A=rand(5,5)
A =
0.8147 0.0975 0.1576 0.1419 0.6557
0.9058 0.2785 0.9706 0.4218 0.0357
0.1270 0.5469 0.9572 0.9157 0.8491
0.9134 0.9575 0.4854 0.7922 0.9340
0.6324 0.9649 0.8003 0.9595 0.6787
A=2*rand(5,5)-1
A =
0.5155 0.4121 0.6469 -0.1225 -0.0205
0.4863 -0.9363 0.3897 -0.2369 -0.1088
-0.2155 -0.4462 -0.3658 0.5310 0.2926
0.3110 -0.9077 0.9004 0.5904 0.4187
-0.6576 -0.8057 -0.9311 -0.6263 0.5094
A(1,2)
ans =
0.4121
A(1,:)
ans =
0.5155 0.4121 0.6469 -0.1225 -0.0205
A(:,3)
ans =
0.6469
0.3897
-0.3658
0.9004
-0.9311
A(:,[3,5])
ans =
0.6469 -0.0205
0.3897 -0.1088
-0.3658 0.2926
0.9004 0.4187
-0.9311 0.5094
A([1,4],[3,5])
ans =
0.6469 -0.0205
0.9004 0.4187
A
A =
0.5155 0.4121 0.6469 -0.1225 -0.0205
0.4863 -0.9363 0.3897 -0.2369 -0.1088
-0.2155 -0.4462 -0.3658 0.5310 0.2926
0.3110 -0.9077 0.9004 0.5904 0.4187
-0.6576 -0.8057 -0.9311 -0.6263 0.5094
A([4,1],[3,5])
ans =
0.9004 0.4187
0.6469 -0.0205
A([4,1],[3,5,1])
ans =
0.9004 0.4187 0.3110
0.6469 -0.0205 0.5155
A((5:-1:1),:)
ans =
-0.6576 -0.8057 -0.9311 -0.6263 0.5094
0.3110 -0.9077 0.9004 0.5904 0.4187
-0.2155 -0.4462 -0.3658 0.5310 0.2926
0.4863 -0.9363 0.3897 -0.2369 -0.1088
0.5155 0.4121 0.6469 -0.1225 -0.0205
A
A =
0.5155 0.4121 0.6469 -0.1225 -0.0205
0.4863 -0.9363 0.3897 -0.2369 -0.1088
-0.2155 -0.4462 -0.3658 0.5310 0.2926
0.3110 -0.9077 0.9004 0.5904 0.4187
-0.6576 -0.8057 -0.9311 -0.6263 0.5094
A'
ans =
0.5155 0.4863 -0.2155 0.3110 -0.6576
0.4121 -0.9363 -0.4462 -0.9077 -0.8057
0.6469 0.3897 -0.3658 0.9004 -0.9311
-0.1225 -0.2369 0.5310 0.5904 -0.6263
-0.0205 -0.1088 0.2926 0.4187 0.5094
(5:-1:1)
ans =
5 4 3 2 1
A'
ans =
0.5155 0.4863 -0.2155 0.3110 -0.6576
0.4121 -0.9363 -0.4462 -0.9077 -0.8057
0.6469 0.3897 -0.3658 0.9004 -0.9311
-0.1225 -0.2369 0.5310 0.5904 -0.6263
-0.0205 -0.1088 0.2926 0.4187 0.5094
[1,2,3]'
ans =
1
2
3
[1+1i,2+2i]
ans =
1.0000 + 1.0000i 2.0000 + 2.0000i
[1+1i,2+2i]'
ans =
1.0000 - 1.0000i
2.0000 - 2.0000i
[1+1i,2+2i].'
