高斯消元法求逆矩阵
A=[1,-1,1,-4;5,-4,3,12;2,1,1,11;2,-1,7,-1]
E=[1,0,0,0;0,1,0,0;0,0,1,0;0,0,0,1]
b=[2,4,3,0]'
%开始消元过程
for k=1:(length(A))
a=A(k,k)
for i=1:(length(A))
A(k,i)=A(k,i)/a
E(k,i)=E(k, i)/a
end
b(k,1)=b(k,1)/ a
for i=k+1:(length(A))
c=-A(i,k)
for j=1: (length(A))
A(i,j)=A(i,j)+c.*A(k,j)
E(i,j)=E(i,j)+c.*E(k,j)
end
b(i,1)= b(i,1)+ c.* b(k,1)
end
end
%开始回代过程
for i=(length(A)):-1:1
for j=i-1:-1:1
c=-A(j,i)
for k=(length(A)):-1:1
A(j,k)= A(j,k)+c.*A(i,k)
E(j,k)= E(j,k)+c.*E(i,k)
end
b(j,1)= b(j,1)+ c.* b(i,1)
end
end
E
A =
1 -1 1 -4
5 -4 3 12
2 1 1 11
2 -1 7 -1
E =
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
b =
2
4
3
0
a =
1
A =
1 -1 1 -4
5 -4 3 12
2 1 1 11
2 -1 7 -1
E =
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
A =
1 -1 1 -4
5 -4 3 12
2 1 1 11
2 -1 7 -1
E =
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
A =
1 -1 1 -4
5 -4 3 12
2 1 1 11
2 -1 7 -1
E =
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
A =
1 -1 1 -4
5 -4 3 12
2 1 1 11
2 -1 7 -1
E =
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
b =
2
4
3
0
c =
-5
A =
1 -1 1 -4
0 -4 3 12
2 1 1 11
2 -1 7 -1
E =
1 0 0 0
-5 1 0 0
0 0 1 0
0 0 0 1
A =
1 -1 1 -4
0 1 3 12
2 1 1 11
2 -1 7 -1
E =
1 0 0 0
-5 1 0 0
0 0 1 0
0 0 0 1
A =
1 -1 1 -4
0 1 -2 12
2 1 1 11
2 -1 7 -1
E =
1 0 0 0
-5 1 0 0
0 0 1 0
0 0 0 1
A =
1 -1 1 -4
0 1 -2 32
2 1 1 11
2 -1 7 -1
E =
1 0 0 0
-5 1 0 0
0 0 1 0
0 0 0 1
b =
2
-6
3
0
c =
-2
A =
1 -1 1 -4
0 1 -2 32
0 1 1 11
2 -1 7 -1
E =
1 0 0 0
-5 1 0 0
-2 0 1 0
0 0 0 1
A =
1 -1 1 -4
0 1 -2 32
0 3 1 11
2 -1 7 -1
E =
1 0 0 0
-5 1 0 0
-2 0 1 0
0 0 0 1
A =
1 -1 1 -4
0 1 -2 32
0 3 -1 11
2 -1 7 -1
E =
1 0 0 0
-5 1 0 0
-2 0 1 0
0 0 0 1
A =
1 -1 1 -4
0 1 -2 32
0 3 -1 19
2 -1 7 -1
E =
1 0 0 0
-5 1 0 0
-2 0 1 0
0 0 0 1
b =
2
-6
-1
0
c =
-2
A =
1 -1 1 -4
0 1 -2 32
0 3 -1 19
0 -1 7 -1
E =
1 0 0 0
-5 1 0 0
-2 0 1 0
-2 0 0 1
A =
1 -1 1 -4
0 1 -2 32
0 3 -1 19
0 1 7 -1
E =
1 0 0 0
-5 1 0 0
-2 0 1 0
-2 0 0 1
A =
1 -1 1 -4
0 1 -2 32
0 3 -1 19
0 1 5 -1
E =
1 0 0 0
-5 1 0 0
-2 0 1 0
-2 0 0 1
A =
1 -1 1 -4
0 1 -2 32
0 3 -1 19
0 1 5 7
E =
1 0 0 0
-5 1 0 0
-2 0 1 0
-2 0 0 1
b =
2
-6
-1
-4
a =
1
A =
1 -1 1 -4
0 1 -2 32
0 3 -1 19
0 1 5 7
E =
1 0 0 0
-5 1 0 0
-2 0 1 0
-2 0 0 1
A =
1 -1 1 -4
0 1 -2 32
0 3 -1 19
0 1 5 7
E =
1 0 0 0
-5 1 0 0
-2 0 1 0
-2 0 0 1
A =
1 -1 1 -4
0 1 -2 32
0 3 -1 19
0 1 5 7
E =
1 0 0 0
-5 1 0 0
-2 0 1 0
-2 0 0 1
A =
1 -1 1 -4
