def plotMatrixSpectrum(model, A, mat_name):
fig_path = os.path.join(model.fig_dir,
"singular_values_{:}.png".format(mat_name))
try:
s = svdvals(A)
except:
s = -np.sort(-sparse_svds(
A, return_singular_vectors=False, k=min(100, min(A.shape))))
plt.plot(s, 'o')
plt.ylabel(r'$\sigma$')
plt.title('Singular values of {:}'.format(mat_name))
plt.savefig(fig_path)
plt.close()
if A.shape[0] == A.shape[1]: #is square
fig_path = os.path.join(model.fig_dir,
"eigenvalues_{:}.png".format(mat_name))
try:
eig = eigvals(A)
except:
eig = sparse_eigs(A,
return_eigenvectors=False,
k=min(1000, min(A.shape)))
plt.plot(eig.real, eig.imag, 'o')
plt.xlabel(r'Re($\lambda$)')
plt.ylabel(r'Im($\lambda$)')
plt.title('Eigenvalues of {:}'.format(mat_name))
plt.savefig(fig_path)
plt.close()
def compute_spectral_radius(X, is_sparse=False):
"""
Computes the spectral radius of a matrix
Parameters
----------
X : np.ndarray or scipy.sparse.csr.csr_matrix
Input matrix
is_sparse : bool, optional, default=False
Whether the passed matrix X is a SciPy sparse matrix or not.
Returns
-------
rho : float
Spectral radius of `X`
"""
# To compute the eigen-values of a matrix, it must be square
assert X.shape[0] == X.shape[1], print('Input matrix must be square')
if is_sparse:
eigvec = sparse_eigs(X, k=1, which='LM', return_eigenvectors=False)
maxeig = np.abs(eigvec[0].real)
else:
eigvals = np.linalg.eigvals(X)
maxeig = np.max(np.abs(eigvals))
return maxeig
def P_calc(A, AT, # LinearOperator and its transpose
tol=1e-6,
maxiter=10000):
"""Calculate the probabilities that maximize the entropy rate for
adjacency matrix A.
"""
from scipy.sparse.linalg import LinearOperator
from scipy.sparse.linalg import eigs as sparse_eigs
w,b = sparse_eigs(A, k=1, tol=tol, maxiter=maxiter)
print('eigenvalue=%f'%(w[0].real,))
w,e = sparse_eigs(AT, k=1, tol=tol, maxiter=maxiter)
e = e[:,-1].real # Left eigenvector
b = b[:,-1].real # Right eigenvector
norm = np.dot(e, A*b)
Ab = A*b # A vector?
mu = e*Ab/norm # Also a vector ?
L = e/mu
R = b/norm
P = LinearOperator(A.shape,
lambda v: L*(A*(R*v)),
rmatvec = lambda v: (A.rmatvec(v*L))*R
)
return mu,P
def correlation_length(B):
"Constructs the mixed transfermatrix and returns correlation length"
from scipy.sparse.linalg import eigs as sparse_eigs
chi = B[0].shape[1]
L = len(B)
T = np.tensordot(B[0], np.conj(B[0]), axes=(0, 0)) # a,b,a*,b*
T = T.transpose(0, 2, 1, 3) # a,a*,b,b*
for i in range(1, L):
T = np.tensordot(T, B[i], axes=(2, 1)) # a,a*,b*,i,b
T = np.tensordot(T, np.conj(B[i]), axes=([2, 3], [1, 0])) #a,a*,b,b*
T = np.reshape(T, (chi**2, chi**2))
# Obtain the 2nd largest eigenvalue
eta = sparse_eigs(T, k=2, which='LM', return_eigenvectors=False, ncv=20)
return -L / np.log(np.min(np.abs(eta)))
def main(argv=None):
'''For looking at sensitivity of time and results to u, dy, n_g, n_h.
'''
import argparse
global DEBUG
if argv is None: # Usual case
argv = sys.argv[1:]
parser = argparse.ArgumentParser(
description='Plot eigenfunction of the integral equation')
parser.add_argument('--u',
type=float,
default=(2.0e-5),
help='log fractional deviation')
parser.add_argument('--dy',
type=float,
default=3.2e-4,
help='y_1-y_0 = log(x_1/x_0)')
parser.add_argument('--n_g',
type=int,
default=200,
help='number of integration elements in value')
parser.add_argument(
'--n_h',
type=int,
default=200,
help=
'number of integration elements in slope. Require n_h > 192 u/(dy^2).'
)
parser.add_argument('--m_g',
type=int,
default=50,
help='number of points for plot in g direction')
parser.add_argument('--m_h',
type=int,
default=50,
help='number of points for plot in h direction')
parser.add_argument('--eigenvalue', action='store_true')
parser.add_argument('--debug', action='store_true')
parser.add_argument('--eigenfunction',
type=str,
default=None,
help="Calculate eigenfunction and write to this file")
parser.add_argument(
'--Av',
type=str,
default=None,
help="Illustrate A applied to some points and write to this file")
args = parser.parse_args(argv)
if args.Av == None and args.eigenfunction == None and not args.eigenvalue:
print('Nothing to do')
return 0
params = {
'axes.labelsize': 18, # Plotting parameters for latex
'text.fontsize': 15,
'legend.fontsize': 15,
'text.usetex': True,
'font.family': 'serif',
'font.serif': 'Computer Modern Roman',
'xtick.labelsize': 15,
'ytick.labelsize': 15
}
mpl.rcParams.update(params)
if args.debug:
DEBUG = True
mpl.rcParams['text.usetex'] = False
else:
mpl.use('PDF')
import matplotlib.pyplot as plt # must be after mpl.use
from scipy.sparse.linalg import LinearOperator
from mpl_toolkits.mplot3d import Axes3D # Mysteriously necessary
#for "projection='3d'".