ans =
1.0000 + 1.0000i
2.0000 + 2.0000i
v=[3,4]
v =
3 4
norm(v)
ans =
5
sqrt(v*v')
ans =
5
v.^2
ans =
9 16
sqrt(sum(v.^2))
ans =
5
realmax
ans =
1.7977e+308
v=[3,4]*1e307
v =
1.0e+307 *
3.0000 4.0000
norm(v)
ans =
5.0000e+307
sqrt(v*v')
ans =
Inf
norm(v,1)
ans =
7.0000e+307
v=[1,-2,3,-4]
v =
1 -2 3 -4
norm(v,1)
ans =
10
sum(abs(v))
ans =
10
norm(v,Inf)
ans =
4
norm(A,1)
ans =
3.5080
A
A =
0.5155 0.4121 0.6469 -0.1225 -0.0205
0.4863 -0.9363 0.3897 -0.2369 -0.1088
-0.2155 -0.4462 -0.3658 0.5310 0.2926
0.3110 -0.9077 0.9004 0.5904 0.4187
-0.6576 -0.8057 -0.9311 -0.6263 0.5094
sum(A)
ans =
0.4395 -2.6838 0.6401 0.1358 1.0914
sum(A,2)
ans =
1.4315
-0.4061
-0.2038
1.3129
-2.5114
sum(abs(A))
ans =
2.1859 3.5080 3.2339 2.1071 1.3500
max(sum(abs(A)))
ans =
3.5080
norm(A,1)
ans =
3.5080
norm(A,Inf)
ans =
3.5301
max(sum(abs(A),2))
ans =
3.5301
norm(A)
ans =
1.9149
lambda=eig(A*A');
lambda
lambda =
0.0331
0.2022
0.9117
2.9912
3.6667
max(lambda)
ans =
3.6667
sqrt(max(lambda))
ans =
1.9149
norm(A)
ans =
1.9149
sqrt(max(eig(A*A')))
ans =
1.9149
inv(A)
ans =
2.6903 1.2738 2.7209 -1.4063 -0.0268
0.7574 -0.5903 0.1600 -0.2258 -0.0020
-1.0630 -0.6767 -2.0380 1.2736 -0.0635
-0.3024 -0.0925 1.0109 0.0186 -0.6279
2.3563 -0.6399 1.2835 0.1782 1.0374
A*inv(A)
ans =
1.0000 0.0000 0.0000 0.0000 -0.0000
0.0000 1.0000 0.0000 0.0000 0.0000
0.0000 0.0000 1.0000 -0.0000 0
-0.0000 -0.0000 0 1.0000 0
0.0000 -0.0000 0 -0.0000 1.0000
format short e
A*inv(A)
ans =
1.0000e+00 7.4593e-17 5.4817e-16 2.9490e-16 -2.0817e-17
1.1102e-16 1.0000e+00 1.3878e-16 1.5613e-16 1.9429e-16
2.2204e-16 8.3267e-17 1.0000e+00 -1.1796e-16 0
-1.1102e-16 -1.1102e-16 0 1.0000e+00 0
2.2204e-16 -1.1102e-16 0 -2.2204e-16 1.0000e+00
norm(A)*norm(inv(A))
ans =
1.0522e+01
cond(A)
ans =
1.0522e+01
hilb(4)
ans =
1.0000e+00 5.0000e-01 3.3333e-01 2.5000e-01
5.0000e-01 3.3333e-01 2.5000e-01 2.0000e-01
3.3333e-01 2.5000e-01 2.0000e-01 1.6667e-01
2.5000e-01 2.0000e-01 1.6667e-01 1.4286e-01
H=hilb(10);
cond(H)
ans =
1.6025e+13
H
H =
Columns 1 through 5
1.0000e+00 5.0000e-01 3.3333e-01 2.5000e-01 2.0000e-01
5.0000e-01 3.3333e-01 2.5000e-01 2.0000e-01 1.6667e-01
3.3333e-01 2.5000e-01 2.0000e-01 1.6667e-01 1.4286e-01
2.5000e-01 2.0000e-01 1.6667e-01 1.4286e-01 1.2500e-01
2.0000e-01 1.6667e-01 1.4286e-01 1.2500e-01 1.1111e-01
1.6667e-01 1.4286e-01 1.2500e-01 1.1111e-01 1.0000e-01
1.4286e-01 1.2500e-01 1.1111e-01 1.0000e-01 9.0909e-02
1.2500e-01 1.1111e-01 1.0000e-01 9.0909e-02 8.3333e-02
1.1111e-01 1.0000e-01 9.0909e-02 8.3333e-02 7.6923e-02
1.0000e-01 9.0909e-02 8.3333e-02 7.6923e-02 7.1429e-02