0 1 -2 32
0 3 -1 19
0 1 5 7
E =
1 0 0 0
-5 1 0 0
-2 0 1 0
-2 0 0 1
b =
2
-6
-1
-4
c =
-3
A =
1 -1 1 -4
0 1 -2 32
0 3 -1 19
0 1 5 7
E =
1 0 0 0
-5 1 0 0
13 0 1 0
-2 0 0 1
A =
1 -1 1 -4
0 1 -2 32
0 0 -1 19
0 1 5 7
E =
1 0 0 0
-5 1 0 0
13 -3 1 0
-2 0 0 1
A =
1 -1 1 -4
0 1 -2 32
0 0 5 19
0 1 5 7
E =
1 0 0 0
-5 1 0 0
13 -3 1 0
-2 0 0 1
A =
1 -1 1 -4
0 1 -2 32
0 0 5 -77
0 1 5 7
E =
1 0 0 0
-5 1 0 0
13 -3 1 0
-2 0 0 1
b =
2
-6
17
-4
c =
-1
A =
1 -1 1 -4
0 1 -2 32
0 0 5 -77
0 1 5 7
E =
1 0 0 0
-5 1 0 0
13 -3 1 0
3 0 0 1
A =
1 -1 1 -4
0 1 -2 32
0 0 5 -77
0 0 5 7
E =
1 0 0 0
-5 1 0 0
13 -3 1 0
3 -1 0 1
A =
1 -1 1 -4
0 1 -2 32
0 0 5 -77
0 0 7 7
E =
1 0 0 0
-5 1 0 0
13 -3 1 0
3 -1 0 1
A =
1 -1 1 -4
0 1 -2 32
0 0 5 -77
0 0 7 -25
E =
1 0 0 0
-5 1 0 0
13 -3 1 0
3 -1 0 1
b =
2
-6
17
2
a =
5
A =
1 -1 1 -4
0 1 -2 32
0 0 5 -77
0 0 7 -25
E =
1.0000 0 0 0
-5.0000 1.0000 0 0
2.6000 -3.0000 1.0000 0
3.0000 -1.0000 0 1.0000
A =
1 -1 1 -4
0 1 -2 32
0 0 5 -77
0 0 7 -25
E =
1.0000 0 0 0
-5.0000 1.0000 0 0
2.6000 -0.6000 1.0000 0
3.0000 -1.0000 0 1.0000
A =
1 -1 1 -4
0 1 -2 32
0 0 1 -77
0 0 7 -25
E =
1.0000 0 0 0
-5.0000 1.0000 0 0
2.6000 -0.6000 0.2000 0
3.0000 -1.0000 0 1.0000
A =
1.0000 -1.0000 1.0000 -4.0000
0 1.0000 -2.0000 32.0000
0 0 1.0000 -15.4000
0 0 7.0000 -25.0000
E =
1.0000 0 0 0
-5.0000 1.0000 0 0
2.6000 -0.6000 0.2000 0
3.0000 -1.0000 0 1.0000
b =
2.0000
-6.0000
3.4000
2.0000
c =
-7
A =
1.0000 -1.0000 1.0000 -4.0000
0 1.0000 -2.0000 32.0000
0 0 1.0000 -15.4000
0 0 7.0000 -25.0000
E =
1.0000 0 0 0
-5.0000 1.0000 0 0
2.6000 -0.6000 0.2000 0
-15.2000 -1.0000 0 1.0000
A =
1.0000 -1.0000 1.0000 -4.0000
0 1.0000 -2.0000 32.0000
0 0 1.0000 -15.4000
0 0 7.0000 -25.0000
E =
1.0000 0 0 0
-5.0000 1.0000 0 0
2.6000 -0.6000 0.2000 0
-15.2000 3.2000 0 1.0000
A =
1.0000 -1.0000 1.0000 -4.0000
0 1.0000 -2.0000 32.0000
0 0 1.0000 -15.4000
0 0 0 -25.0000
E =
1.0000 0 0 0
-5.0000 1.0000 0 0
2.6000 -0.6000 0.2000 0
-15.2000 3.2000 -1.4000 1.0000
A =
1.0000 -1.0000 1.0000 -4.0000
0 1.0000 -2.0000 32.0000
0 0 1.0000 -15.4000
0 0 0 82.8000
E =
1.0000 0 0 0
-5.0000 1.0000 0 0
2.6000 -0.6000 0.2000 0
-15.2000 3.2000 -1.4000 1.0000
b =
2.0000
-6.0000
3.4000
-21.8000
a =
82.8000
A =
1.0000 -1.0000 1.0000 -4.0000
0 1.0000 -2.0000 32.0000
0 0 1.0000 -15.4000
0 0 0 82.8000
E =
1.0000 0 0 0
-5.0000 1.0000 0 0
2.6000 -0.6000 0.2000 0
-0.1836 3.2000 -1.4000 1.0000
