def jacobi_method(A, b, iterations=5, startvector=None, w=1, threshold=0.1, full_output=False):
if isinstance(A, SparseMatrix):
Dinvers_content = {key: 1/A[key] for key in A.content.keys() if key[0] == key[1]}
D_invers = SparseMatrix([], [], [], shape=A.shape)
D_invers.content = Dinvers_content
Diw = D_invers * w
else:
Dinvers = np.zeros(A.shape)
np.fill_diagonal(Dinvers, 1/4)
Diw = Dinvers * w
if startvector is not None:
x_old = startvector
else:
x_old = np.zeros(b.shape)
if full_output:
residua = []
for k in range(iterations):
# Vorlesung 21.04 S. 4
residuum = (b - (A @ x_old))
if full_output:
residua.append(np.linalg.norm(residuum))
x_new = x_old + Diw @ residuum
if residuum.max() < threshold and threshold > 0:
print(residuum.max(), k)
break
x_old = x_new
if full_output:
return x_new, residua
return x_new
def test_transposed(self):
a = SparseMatrix([1, 1, 1], [0, 1, 2], [0, 1, 2])
self.assertEqual(a, a.T)
b = SparseMatrix([1, 1, 1], [0, 0, 0], [0, 1, 2])
c = SparseMatrix([1, 1, 1], [0, 1, 2], [0, 0, 0])
self.assertEqual(b, c.T)
def test_mul(self):
a = SparseMatrix([1, 2, 3], [0, 1, 0], [1, 2, 0])
b = 2
c = SparseMatrix([2, 4, 6], [0, 1, 0], [1, 2, 0])
self.assertEqual(a * b, c)
print(b * a)
self.assertEqual(b * a, c)
def test_radd(self):
a = SparseMatrix([1, 1, 1], [0, 1, 2], [0, 1, 2], shape=(3, 3))
b = SparseMatrix([1, 1, 1], [1, 1, 1], [0, 1, 2], shape=(3, 3))
c = SparseMatrix([1, 1, 2, 1, 1], [0, 1, 1, 1, 2], [0, 0, 1, 2, 2])
self.assertEqual(a, 0 + a)
d = sum([a, b])
self.assertEqual(c, d)
def test_jacobi(self):
A = SparseMatrix([4, -2, 1, -1, 5, -2, 1, 1, 3],
[0, 0, 0, 1, 1, 1, 2, 2, 2],
[0, 1, 2, 0, 1, 2, 0, 1, 2])
b = SparseMatrix([2, 4, 6], [0, 1, 2], [0, 0, 0])
x0 = SparseMatrix([1, 2, 1], [0, 1, 2], [0, 0, 0])
x1 = SparseMatrix([0.25, 1.4, 1], [0, 1, 2], [0, 0, 0])
res = jacobi_method(A, b, iterations=1, startvector=x0, w=1)
print(res)
self.assertEqual(x1, res)
def question2():
first_image1 = np.array(MNIST_DATASET[0].tolist())
n, m, coord = array_to_coord(first_image1)
mat = SparseMatrix(fromiter=coord, shape=(n, m))
first_image2 = mat.todense()
for i in range(n):
for j in range(m):
if not first_image1[i, j] == first_image2[i, j]:
print('Erreur, les images sont differentes')
return None
print('Les deux images sont identiques')
return None
def SOR(A, b, iterations=5, startvector=None, w=1/2, threshold=0.1, full_output=False):
