def gram_schmidt(self):
"""Computes QR using the Gram-Schmidt method.
Suffers from numerical instabilities when 2 vectors
are nearly orthogonal which may cause loss of
orthogonality between the columns of Q.
Since we compute a unique QR decomposition, we force
the diagonal elements of R to be positive.
"""
self.A = np.array(self.backup)
M, N = self.A.shape
self.R = np.array(self.A)
self.A[:, 0] /= utils.l2_norm(self.A[:, 0])
for i in range(1, N):
for j in range(i):
self.A[:, i] -= utils.projection(self.A[:, i], self.A[:, j])
utils.normalize(self.A[:, i], inplace=True)
self.Q = self.A
self.R = np.dot(self.Q.T, self.R)
return self.Q, self.R
def projected_iteration(A, k, max_iter=1000, sort=True):
"""Sequentially find the k eigenpairs of a symmetric matrix.
Concretely, combines power iteration and deflation to find
eigenpairs in order of decreasing magnitude.
Args:
A: a square symmetric array of shape (N, N).
k (int): the number of eigenpairs to return.
sort (bool): Whether to sort by decreasing eigenvalue magnitude.
Returns:
e, v: eigenvalues and eigenvectors. The eigenvectors are
stacked column-wise.
"""
assert utils.is_symmetric(A), "[!] Matrix must be symmetric."
assert k > 0 and k <= A.shape[0], "[!] k must be between 1 and {}.".format(
A.shape[0])
eigvecs = np.zeros((A.shape[0], k))
eigvals = np.zeros(A.shape[0])
for i in range(k):
v = np.random.randn(A.shape[0])
for _ in range(max_iter):
# project out computed eigenvectors
proj_sum = np.zeros_like(v)
for j in range(i):
proj_sum += utils.projection(v, eigvecs[:, j])
v -= proj_sum
v_new = A @ v
v_new = utils.normalize(v_new)
if np.all(np.abs(v_new - v) < 1e-8):
break
v = v_new
e = single.rayleigh_quotient(A, v)
# store eigenpair
eigvecs[:, i] = np.array(v)
eigvals[i] = e
# sort by largest absolute eigenvalue
if sort:
idx = np.abs(eigvals).argsort()[::-1]
eigvecs = eigvecs[:, idx]
eigvals = eigvals[idx]
return eigvals, eigvecs
def power_iteration(A, max_iter=1000):
"""Finds the largest eigenpair of a symmetric matrix.
Args:
A: a square symmetric array of shape (N, N).
Returns:
e, v: eigenvalue and right eigenvector.
"""
assert utils.is_symmetric(A), "[!] Matrix must be symmetric."
v = np.random.randn(A.shape[0])
for i in range(max_iter):
v_new = A @ v
v_new = utils.normalize(v_new)
if np.all(np.abs(v_new - v) < 1e-8):
break
v = v_new
e = rayleigh_quotient(A, v)
return e, v
def rayleigh_quotient_iteration(A, mu, max_iter=1000):
"""Finds an eigenpair closest to an initial eigenvalue guess.
Args:
A: a square symmetric array of shape (N, N).
mu: an initial eigenvalue guess.
Returns:
e, v: eigenvalue and right eigenvector.
"""
assert utils.is_symmetric(A), "[!] Matrix must be symmetric."
v = np.random.randn(A.shape[0])
for i in range(max_iter):
v_new = solve(A - mu * np.eye(A.shape[0]), v)
v_new = utils.normalize(v_new)
if np.all(np.abs(v_new - v) < 1e-8):
break
v = v_new
mu = rayleigh_quotient(A, v)
return mu, v
def inverse_iteration(A, max_iter=1000):
"""Finds the smallest eigenpair of a symmetric matrix.
Args:
A: a square symmetric array of shape (N, N).
Returns:
e, v: eigenvalue and right eigenvector.
"""
assert utils.is_symmetric(A), "[!] Matrix must be symmetric."
v = np.random.randn(A.shape[0])
PLU = LU(A, pivoting='partial').decompose()
for i in range(max_iter):
v_new = solve(PLU, v)
v_new = utils.normalize(v_new)
if np.all(np.abs(v_new - v) < 1e-8):
break
v = v_new
e = rayleigh_quotient(A, v)
return e, v