Columns 6 through 10
1.6667e-01 1.4286e-01 1.2500e-01 1.1111e-01 1.0000e-01
1.4286e-01 1.2500e-01 1.1111e-01 1.0000e-01 9.0909e-02
1.2500e-01 1.1111e-01 1.0000e-01 9.0909e-02 8.3333e-02
1.1111e-01 1.0000e-01 9.0909e-02 8.3333e-02 7.6923e-02
1.0000e-01 9.0909e-02 8.3333e-02 7.6923e-02 7.1429e-02
9.0909e-02 8.3333e-02 7.6923e-02 7.1429e-02 6.6667e-02
8.3333e-02 7.6923e-02 7.1429e-02 6.6667e-02 6.2500e-02
7.6923e-02 7.1429e-02 6.6667e-02 6.2500e-02 5.8824e-02
7.1429e-02 6.6667e-02 6.2500e-02 5.8824e-02 5.5556e-02
6.6667e-02 6.2500e-02 5.8824e-02 5.5556e-02 5.2632e-02
x=(1:n)'
{Undefined function or variable 'n'.
}
n=10
n =
10
x=(1:n)'
x =
1
2
3
4
5
6
7
8
9
10
b=H*x;
\
\
↑
{Error: Unexpected MATLAB operator.
}
H
H =
Columns 1 through 5
1.0000e+00 5.0000e-01 3.3333e-01 2.5000e-01 2.0000e-01
5.0000e-01 3.3333e-01 2.5000e-01 2.0000e-01 1.6667e-01
3.3333e-01 2.5000e-01 2.0000e-01 1.6667e-01 1.4286e-01
2.5000e-01 2.0000e-01 1.6667e-01 1.4286e-01 1.2500e-01
2.0000e-01 1.6667e-01 1.4286e-01 1.2500e-01 1.1111e-01
1.6667e-01 1.4286e-01 1.2500e-01 1.1111e-01 1.0000e-01
1.4286e-01 1.2500e-01 1.1111e-01 1.0000e-01 9.0909e-02
1.2500e-01 1.1111e-01 1.0000e-01 9.0909e-02 8.3333e-02
1.1111e-01 1.0000e-01 9.0909e-02 8.3333e-02 7.6923e-02
1.0000e-01 9.0909e-02 8.3333e-02 7.6923e-02 7.1429e-02
Columns 6 through 10
1.6667e-01 1.4286e-01 1.2500e-01 1.1111e-01 1.0000e-01
1.4286e-01 1.2500e-01 1.1111e-01 1.0000e-01 9.0909e-02
1.2500e-01 1.1111e-01 1.0000e-01 9.0909e-02 8.3333e-02
1.1111e-01 1.0000e-01 9.0909e-02 8.3333e-02 7.6923e-02
1.0000e-01 9.0909e-02 8.3333e-02 7.6923e-02 7.1429e-02
9.0909e-02 8.3333e-02 7.6923e-02 7.1429e-02 6.6667e-02
8.3333e-02 7.6923e-02 7.1429e-02 6.6667e-02 6.2500e-02
7.6923e-02 7.1429e-02 6.6667e-02 6.2500e-02 5.8824e-02
7.1429e-02 6.6667e-02 6.2500e-02 5.8824e-02 5.5556e-02
6.6667e-02 6.2500e-02 5.8824e-02 5.5556e-02 5.2632e-02
b
b =
1.0000e+01
7.9801e+00
6.7936e+00
5.9596e+00
5.3271e+00
4.8255e+00
4.4156e+00
4.0731e+00
3.7820e+00
3.5311e+00
H\b
ans =
1.0000e+00
2.0000e+00
3.0000e+00
4.0002e+00
4.9993e+00
6.0021e+00
6.9966e+00
8.0033e+00
8.9983e+00
1.0000e+01
xtilde=H\b
xtilde =
1.0000e+00
2.0000e+00
3.0000e+00
4.0002e+00
4.9993e+00
6.0021e+00
6.9966e+00
8.0033e+00
8.9983e+00
1.0000e+01
format long e
xtilde=H\b
xtilde =
9.999999909306136e-01
2.000000790940804e+00
2.999983029621682e+00
4.000155189181782e+00
4.999256190243317e+00
6.002052991410876e+00
6.996620157939202e+00
8.003275780790171e+00
8.998275889376139e+00
1.000037999640940e+01
r=b-A*xtilde;
{Error using *
Inner matrix dimensions must agree.