A =
1.0000 -1.0000 1.0000 -4.0000
0 1.0000 -2.0000 32.0000
0 0 1.0000 -15.4000
0 0 0 82.8000
E =
1.0000 0 0 0
-5.0000 1.0000 0 0
2.6000 -0.6000 0.2000 0
-0.1836 0.0386 -1.4000 1.0000
A =
1.0000 -1.0000 1.0000 -4.0000
0 1.0000 -2.0000 32.0000
0 0 1.0000 -15.4000
0 0 0 82.8000
E =
1.0000 0 0 0
-5.0000 1.0000 0 0
2.6000 -0.6000 0.2000 0
-0.1836 0.0386 -0.0169 1.0000
A =
1.0000 -1.0000 1.0000 -4.0000
0 1.0000 -2.0000 32.0000
0 0 1.0000 -15.4000
0 0 0 1.0000
E =
1.0000 0 0 0
-5.0000 1.0000 0 0
2.6000 -0.6000 0.2000 0
-0.1836 0.0386 -0.0169 0.0121
b =
2.0000
-6.0000
3.4000
-0.2633
c =
15.4000
A =
1 -1 1 -4
0 1 -2 32
0 0 1 0
0 0 0 1
E =
1.0000 0 0 0
-5.0000 1.0000 0 0
2.6000 -0.6000 0.2000 0.1860
-0.1836 0.0386 -0.0169 0.0121
A =
1 -1 1 -4
0 1 -2 32
0 0 1 0
0 0 0 1
E =
1.0000 0 0 0
-5.0000 1.0000 0 0
2.6000 -0.6000 -0.0604 0.1860
-0.1836 0.0386 -0.0169 0.0121
A =
1 -1 1 -4
0 1 -2 32
0 0 1 0
0 0 0 1
E =
1.0000 0 0 0
-5.0000 1.0000 0 0
2.6000 -0.0048 -0.0604 0.1860
-0.1836 0.0386 -0.0169 0.0121
A =
1 -1 1 -4
0 1 -2 32
0 0 1 0
0 0 0 1
E =
1.0000 0 0 0
-5.0000 1.0000 0 0
-0.2271 -0.0048 -0.0604 0.1860
-0.1836 0.0386 -0.0169 0.0121
b =
2.0000
-6.0000
-0.6546
-0.2633
c =
-32
A =
1 -1 1 -4
0 1 -2 0
0 0 1 0
0 0 0 1
E =
1.0000 0 0 0
-5.0000 1.0000 0 -0.3865
-0.2271 -0.0048 -0.0604 0.1860
-0.1836 0.0386 -0.0169 0.0121
A =
1 -1 1 -4
0 1 -2 0
0 0 1 0
0 0 0 1
E =
1.0000 0 0 0
-5.0000 1.0000 0.5411 -0.3865
-0.2271 -0.0048 -0.0604 0.1860
-0.1836 0.0386 -0.0169 0.0121
A =
1 -1 1 -4
0 1 -2 0
0 0 1 0
0 0 0 1
E =
1.0000 0 0 0
-5.0000 -0.2367 0.5411 -0.3865
-0.2271 -0.0048 -0.0604 0.1860
-0.1836 0.0386 -0.0169 0.0121
A =
1 -1 1 -4
0 1 -2 0
0 0 1 0
0 0 0 1
E =
1.0000 0 0 0
0.8744 -0.2367 0.5411 -0.3865
-0.2271 -0.0048 -0.0604 0.1860
-0.1836 0.0386 -0.0169 0.0121
b =
2.0000
2.4251
-0.6546
-0.2633
c =
4
A =
1 -1 1 0
0 1 -2 0
0 0 1 0
0 0 0 1
E =
1.0000 0 0 0.0483
0.8744 -0.2367 0.5411 -0.3865
-0.2271 -0.0048 -0.0604 0.1860
-0.1836 0.0386 -0.0169 0.0121
A =
1 -1 1 0
0 1 -2 0
0 0 1 0
0 0 0 1
E =
1.0000 0 -0.0676 0.0483
0.8744 -0.2367 0.5411 -0.3865
-0.2271 -0.0048 -0.0604 0.1860
-0.1836 0.0386 -0.0169 0.0121
A =
1 -1 1 0
0 1 -2 0
0 0 1 0
0 0 0 1
E =
1.0000 0.1546 -0.0676 0.0483
0.8744 -0.2367 0.5411 -0.3865
-0.2271 -0.0048 -0.0604 0.1860
-0.1836 0.0386 -0.0169 0.0121
A =
1 -1 1 0
0 1 -2 0
0 0 1 0
0 0 0 1
E =
0.2657 0.1546 -0.0676 0.0483