from matplotlib import cm
from matplotlib.ticker import LinearLocator, FormatStrFormatter
from first_c import LO_step
#from first import LO_step # Factor of 9 slower
g_step = 2 * args.u / (args.n_g - 1)
h_max = (24 * args.u)**.5
h_step = 2 * h_max / (args.n_h - 1)
A = LO_step(args.u, args.dy, g_step, h_step)
if args.eigenvalue:
from scipy.sparse.linalg import eigs as sparse_eigs
from numpy.linalg import norm
tol = 1e-10
w, b = sparse_eigs(A, tol=tol, k=10)
print('Eigenvalues from scipy.sparse.linalg.eigs:')
for val in w:
print('norm:%e, %s' % (norm(val), val))
A.power(small=tol)
print('From A.power() %e' % (A.eigenvalue, ))
if args.eigenfunction != None:
# Calculate and plot the eigenfunction
tol = 5e-6
maxiter = 1000
A.power(small=tol, n_iter=maxiter, verbose=True)
floor = 1e-20 * max(A.eigenvector).max()
fig = plt.figure(figsize=(24, 12))
ax1 = fig.add_subplot(1, 2, 1, projection='3d', elev=15, azim=135)
ax2 = fig.add_subplot(1, 2, 2, projection='3d', elev=15, azim=45)
plot_eig(A, floor, ax1, args)
plot_eig(A, floor, ax2, args)
if not DEBUG:
fig.savefig(open(args.eigenfunction, 'wb'), format='pdf')
if DEBUG:
plt.show()
return 0
def main(argv=None):
'''For looking at sensitivity of time and results to u, dy, n_g, n_h.
'''
import argparse
global DEBUG
if argv is None: # Usual case
argv = sys.argv[1:]
parser = argparse.ArgumentParser(
description='Plot eigenfunction of the integral equation')
parser.add_argument('--u', type=float, default=(2.0e-5),
help='log fractional deviation')
parser.add_argument('--dy', type=float, default=3.2e-4,
help='y_1-y_0 = log(x_1/x_0)')
parser.add_argument('--n_g', type=int, default=200,
help='number of integration elements in value')
parser.add_argument('--n_h', type=int, default=200,
help='number of integration elements in slope. Require n_h > 192 u/(dy^2).')
parser.add_argument('--m_g', type=int, default=50,
help='number of points for plot in g direction')
parser.add_argument('--m_h', type=int, default=50,
help='number of points for plot in h direction')
parser.add_argument('--eigenvalue', action='store_true')
parser.add_argument('--debug', action='store_true')
parser.add_argument('--eigenfunction', type=str, default=None,
help="Calculate eigenfunction and write to this file")
parser.add_argument('--Av', type=str, default=None,
help="Illustrate A applied to some points and write to this file")
args = parser.parse_args(argv)
if args.Av == None and args.eigenfunction == None and not args.eigenvalue:
print('Nothing to do')
return 0
params = {'axes.labelsize': 18, # Plotting parameters for latex
'text.fontsize': 15,
'legend.fontsize': 15,
'text.usetex': True,
'font.family':'serif',
'font.serif':'Computer Modern Roman',
'xtick.labelsize': 15,
'ytick.labelsize': 15}
mpl.rcParams.update(params)
if args.debug:
DEBUG = True
mpl.rcParams['text.usetex'] = False
else:
mpl.use('PDF')
import matplotlib.pyplot as plt # must be after mpl.use
from scipy.sparse.linalg import LinearOperator
from mpl_toolkits.mplot3d import Axes3D # Mysteriously necessary
#for "projection='3d'".
from matplotlib import cm
from matplotlib.ticker import LinearLocator, FormatStrFormatter
from first_c import LO_step
#from first import LO_step # Factor of 9 slower
g_step = 2*args.u/(args.n_g-1)
h_max = (24*args.u)**.5
h_step = 2*h_max/(args.n_h-1)
A = LO_step( args.u, args.dy, g_step, h_step)
if args.eigenvalue:
from scipy.sparse.linalg import eigs as sparse_eigs
from numpy.linalg import norm
tol = 1e-10
w,b = sparse_eigs(A, tol=tol, k=10)
print('Eigenvalues from scipy.sparse.linalg.eigs:')
for val in w:
print('norm:%e, %s'%(norm(val), val))
A.power(small=tol)
print('From A.power() %e'%(A.eigenvalue,))
if args.eigenfunction != None:
# Calculate and plot the eigenfunction
tol = 5e-6
maxiter = 1000
A.power(small=tol, n_iter=maxiter, verbose=True)
floor = 1e-20*max(A.eigenvector).max()
fig = plt.figure(figsize=(24,12))
ax1 = fig.add_subplot(1,2,1, projection='3d', elev=15, azim=135)
ax2 = fig.add_subplot(1,2,2, projection='3d', elev=15, azim=45)
plot_eig(A, floor, ax1, args)
plot_eig(A, floor, ax2, args)
if not DEBUG:
fig.savefig( open(args.eigenfunction, 'wb'), format='pdf')
if DEBUG:
plt.show()
return 0