def solve_gen_eig_prob(A, B, eps=1e-6):
"""
Solves the generalised eigenvalue problem of the form:
Aw = \lambda*Bw
Note: can be validated against `scipy.linalg.eig(A, b=B)`
Ref:
'Eigenvalue and Generalized Eigenvalue Problems: Tutorial (2019)'
Benyamin Ghojogh and Fakhri Karray and Mark Crowley
arXiv 1903.11240
"""
Lam_b, Phi_b = np.linalg.eig(B) # eig decomp of B alone
Lam_b = np.eye(
len(Lam_b)) * Lam_b # convert to diagonal matrix of eig vals
Lam_b_sq = replace_nan(Lam_b**0.5) + np.eye(len(Lam_b)) * eps
Phi_b_hat = np.dot(Phi_b, np.linalg.inv(Lam_b_sq))
A_hat = np.dot(np.dot(Phi_b_hat.transpose(), A), Phi_b_hat)
Lam_a, Phi_a = np.linalg.eig(A_hat)
Lam_a = np.eye(len(Lam_a)) * Lam_a
Lam = Lam_a
Phi = np.dot(Phi_b_hat, Phi_a)
return np.diag(Lam), Phi
def solve_eig_qr(A, iterations=30):
"""
Use the QR iteration algorithm to iteratively solve for the eigenvectors and eigenvalues
of a matrix A. Note: only guaranteed to recover exactly for symmetric matrices
with real eigenvalues. May work partially for asymmetric matrices (no complex support yet).
Returns:
`lam`: vector of eigenvalues
`Q_bar`: matrix of eigenvectors (columns)
"""
Ak = A
Q_bar = np.eye(len(Ak))
for _ in range(iterations):
Qk, Rk = np.linalg.qr(Ak)
Ak = np.dot(Rk, Qk)
Q_bar = np.dot(Q_bar, Qk)
lam = np.diag(Ak)
return lam, Q_bar
from ulab import numpy as np print(np.eye(3))
c = np.array([[19, 22], [43, 50]]) matrix_is_close(np.dot(a, b), c, 2) c = np.array([[26, 30], [38, 44]]) matrix_is_close(np.dot(a.transpose(), b), c, 2) c = np.array([[17, 23], [39, 53]]) matrix_is_close(np.dot(a, b.transpose()), c, 2) c = np.array([[23, 31], [34, 46]]) matrix_is_close(np.dot(a.transpose(), b.transpose()), c, 2) a = np.array([[1., 2.], [3., 4.]]) b = np.linalg.inv(a) ab = np.dot(a, b) c = np.eye(2) matrix_is_close(ab, c, 2) a = np.array([[1, 2, 3, 4], [4, 5, 6, 4], [7, 8.6, 9, 4], [3, 4, 5, 6]]) b = np.linalg.inv(a) ab = np.dot(a, b) c = np.eye(4) matrix_is_close(ab, c, 4) a = np.array([[1, 2, 3, 4], [4, 5, 6, 4], [7, 8.6, 9, 4], [3, 4, 5, 6]]) result = (np.linalg.det(a)) ref_result = 7.199999999999995 print(math.isclose(result, ref_result, rel_tol=1E-9, abs_tol=1E-9)) a = np.array([1, 2, 3]) w, v = np.linalg.eig(np.diag(a))
from ulab import numpy as np
print('isinf(0): ', np.isinf(0))
a = np.array([1, 2, np.nan])
print('\n' + '=' * 20)
print('a:\n', a)
print('\nisinf(a):\n', np.isinf(a))
b = np.array([1, 2, np.inf])
print('\n' + '=' * 20)
print('b:\n', b)
print('\nisinf(b):\n', np.isinf(b))
c = np.array([1, 2, 3], dtype=np.uint16)
print('\n' + '=' * 20)
print('c:\n', c)
print('\nisinf(c):\n', np.isinf(c))
d = np.eye(5) * 1e999
print('\n' + '=' * 20)
print('d:\n', d)
print('\nisinf(d):\n', np.isinf(d))
try:
from ulab import numpy as np
except:
import numpy as np
def print_as_buffer(a):
print(len(memoryview(a)), list(memoryview(a)))
print_as_buffer(np.ones(3))
print_as_buffer(np.zeros(3))
print_as_buffer(np.eye(4))
print_as_buffer(np.ones(1, dtype=np.int8))
print_as_buffer(np.ones(2, dtype=np.uint8))
print_as_buffer(np.ones(3, dtype=np.int16))
print_as_buffer(np.ones(4, dtype=np.uint16))
print_as_buffer(np.ones(5, dtype=np.float))
print_as_buffer(np.linspace(0, 1, 9))
print(np.arange(2, 10))
print(np.arange(2, 10, 3))
print(np.arange(2, 10, 3, dtype=np.float))
print("Array concatenation:")
a = np.array([1,2,3], dtype=np.float)
b = np.array([4,5,6], dtype=np.float)
print(np.concatenate((a,b)))
print(np.concatenate((a,b), axis=0))
a = np.array([[1,2,3],[4,5,6],[7,8,9]], dtype=np.float)
b = np.array([[1,2,3],[4,5,6],[7,8,9]], dtype=np.float)
print(np.concatenate((a,b), axis=0))
print(np.concatenate((a,b), axis=1))
print(np.concatenate((b,a), axis=0))
print(np.concatenate((b,a), axis=1))
print("Identity array creation:")
print(np.eye(3))
print(np.eye(3, M=4))
print(np.eye(3, M=4, k=0))
print(np.eye(3, M=4, k=-1))
print(np.eye(3, M=4, k=-2))
print(np.eye(3, M=4, k=-3))
print(np.eye(3, M=4, k=1))
print(np.eye(3, M=4, k=2))
print(np.eye(3, M=4, k=3))
print(np.eye(4, M=4))
print(np.eye(4, M=3, k=0))
print(np.eye(4, M=3, k=-1))
print(np.eye(4, M=3, k=-2))
print(np.eye(4, M=3, k=-3))
print(np.eye(4, M=3, k=1))
print(np.eye(4, M=3, k=2))
if i == j:
print(math.isclose(v[i][j], 1.0, rel_tol=1E-9, abs_tol=1E-9))
else:
print(math.isclose(v[i][j], 0.0, rel_tol=1E-9, abs_tol=1E-9))
a = np.array([[25, 15, -5], [15, 18, 0], [-5, 0, 11]])
result = (np.linalg.cholesky(a))
ref_result = np.array([[5., 0., 0.], [3., 3., 0.], [-1., 1., 3.]])
for i in range(3):
for j in range(3):
print(
math.isclose(result[i][j],
ref_result[i][j],
rel_tol=1E-9,
abs_tol=1E-9))
a = np.array([1, 2, 3, 4, 5], dtype=np.float)
result = (np.linalg.norm(a))
ref_result = 7.416198487095663
print(math.isclose(result, ref_result, rel_tol=1E-9, abs_tol=1E-9))
a = np.array([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
result = (np.linalg.norm(a)) ## Here is a problem
ref_result = 16.881943016134134
print(math.isclose(result, ref_result, rel_tol=1E-6, abs_tol=1E-6))
if use_ulab:
print(np.linalg.trace(np.eye(3)))
else:
print(np.trace(np.eye(3)))