# D = {key: A[key] for key in A.content.keys() if key[0] == key[1]}
# R = {key: A[key] for key in A.content.keys() if key[0] < key[1]}
# L = {key: A[key] for key in A.content.keys() if key[0] > key[1]}
# I = {key: 1 for key in A.content.keys() if key[0] == key[1]}
# for i in 1 ... laenge x
# berechne si
# berechne xi
# Gauß-Seidel: -> berechne die si mit den geupdateten xi (in jacobi würde man ERST ALLE si berechnen)
# für k+1: s_i = A_zeile_i * x^k
# für k+1 für die i-te komponente: x^k+1_i = x^k_i - w/(A_i,i) * (s_i - b_i)
# Jacobi si, si, si, ..., xi, xi, xi, ...
# Gauß-Seidel si, xi, si, xi, ...
# SOR -> 1/(A-i,i) wird zu w/(A_i,i)
# aus dem video
x_old = startvector or SparseMatrix([], [], [], shape=b.shape)
x_new = startvector or SparseMatrix([], [], [], shape=b.shape)
if full_output:
residua = []
for k in range(iterations):
for i in range(b.shape[0]):
#A_i_content = {key: A[key] for key in A.content.keys() if key[0] == i}
A_i_content = {(0, key[1]): A[key] for key in A.content.keys() if key[0] == i}# hier war der fehler
A_i = SparseMatrix([], [], [], shape=(1, A.shape[1]))
A_i.content = A_i_content
# print("Ai ", A_i)
s_i = (A_i @ x_old)
wert = s_i[0, 0]
# print(A_i, " @ ", x_old, "=", s_i)
# print("s", i, " = ", wert)
x_new[i, 0] = x_old[i, 0] - w/(A[i,i]) * (wert - b[i, 0])
x_old = 1 * x_new # TODO: copy methode implementieren
residuum = (b - (A @ x_old))
if full_output:
residua.append(max(residuum.content.values()))
if max(residuum.content.values()) < threshold:
break
# eine Iteration, ein mal den Lösungsvektor
# if ...:
# break
# k Iterations
if full_output:
return x_new, residua
return x_old
def test_matmul(self):
# test shape
a = SparseMatrix([], [], [], shape=(3, 5))
b = SparseMatrix([], [], [], shape=(5, 7))
self.assertEqual((3, 7), (a @ b).shape)
#example from Wikipedia
a = SparseMatrix([3, 2, 1, 1, 2], [0, 0, 0, 1, 1], [0, 1, 2, 0, 2])
b = SparseMatrix([1, 2, 1, 4], [0, 0, 1, 2], [0, 1, 1, 0])
c = SparseMatrix([7, 8, 9, 2], [0, 0, 1, 1], [0, 1, 0, 1])
self.assertEqual(c, a @ b)
def test_add(self):
a = SparseMatrix([1, 1, 1], [0, 1, 2], [0, 1, 2], shape=(3, 3))
b = SparseMatrix([1, 1, 1], [1, 1, 1], [0, 1, 2], shape=(3, 3))
c = SparseMatrix([1, 1, 2, 1, 1], [0, 1, 1, 1, 2], [0, 0, 1, 2, 2])
d = SparseMatrix([1, -1, -1, 1], [0, 1, 1, 2], [0, 0, 2, 2],
shape=(3, 3))
self.assertEqual(c, a + b)
self.assertEqual(d, a - b)
e = SparseMatrix([], [], [], shape=(10, 10))
self.assertRaises(ArithmeticError, a.__add__, e)
def test_shape_parameter(self):
values = [1, 2]
row = [0, 1]
col = [0, 1]
a = SparseMatrix(values, row, col)
b = SparseMatrix(values, row, col, shape=(3, 4))
c = SparseMatrix([1, 1], [12, 23], [7, 9])
self.assertEqual(a.shape, (2, 2))
self.assertEqual(b.shape, (3, 4))
self.assertRaises(IndexError, a.__getitem__, (2, 3))
self.assertEqual(b[2, 3], 0)
self.assertEqual(c.shape, (24, 10))
def test_equals(self):
a = SparseMatrix([1, 1, 1], [0, 1, 2], [0, 1, 2])
b = SparseMatrix([1, 1, 1], [2, 1, 0], [2, 1, 0])
c = SparseMatrix([1, 1, 1], [2, 0, 1], [2, 0, 1])
d = SparseMatrix([1, 1, 1, 0], [0, 1, 2, 1], [0, 1, 2, 2])
e = SparseMatrix([1, 1, 1], [0, 1, 2], [0, 1, 2], shape=(4, 4))
self.assertEqual(a, b)
self.assertEqual(a, c)
self.assertEqual(b, c)
self.assertEqual(a, d)
self.assertNotEqual(a, e)
def test_non_zero_indexing(self):
values = [1, 2, 3, 4]
row = [1, 2, 3, 4]
col = [1, 2, 3, 4]
mat = SparseMatrix(values, row, col)
self.assertEqual(mat[1, 1], 1)
self.assertEqual(mat[2, 2], 2)
self.assertEqual(mat[3, 3], 3)
self.assertEqual(mat[4, 4], 4)
def test_zero_indexing(self):
values = [1, 2, 3, 4]
row = [1, 2, 3, 4]
col = [1, 2, 3, 4]
mat = SparseMatrix(values, row, col)
self.assertEqual(mat[0, 0], 0)
self.assertEqual(mat[1, 2], 0)
self.assertEqual(mat[3, 4], 0)
self.assertEqual(mat[4, 3], 0)