}
r=b-H*xtilde;
norm(r)
ans =
3.233018248352212e-15
cond(H)*norm(r)/norm(b)
ans =
2.731015350781805e-03
norm(xtilde-x)/norm(x)
ans =
2.794401910521555e-04
xtilde=invhilb(10)*b
xtilde =
9.999999986030161e-01
1.999999582767487e+00
3.000005722045898e+00
3.999877929687500e+00
5.000488281250000e+00
5.999511718750000e+00
7.001953125000000e+00
7.998779296875000e+00
9.000732421875000e+00
9.999786376953125e+00
norm(xtilde-x)/norm(x)
ans =
1.287165538898834e-04
A=randn(4)
A =
Columns 1 through 2
-8.636528219887144e-01 -6.849328103348064e-03
7.735909113042493e-02 1.532630308284750e+00
-1.214117043615409e+00 -7.696659137536819e-01
-1.113500741486764e+00 3.713788127600577e-01
Columns 3 through 4
-2.255844022712519e-01 5.525270211122237e-01
1.117356138814467e+00 1.100610217880866e+00
-1.089064295052236e+00 1.544211895503951e+00
3.255746416497347e-02 8.593113317542546e-02
format
A=randn(4)
A =
-1.4916 -0.6156 -0.7648 -0.1774
-0.7423 0.7481 -1.4023 -0.1961
-1.0616 -0.1924 -1.4224 1.4193
2.3505 0.8886 0.4882 0.2916
triu(A)
ans =
-1.4916 -0.6156 -0.7648 -0.1774
0 0.7481 -1.4023 -0.1961
0 0 -1.4224 1.4193
0 0 0 0.2916
tril(A)
ans =
-1.4916 0 0 0
-0.7423 0.7481 0 0
-1.0616 -0.1924 -1.4224 0
2.3505 0.8886 0.4882 0.2916
A([2,1,3,4],:)
ans =
-0.7423 0.7481 -1.4023 -0.1961
-1.4916 -0.6156 -0.7648 -0.1774
-1.0616 -0.1924 -1.4224 1.4193
2.3505 0.8886 0.4882 0.2916
A=tril(rand(4))
A =
0.5497 0 0 0
0.9172 0.3804 0 0
0.2858 0.5678 0.7792 0
0.7572 0.0759 0.9340 0.0119
A([2,1,4,3],:)
ans =
0.9172 0.3804 0 0
0.5497 0 0 0
0.7572 0.0759 0.9340 0.0119
0.2858 0.5678 0.7792 0
A=triu(rand(3))
A =
0.3371 0.3112 0.6020
0 0.5285 0.2630
0 0 0.6541
b=rand(3,1)
b =
0.6892
0.7482
0.4505
x=bs(A,b)
{Undefined function or variable 'n'.
Error in bs (line 3)
x(n) = x(n) / A(n,n);
}
x=bs(A,b)
{Undefined function or variable 'n'.
Error in bs (line 3)
x(n) = x(n) / A(n,n);
}
x=bs(A,b)
x =
-0.1759
1.0728
0.6888
A*x
ans =
0.6892
0.7482
0.4505
b
b =
0.6892
0.7482
0.4505
diary off