In [1]:
import numpy as np
In [2]:
S = np.matrix([[1, 3, 2, 5], [7, 4, -1, 0], [3, 8, -1, 3], [1, -5, 4, 6]])
print(S)
[[ 1 3 2 5] [ 7 4 -1 0] [ 3 8 -1 3] [ 1 -5 4 6]]
In [3]:
S[2,1]
Out[3]:
8
In [4]:
S[2,1] = -2
print(S)
[[ 1 3 2 5] [ 7 4 -1 0] [ 3 -2 -1 3] [ 1 -5 4 6]]
In [5]:
S[:,2]
Out[5]:
matrix([[ 2],
[-1],
[-1],
[ 4]])
In [6]:
S[1,:]
Out[6]:
matrix([[ 7, 4, -1, 0]])
In [7]:
S[1,:] = S[1,:] - 7 * S[0,]
print(S)
[[ 1 3 2 5] [ 0 -17 -15 -35] [ 3 -2 -1 3] [ 1 -5 4 6]]
In [8]:
S.shape
Out[8]:
(4, 4)
In [9]:
S.shape[0]
Out[9]:
4
In [10]:
S.shape[1]
Out[10]:
4
In [11]:
I = np.eye(4)
print(I)
[[1. 0. 0. 0.] [0. 1. 0. 0.] [0. 0. 1. 0.] [0. 0. 0. 1.]]
In [12]:
A = np.matrix([[1, 3, 2, 5], [-1, -1, 1, 2], [2, 4, -1, 3], [3, 5, 4, 6]])
print(A)
[[ 1 3 2 5] [-1 -1 1 2] [ 2 4 -1 3] [ 3 5 4 6]]
In [13]:
def zeroColSousDiag(j, M, N) :
"zerocolSousDiag(j : int, M : np.matrix, N : np.matrix) -> [G : np.matrix, D : np.matrix]"
G = np.copy(M) # sinon Python modifie aussi la matrice initiale
D = np.copy(N)
n = M.shape[0]
for i in range(j+1, n) :
D[i,:] = G[j,j] * D[i,:] - G[i,j] * D[j,:]
G[i,:] = G[j,j] * G[i,:] - G[i,j] * G[j,:]
return [G, D]
In [14]:
[G, D] = zeroColSousDiag(0, A, I)
print([G, D])
[array([[ 1, 3, 2, 5],
[ 0, 2, 3, 7],
[ 0, -2, -5, -7],
[ 0, -4, -2, -9]]), array([[ 1., 0., 0., 0.],
[ 1., 1., 0., 0.],
[-2., 0., 1., 0.],
[-3., 0., 0., 1.]])]
In [15]:
zeroColSousDiag(1, G, D)
Out[15]:
[array([[ 1, 3, 2, 5],
[ 0, 2, 3, 7],
[ 0, 0, -4, 0],
[ 0, 0, 8, 10]]),
array([[ 1., 0., 0., 0.],
[ 1., 1., 0., 0.],
[-2., 2., 2., 0.],
[-2., 4., 0., 2.]])]
In [16]:
def transfTriangSup(M, N) :
"transfTriangSup(M : np.matrix, N : np.matrix) -> [G : np.matrix, D : np.matrix]"
G = np.copy(M)
D = np.copy(N)
n = M.shape[0]
for j in range(n) :
[G, D] = zeroColSousDiag(j, G, D)
return [G, D]
In [17]:
transfTriangSup(A, I)
Out[17]:
[array([[ 1, 3, 2, 5],
[ 0, 2, 3, 7],
[ 0, 0, -4, 0],
[ 0, 0, 0, -40]]),
array([[ 1., 0., 0., 0.],
[ 1., 1., 0., 0.],
[ -2., 2., 2., 0.],
[ 24., -32., -16., -8.]])]
In [18]:
def zeroColSurDiag(j, T, P) :
"zeroColSurDiag(j : int, T : np.matrix, P : np.matrix) -> [G : np.matrix, D : np.matrix]"
G = np.copy(T)
D = np.copy(P)
for i in range(j) :
D[i,:] = G[j,j] * D[i,:] - G[i,j] * D[j,:]
G[i,:] = G[j,j] * G[i,:] - G[i,j] * G[j,:]
return [G, D]
In [19]:
[T, P] = transfTriangSup(A, I)
[G, D] = zeroColSurDiag(3, T, P)
print([G, D])
[array([[ -40, -120, -80, 0],
[ 0, -80, -120, 0],
[ 0, 0, 160, 0],
[ 0, 0, 0, -40]]), array([[-160., 160., 80., 40.],
[-208., 184., 112., 56.],
[ 80., -80., -80., 0.],
[ 24., -32., -16., -8.]])]