0.8744 -0.2367 0.5411 -0.3865
-0.2271 -0.0048 -0.0604 0.1860
-0.1836 0.0386 -0.0169 0.0121
b =
0.9469
2.4251
-0.6546
-0.2633
c =
2
A =
1 -1 1 0
0 1 -2 0
0 0 1 0
0 0 0 1
E =
0.2657 0.1546 -0.0676 0.0483
0.8744 -0.2367 0.5411 -0.0145
-0.2271 -0.0048 -0.0604 0.1860
-0.1836 0.0386 -0.0169 0.0121
A =
1 -1 1 0
0 1 0 0
0 0 1 0
0 0 0 1
E =
0.2657 0.1546 -0.0676 0.0483
0.8744 -0.2367 0.4203 -0.0145
-0.2271 -0.0048 -0.0604 0.1860
-0.1836 0.0386 -0.0169 0.0121
A =
1 -1 1 0
0 1 0 0
0 0 1 0
0 0 0 1
E =
0.2657 0.1546 -0.0676 0.0483
0.8744 -0.2464 0.4203 -0.0145
-0.2271 -0.0048 -0.0604 0.1860
-0.1836 0.0386 -0.0169 0.0121
A =
1 -1 1 0
0 1 0 0
0 0 1 0
0 0 0 1
E =
0.2657 0.1546 -0.0676 0.0483
0.4203 -0.2464 0.4203 -0.0145
-0.2271 -0.0048 -0.0604 0.1860
-0.1836 0.0386 -0.0169 0.0121
b =
0.9469
1.1159
-0.6546
-0.2633
c =
-1
A =
1 -1 1 0
0 1 0 0
0 0 1 0
0 0 0 1
E =
0.2657 0.1546 -0.0676 -0.1377
0.4203 -0.2464 0.4203 -0.0145
-0.2271 -0.0048 -0.0604 0.1860
-0.1836 0.0386 -0.0169 0.0121
A =
1 -1 0 0
0 1 0 0
0 0 1 0
0 0 0 1
E =
0.2657 0.1546 -0.0072 -0.1377
0.4203 -0.2464 0.4203 -0.0145
-0.2271 -0.0048 -0.0604 0.1860
-0.1836 0.0386 -0.0169 0.0121
A =
1 -1 0 0
0 1 0 0
0 0 1 0
0 0 0 1
E =
0.2657 0.1594 -0.0072 -0.1377
0.4203 -0.2464 0.4203 -0.0145
-0.2271 -0.0048 -0.0604 0.1860
-0.1836 0.0386 -0.0169 0.0121
A =
1 -1 0 0
0 1 0 0
0 0 1 0
0 0 0 1
E =
0.4928 0.1594 -0.0072 -0.1377
0.4203 -0.2464 0.4203 -0.0145
-0.2271 -0.0048 -0.0604 0.1860
-0.1836 0.0386 -0.0169 0.0121
b =
1.6014
1.1159
-0.6546
-0.2633
c =
1
A =
1 -1 0 0
0 1 0 0
0 0 1 0
0 0 0 1
E =
0.4928 0.1594 -0.0072 -0.1522
0.4203 -0.2464 0.4203 -0.0145
-0.2271 -0.0048 -0.0604 0.1860
-0.1836 0.0386 -0.0169 0.0121
A =
1 -1 0 0
0 1 0 0
0 0 1 0
0 0 0 1
E =
0.4928 0.1594 0.4130 -0.1522
0.4203 -0.2464 0.4203 -0.0145
-0.2271 -0.0048 -0.0604 0.1860
-0.1836 0.0386 -0.0169 0.0121
A =
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
E =
0.4928 -0.0870 0.4130 -0.1522
0.4203 -0.2464 0.4203 -0.0145
-0.2271 -0.0048 -0.0604 0.1860
-0.1836 0.0386 -0.0169 0.0121
A =
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
E =
0.9130 -0.0870 0.4130 -0.1522
0.4203 -0.2464 0.4203 -0.0145
-0.2271 -0.0048 -0.0604 0.1860
-0.1836 0.0386 -0.0169 0.0121
b =
2.7174
1.1159
-0.6546
-0.2633
E =
0.9130 -0.0870 0.4130 -0.1522
0.4203 -0.2464 0.4203 -0.0145
-0.2271 -0.0048 -0.0604 0.1860
-0.1836 0.0386 -0.0169 0.0121
