(2)+x2(3)*nx2.^2;

plot(A',y,'*',nx1,ny1,'+b',nx2,ny2,'r')

输出

>> lind.m

x1 =

18.8820

-0.0861

rmse1 =

2.5105

A2 =

1 1 1

1 1 1

1 2 4

1 2 4

1 2 4

1 3 9

1 3 9

1 4 16

1 4 16

1 4 16

1 5 25

1 6 36

x2 =

10.3924

6.3994

-0.9793

rmse2 =

1.0960