LU/QR Class Demonstrations¶

In [1]:
import numpy as np
import scipy
import matplotlib.pyplot as plt
# This next line sets the default output precission for Numpy stuff.
np.set_printoptions(precision=4, suppress=True)

LU Factorization¶

In [2]:
A = np.array([[1.,2,3],[4,5,6],[7,8,1]])
print(A)

[[1. 2. 3.]
 [4. 5. 6.]
 [7. 8. 1.]]
In [3]:
P, L, U = scipy.linalg.lu(A)
print(P,'\n')
print(L,'\n')
print(U)

[[0. 1. 0.]
 [0. 0. 1.]
 [1. 0. 0.]] 

[[1.     0.     0.    ]
 [0.1429 1.     0.    ]
 [0.5714 0.5    1.    ]] 

[[7.     8.     1.    ]
 [0.     0.8571 2.8571]
 [0.     0.     4.    ]]
In [4]:
A
Out[4]:
array([[1., 2., 3.],
       [4., 5., 6.],
       [7., 8., 1.]])
In [5]:
P @ L @ U
Out[5]:
array([[1., 2., 3.],
       [4., 5., 6.],
       [7., 8., 1.]])
In [6]:
U
Out[6]:
array([[7.    , 8.    , 1.    ],
       [0.    , 0.8571, 2.8571],
       [0.    , 0.    , 4.    ]])
In [7]:
L
Out[7]:
array([[1.    , 0.    , 0.    ],
       [0.1429, 1.    , 0.    ],
       [0.5714, 0.5   , 1.    ]])
In [8]:
myL = np.array([[1,0,0],[4,1.,0],[7,2,1]])
myU = np.array([[1.,2,3],[0,-3,-6],[0,0,-8]])
print(myL,'\n')
print(myU,'\n')
print(myL@myU)

[[1. 0. 0.]
 [4. 1. 0.]
 [7. 2. 1.]] 

[[ 1.  2.  3.]
 [ 0. -3. -6.]
 [ 0.  0. -8.]] 

[[1. 2. 3.]
 [4. 5. 6.]
 [7. 8. 1.]]

QR Factorization¶

In [9]:
A = np.random.rand(7,4)
#A[:,2] = A[:,0]
In [10]:
print(A)

[[0.5027 0.9672 0.0954 0.9943]
 [0.196  0.8704 0.0372 0.4618]
 [0.6588 0.4099 0.1037 0.0735]
 [0.0973 0.4428 0.8303 0.4044]
 [0.9057 0.9722 0.2053 0.6316]
 [0.53   0.8138 0.3113 0.2674]
 [0.8983 0.919  0.1748 0.2062]]
In [11]:
Q, R = np.linalg.qr(A, 'reduced')
print(Q, '\n')
print(R)

[[-0.3093 -0.4066  0.233  -0.6765]
 [-0.1206 -0.6918  0.3272  0.3701]
 [-0.4053  0.3944 -0.0774  0.0098]
 [-0.0598 -0.3551 -0.9028 -0.0906]
 [-0.5571  0.1002  0.0068 -0.3112]
 [-0.326  -0.2062 -0.1308  0.3915]
 [-0.5526  0.1483  0.023   0.3832]] 

[[-1.6256 -1.9114 -0.4381 -0.9702]
 [ 0.     -0.925  -0.3361 -0.7996]
 [ 0.      0.     -0.7585 -0.0139]
 [ 0.      0.      0.     -0.5505]]
In [12]:
Q, R = np.linalg.qr(A, 'complete')
print(Q, '\n')
print(R)

[[-0.3093 -0.4066  0.233  -0.6765 -0.4636  0.0981  0.0506]
 [-0.1206 -0.6918  0.3272  0.3701  0.1997 -0.4412 -0.1682]
 [-0.4053  0.3944 -0.0774  0.0098 -0.2922 -0.4529 -0.6194]
 [-0.0598 -0.3551 -0.9028 -0.0906 -0.005  -0.2035  0.0756]
 [-0.5571  0.1002  0.0068 -0.3112  0.759   0.0806  0.0055]
 [-0.326  -0.2062 -0.1308  0.3915 -0.125   0.7276 -0.3685]
 [-0.5526  0.1483  0.023   0.3832 -0.2608 -0.115   0.6663]] 

[[-1.6256 -1.9114 -0.4381 -0.9702]
 [ 0.     -0.925  -0.3361 -0.7996]
 [ 0.      0.     -0.7585 -0.0139]
 [ 0.      0.      0.     -0.5505]
 [ 0.      0.      0.      0.    ]
 [ 0.      0.      0.      0.    ]
 [ 0.      0.      0.      0.    ]]
In [13]:
print(A,'\n')
print(Q@R)

[[0.5027 0.9672 0.0954 0.9943]
 [0.196  0.8704 0.0372 0.4618]
 [0.6588 0.4099 0.1037 0.0735]
 [0.0973 0.4428 0.8303 0.4044]
 [0.9057 0.9722 0.2053 0.6316]
 [0.53   0.8138 0.3113 0.2674]
 [0.8983 0.919  0.1748 0.2062]] 

[[0.5027 0.9672 0.0954 0.9943]
 [0.196  0.8704 0.0372 0.4618]
 [0.6588 0.4099 0.1037 0.0735]
 [0.0973 0.4428 0.8303 0.4044]
 [0.9057 0.9722 0.2053 0.6316]
 [0.53   0.8138 0.3113 0.2674]
 [0.8983 0.919  0.1748 0.2062]]
In [14]:
print(Q.T@Q)

[[ 1.  0. -0.  0.  0. -0.  0.]
 [ 0.  1. -0.  0.  0. -0.  0.]
 [-0. -0.  1.  0. -0. -0. -0.]
 [ 0.  0.  0.  1.  0.  0.  0.]
 [ 0.  0. -0.  0.  1.  0. -0.]
 [-0. -0. -0.  0.  0.  1. -0.]
 [ 0.  0. -0.  0. -0. -0.  1.]]
In [15]:
plt.spy(Q.T@Q, precision=0.0001);
In [ ]: