def decompose(self):
"""Compute the SVD in 2 steps:
1. Compute the eigenvalue decomposition of `A.T @ A`
using the QR algorithm. From this decomposition,
we can derive V and and Σ.
2. Solve the system `UΣ = AV` for U using a QR
factorization.
This algorithm isn't usually used in practice
because of its sensitivity to the gram matrix
`A.T @ A`.
Returns:
U: A numpy array of shape (M, M)
S: A numpy array of shape (M, N).
V: A numpy array of shape (N, N).
"""
A = np.array(self.backup)
M, N = A.shape
eigvals, V = qr_algorithm(A.T @ A)
S = create_diag(np.sqrt(eigvals), (M, N))
U = QR(S.T, reduce=False).solve(V.T @ A.T)
U = U.T
return U, S, V
def test_qr_algorithm_with_hessenberg(self):
M = random_symmetric(4)
eigvals, eigvecs = LA.eig(M)
idx = np.abs(eigvals).argsort()[::-1]
expected_eigvecs = eigvecs[:, idx]
expected_eigvals = eigvals[idx]
actual_eigvals, actual_eigvecs = multi.qr_algorithm(M)
self.assertTrue(self.absallclose(actual_eigvecs, expected_eigvecs))
self.assertTrue(self.absallclose(actual_eigvals, expected_eigvals))
"""QR algorithm vs projected iteration.
"""
import time
import numpy as np
import numpy.linalg as LA
from linalg.eigen import multi
from linalg import utils
if __name__ == "__main__":
M = utils.random_spd(10)
tic = time.time()
eigvals, eigvecs = multi.qr_algorithm(M, hess=False)
toc = time.time()
time_qr = toc - tic
print("QR: {}s".format(time_qr))
tic = time.time()
eigvals, eigvecs = multi.projected_iteration(M, len(M))
toc = time.time()
time_proj = toc - tic
print("Projected Iteration: {}s".format(time_proj))
Specifically, the QR decomposition doesn't take advantage
of the Hessenberg form by skipping row elements that are
already zero when applying reflectors to each column
of the matrix.
"""
import time
import numpy as np
import scipy.linalg as LA
from linalg.eigen import multi
from linalg import utils
if __name__ == "__main__":
M = utils.random_spd(10)
tic = time.time()
eigvals, eigvecs = multi.qr_algorithm(LA.hessenberg(M), hess=False)
toc = time.time()
time_with_hess = toc - tic
print("With hessenberg: {}s".format(time_with_hess))
tic = time.time()
eigvals, eigvecs = multi.qr_algorithm(M, hess=False)
toc = time.time()
time_without_hess = toc - tic
print("Without hessenberg: {}s".format(time_without_hess))
speedup = time_without_hess / time_with_hess
print("Speedup: {}x".format(speedup))