In [20]:
[G, D] = zeroColSurDiag(2, G, D)
print([G, D])
[array([[ -6400, -19200, 0, 0],
[ 0, -12800, 0, 0],
[ 0, 0, 160, 0],
[ 0, 0, 0, -40]]), array([[-1.920e+04, 1.920e+04, 6.400e+03, 6.400e+03],
[-2.368e+04, 1.984e+04, 8.320e+03, 8.960e+03],
[ 8.000e+01, -8.000e+01, -8.000e+01, 0.000e+00],
[ 2.400e+01, -3.200e+01, -1.600e+01, -8.000e+00]])]
In [21]:
def transfDiag(T, P) :
"transfDiag(T : np.matrix, P : np.matrix) -> [G : np.matrix, D : np.matrix]"
G = np.copy(T)
D = np.copy(P)
n = T.shape[0]
for j in range(n) :
[G, D] = zeroColSurDiag(j, G, D)
return [G, D]
In [22]:
[Delta, R] = transfDiag(T, P)
print([Delta, R])
[array([[320, 0, 0, 0],
[ 0, 320, 0, 0],
[ 0, 0, 160, 0],
[ 0, 0, 0, -40]]), array([[-816., 528., 304., 352.],
[ 592., -496., -208., -224.],
[ 80., -80., -80., 0.],
[ 24., -32., -16., -8.]])]
In [23]:
# Delta[0,:] = (1/2) * Delta[0,:]
# print(Delta)
In [24]:
def transfId(Delta, R) :
"transfId(Delta : np.matrix, R : np.matrix) -> [G : np.matrix, D : np.matrix]"
G = np.copy(Delta)
D = np.copy(R)
n = Delta.shape[0]
for i in range(n) :
D[i,:] = (1 / G[i,i]) * D[i,:]
G[i,:] = (1 / G[i,i]) * G[i,:]
return [G, D]
In [25]:
[L, U] = transfId(Delta, R)
print([L, U])
[array([[1, 0, 0, 0],
[0, 1, 0, 0],
[0, 0, 1, 0],
[0, 0, 0, 1]]), array([[-2.55, 1.65, 0.95, 1.1 ],
[ 1.85, -1.55, -0.65, -0.7 ],
[ 0.5 , -0.5 , -0.5 , 0. ],
[-0.6 , 0.8 , 0.4 , 0.2 ]])]
In [26]:
np.dot(A, U)
Out[26]:
matrix([[ 1.00000000e+00, 0.00000000e+00, 0.00000000e+00,
0.00000000e+00],
[ 0.00000000e+00, 1.00000000e+00, 0.00000000e+00,
0.00000000e+00],
[-8.88178420e-16, 4.44089210e-16, 1.00000000e+00,
0.00000000e+00],
[-8.88178420e-16, 4.44089210e-16, 2.22044605e-16,
1.00000000e+00]])
In [27]:
def calcInv(M) :
"calcInv(M : np.matrix) -> Inv : np.matrix"
G = np.copy(M)
n = M.shape[0]
D = np.eye(n)
[G, D] = transfTriangSup(G, D)
[G, D] = transfDiag(G, D)
[G, D] = transfId(G, D)
Inv = D
return Inv
In [28]:
V = calcInv(A)
print(V)
[[-2.55 1.65 0.95 1.1 ] [ 1.85 -1.55 -0.65 -0.7 ] [ 0.5 -0.5 -0.5 0. ] [-0.6 0.8 0.4 0.2 ]]
In [29]:
np.dot(A, V)
Out[29]:
matrix([[ 1.00000000e+00, 0.00000000e+00, 0.00000000e+00,
0.00000000e+00],
[ 0.00000000e+00, 1.00000000e+00, 0.00000000e+00,
0.00000000e+00],
[-8.88178420e-16, 4.44089210e-16, 1.00000000e+00,
0.00000000e+00],
[-8.88178420e-16, 4.44089210e-16, 2.22044605e-16,
1.00000000e+00]])
In [30]:
J = np.ones([4,4])
print(J)
[[1. 1. 1. 1.] [1. 1. 1. 1.] [1. 1. 1. 1.] [1. 1. 1. 1.]]