def test_out_of_range_indexing(self):
values = [1, 2, 3, 4]
row = [1, 2, 3, 4]
col = [1, 2, 3, 4]
mat = SparseMatrix(values, row, col)
self.assertRaises(IndexError, mat.__getitem__, (0, -1))
self.assertRaises(IndexError, mat.__getitem__, (-1, 0))
self.assertRaises(IndexError, mat.__getitem__, (0, 5))
self.assertRaises(IndexError, mat.__getitem__, (5, 0))
def test_setitem(self):
a = SparseMatrix([1, 1, 1], [0, 1, 2], [0, 1, 2])
self.assertEqual(a[0, 0], 1)
a[0, 0] = 5
self.assertEqual(a[0, 0], 5)
self.assertEqual(a[0, 2], 0)
a[0, 2] = 5
self.assertEqual(a[0, 2], 5)
self.assertRaises(IndexError, a.__setitem__, (0, 3), 5)
def calc_pmi(X_Y_C):
P = SparseMatrix()
X_C, Y_C = Counter(), Counter()
x_y_sum = 0
for x, x_Y_C in X_Y_C.items():
for y, c in x_Y_C.items():
X_C[x] += c
Y_C[y] += c
x_y_sum += c
for x, x_Y_C in X_Y_C.items():
for y, c in x_Y_C.items():
P[x, y] = log2((c * x_y_sum) / (X_C[x] * Y_C[y]))
return P
# k Iterations
if full_output:
return x_new, residua
return x_old
if __name__ == '__main__':
## example from the english wikipedia
# https://en.wikipedia.org/wiki/Jacobi_method
# A = SparseMatrix([2, 1, 5, 7], [0, 0, 1, 1], [0, 1, 0, 1])
# b = SparseMatrix([11, 13], [0, 1], [0, 0])
# x0 = SparseMatrix([1, 1], [0, 1], [0, 0])
A = SparseMatrix([4, -2, 1, -1, 5, -2, 1, 1, 3], [0, 0, 0, 1, 1, 1, 2, 2, 2], [0, 1, 2, 0, 1, 2, 0, 1, 2])
b = SparseMatrix([2, 4, 6], [0, 1, 2], [0, 0, 0])
x0 = SparseMatrix([1, 2, 1], [0, 1, 2], [0, 0, 0])
x_jac1 = SparseMatrix([1.25, 1.4, 1], [0, 1, 2], [0, 0, 0])
x_gasei1 = SparseMatrix([1.25, 1.45, 1.1], [0, 1, 2], [0, 0, 0])
print("--- JACOBI ---")
# res = jacobi_method(A, b, iterations=1, w=1)
# print("1 iterations: ", res)
# # print("EXPECTED: 1 iterations: ", x_jac1)
# res = jacobi_method(A, b, iterations=2, w=1)
# print("2 iterations: ", res)
res, residua = jacobi_method(A, b, iterations=25, w=1, full_output=True)
def _new(cls, *args, **kwargs):
s = SparseMatrix(*args)
rows = Integer(s.rows)
cols = Integer(s.cols)
mat = Dict(s._smat)
return Basic.__new__(cls, rows, cols, mat)
from matplotlib import pyplot as plt
#%% load data
# a_ij = (np.loadtxt("./test_data/a_ij.dat")).tolist()
# i_index = (np.loadtxt("./test_data/i.dat", dtype=int)).tolist()
# j_index = (np.loadtxt("./test_data/j.dat", dtype=int)).tolist()
# b_vector = (np.loadtxt("./test_data/b.dat")).tolist()
# %% generate sparse matrizes
# A = SparseMatrix(a_ij, i_index, j_index)
# b = SparseMatrix(b_vector, list(range(A.shape[0])), [0 for i in range(A.shape[1])])
#%%
A = SparseMatrix([4, -2, 1, -1, 5, -2, 1, 1, 3], [0, 0, 0, 1, 1, 1, 2, 2, 2],
[0, 1, 2, 0, 1, 2, 0, 1, 2])
b = SparseMatrix([2, 4, 6], [0, 1, 2], [0, 0, 0])
#x0 = SparseMatrix([1, 2, 1], [0, 1, 2], [0, 0, 0])
# %% run linsolve Methods
th = 0.0001
max_its = 10
print('jacobi')
jacobi_res, jacobi_resids = jacobi_method(A,
b,
threshold=th,
iterations=max_its,
full_output=True)
print('gaus seidel')
import numpy as np
import math as math
import matplotlib.pyplot as plt
from sparse import SparseMatrix
%matplotlib inline
# Importation des donnes MNIST
mnist_dataset = np.memmap('train-images-idx3-ubyte', offset=16, shape=(60000, 28, 28))
premiere_image = mnist_dataset[0].tolist() # first_image est de taille (28, 28)
encodage_premiere_image = SparseMatrix(premiere_image,(28,28))
premiere_image_encode_decode = encodage_premiere_image.todense()
plt.imshow(premiere_image, cmap='gray_r')
plt.show()
plt.imshow(encodage_premiere_image.todense(), cmap='gray_r')
plt.show()
# Comparaison Pixel par Pixel
pas_pareil = False
for i in range(len(premiere_image)):
for j in range(len(premiere_image[i])):
if premiere_image[i][j] != premiere_image_encode_decode[i][j]:
print(premiere_image[i][j])
print(premiere_image_encode_decode[i][j])
print(i)
print(j)
print("\n")
pas_pareil = True