In [31]:
calcInv(J)
C:\Users\roxan\AppData\Local\Temp\ipykernel_22272\1601645827.py:7: RuntimeWarning: divide by zero encountered in scalar divide D[i,:] = (1 / G[i,i]) * D[i,:] C:\Users\roxan\AppData\Local\Temp\ipykernel_22272\1601645827.py:7: RuntimeWarning: invalid value encountered in multiply D[i,:] = (1 / G[i,i]) * D[i,:] C:\Users\roxan\AppData\Local\Temp\ipykernel_22272\1601645827.py:8: RuntimeWarning: divide by zero encountered in scalar divide G[i,:] = (1 / G[i,i]) * G[i,:] C:\Users\roxan\AppData\Local\Temp\ipykernel_22272\1601645827.py:8: RuntimeWarning: invalid value encountered in multiply G[i,:] = (1 / G[i,i]) * G[i,:]
Out[31]:
array([[nan, nan, nan, nan],
[nan, nan, nan, nan],
[nan, nan, nan, nan],
[nan, nan, nan, nan]])
In [32]:
P = np.matrix([[0, 1], [1, 0]])
print(P)
[[0 1] [1 0]]
In [33]:
calcInv(P)
C:\Users\roxan\AppData\Local\Temp\ipykernel_22272\1601645827.py:7: RuntimeWarning: divide by zero encountered in scalar divide D[i,:] = (1 / G[i,i]) * D[i,:] C:\Users\roxan\AppData\Local\Temp\ipykernel_22272\1601645827.py:7: RuntimeWarning: invalid value encountered in multiply D[i,:] = (1 / G[i,i]) * D[i,:] C:\Users\roxan\AppData\Local\Temp\ipykernel_22272\1601645827.py:8: RuntimeWarning: divide by zero encountered in scalar divide G[i,:] = (1 / G[i,i]) * G[i,:] C:\Users\roxan\AppData\Local\Temp\ipykernel_22272\1601645827.py:8: RuntimeWarning: invalid value encountered in multiply G[i,:] = (1 / G[i,i]) * G[i,:] C:\Users\roxan\AppData\Local\Temp\ipykernel_22272\1601645827.py:8: RuntimeWarning: invalid value encountered in cast G[i,:] = (1 / G[i,i]) * G[i,:]
Out[33]:
array([[nan, nan],
[ 1., -0.]])
In [34]:
def trouvePivot(j, M) :
"trouvePivot(j : int, M : np.matrix) -> p : int"
n = M.shape[0]
for p in range(j, n) :
if M[p, j] != 0 :
return p
return n
## Autre version
# def trouvePivot(j, M) :
# n = M.shape[0]
# p = j
# while M[p,j] == 0 and p < (n-1) :
# p = p + 1
# if M[p, j] == 0 and p == (n-1) :
# p = n
# return p
In [35]:
trouvePivot(0, P)
Out[35]:
1
In [36]:
def echangeLignes(M, N, i1, i2) :
G = np.copy(M)
D = np.copy(N)
auxG = np.copy(M)
auxD = np.copy(N)
G[i1,:] = auxG[i2,:]
G[i2,:] = auxG[i1,:]
D[i1,:] = auxD[i2,:]
D[i2,:] = auxD[i1,:]
return [G, D]
In [37]:
[B, C] = echangeLignes(U, J, 1, 3)
print([B, C])
[array([[-2.55, 1.65, 0.95, 1.1 ],
[-0.6 , 0.8 , 0.4 , 0.2 ],
[ 0.5 , -0.5 , -0.5 , 0. ],
[ 1.85, -1.55, -0.65, -0.7 ]]), array([[1., 1., 1., 1.],
[1., 1., 1., 1.],
[1., 1., 1., 1.],
[1., 1., 1., 1.]])]
In [38]:
def transfTriangSup(M, N) :
"transfTriangSup(M : np.matrix, N : np.matrix) -> [G : np.matrix, D : np.matrix, boolInv : bool]"
G = np.copy(M)
D = np.copy(N)
n = M.shape[0]
boolInv = True
j = 0
p = trouvePivot(j, G)
while j < (n-1) and p < n :
[G, D] = echangeLignes(G, D, j, p)
[G, D] = zeroColSousDiag(j, G, D)
j = j + 1
p = trouvePivot(j, G)
if p == n :
boolInv = False
return [G, D, boolInv]
In [39]:
def calcInv(M) :
"calcInv(M : np.matrix) -> Inv : np.matrix ou str"
G = np.copy(M)
n = M.shape[0]
D = np.eye(n)
[G, D, boolInv] = transfTriangSup(G, D)
if boolInv == True :
[G, D] = transfDiag(G, D)
[G, D] = transfId(G, D)
Inv = D
else :
Inv = 'La matrice n est pas inversible'
return Inv
In [ ]: