def process(seed, K):
"""
K is model order / number of zeros
"""
# create the dirac locations with many, many points
rng = np.random.RandomState(seed)
tk = np.sort(rng.rand(K)*period)
# true zeros
uk = np.exp(-1j*2*np.pi*tk/period)
coef_poly = poly.polyfromroots(uk) # more accurate than np.poly
# estimate zeros
uk_hat = np.roots(np.flipud(coef_poly))
uk_hat_poly = poly.polyroots(coef_poly)
uk_hat_mpmath = mpmath.polyroots(np.flipud(coef_poly), maxsteps=100,
cleanup=True, error=False, extraprec=50)
# compute error
min_dev_norm = distance(uk, uk_hat)[0]
_err_roots = 20*np.log10(np.linalg.norm(uk)/min_dev_norm)
min_dev_norm = distance(uk, uk_hat_poly)[0]
_err_poly = 20*np.log10(np.linalg.norm(uk)/min_dev_norm)
# for mpmath, need to compute error with its precision
uk = np.sort(uk)
uk_mpmath = [mpmath.mpc(z) for z in uk]
uk_hat_mpmath = sorted(uk_hat_mpmath, key=cmp_to_key(compare_mpc))
dev = [uk_mpmath[k] - uk_hat_mpmath[k] for k in range(len(uk_mpmath))]
_err_mpmath = 20*mpmath.log(mpmath.norm(uk_mpmath) / mpmath.norm(dev),
b=10)
return _err_roots, _err_poly, _err_mpmath
def visibility_first_derivative(self, posix_time):
"""Calculate the derivative of the visibility function of the satellite and the site at a
given time.
Args:
posix_time (float): The UNIX time to evaluate the derivative visibility function at.
Returns:
The value of the visibility function evaluated at the provided time.
"""
# Since most helper functions don't play well with mpmath floats we have to perform a lossy
# conversion.
posix_time = float(posix_time)
sat_pos_vel = np.array(
self.sat_irp.interpolate(posix_time)) * mp.mpf(1.0)
site_pos = np.array(self.site_ecef) * mp.mpf(1.0)
pos_diff = np.subtract(sat_pos_vel[0], site_pos)
vel_diff = sat_pos_vel[1]
site_normal_pos = site_pos / mp.norm(site_pos)
site_normal_vel = [0, 0, 0]
first_term = mp.mpf(
((1.0 / mp.norm(pos_diff)) * (mp.fdot(vel_diff, site_normal_pos) +
mp.fdot(pos_diff, site_normal_vel))))
second_term = mp.mpf(((1.0 / mp.power(
(mp.norm(pos_diff)), 3)) * mp.fdot(pos_diff, vel_diff) *
mp.fdot(pos_diff, site_normal_pos)))
return first_term - second_term
def run_stoch_eig(P, verbose=0):
"""
stoch_eig returns a stochastic vector x such that x P = x
for an irreducible stochstic matrix P.
"""
if verbose > 1:
print("original matrix (stoch_eig):\n", P)
x = mp.stoch_eig(P)
if verbose > 1:
print("x\n", x)
eps = mp.exp(0.8 * mp.log(mp.eps)) # From test_eigen.py
# x is a left eigenvector of P with eigenvalue unity
err0 = mp.norm(x*P-x, p=1)
if verbose > 0:
print("|xP - x| (stoch_eig):", err0)
assert err0 < eps
# x is a nonnegative vector
if verbose > 0:
print("min(x) (stoch_eig):", min(x))
assert min(x) >= 0 - eps
# 1-norm of x is one
err1 = mp.fabs(mp.norm(x, p=1) - 1)
if verbose > 0:
print("||x| - 1| (stoch_eig):", err1)
assert err1 < eps
def _view_cone_calc(lat_geoc, lon_geoc, sat_pos, sat_vel, q_max, m):
"""Semi-private: Performs the viewing cone visibility calculation for the day defined by m.
Note: This function is based on a paper titled "rapid satellite-to-site visibility determination
based on self-adaptive interpolation technique" with some variation to account for interaction
of viewing cone with the satellite orbit.
Args:
lat_geoc (float): site location in degrees at the start of POI
lon_geoc (float): site location in degrees at the start of POI
sat_pos (Vector3D): position of satellite (at the same time as sat_vel)
sat_vel (Vector3D): velocity of satellite (at the same time as sat_pos)
q_max (float): maximum orbital radius
m (int): interval offsets (number of days after initial condition)
Returns:
Returns 4 numbers representing times at which the orbit is tangent to the viewing cone,
Raises:
ValueError: if any of the 4 formulas has a complex answer. This happens when the orbit and
viewing cone do not intersect or only intersect twice.
Note: With more analysis it should be possible to find a correct interval even in the case
where there are only two intersections but this is beyond the current scope of the project.
"""
lat_geoc = (lat_geoc * mp.pi) / 180
lon_geoc = (lon_geoc * mp.pi) / 180
# P vector (also referred to as orbital angular momentum in the paper) calculations
p_unit_x, p_unit_y, p_unit_z = cross(sat_pos, sat_vel) / (mp.norm(sat_pos) * mp.norm(sat_vel))
# Following are equations from Viewing cone section of referenced paper
r_site_magnitude = earth_radius_at_geocetric_lat(lat_geoc)
gamma1 = THETA_NAUGHT + mp.asin((r_site_magnitude * mp.sin((mp.pi / 2) + THETA_NAUGHT)) / q_max)
gamma2 = mp.pi - gamma1
# Note: atan2 instead of atan to get the correct quadrant.
arctan_term = mp.atan2(p_unit_x, p_unit_y)
arcsin_term_gamma, arcsin_term_gamma2 = [(mp.asin((mp.cos(gamma) - p_unit_z * mp.sin(lat_geoc))
/ (mp.sqrt((p_unit_x ** 2) + (p_unit_y ** 2)) *
mp.cos(lat_geoc)))) for gamma in [gamma1, gamma2]]
angle_1 = (arcsin_term_gamma - lon_geoc - arctan_term + 2 * mp.pi * m)
angle_2 = (mp.pi - arcsin_term_gamma - lon_geoc - arctan_term + 2 * mp.pi * m)
angle_3 = (arcsin_term_gamma2 - lon_geoc - arctan_term + 2 * mp.pi * m)
angle_4 = (mp.pi - arcsin_term_gamma2 - lon_geoc - arctan_term + 2 * mp.pi * m)
angles = [angle_1, angle_2, angle_3, angle_4]
# Check for complex answers
if any([not isinstance(angle, mp.mpf) for angle in angles]):
raise ValueError()
# Map all angles to 0 to 2*pi
for idx in range(len(angles)):
while angles[idx] < 0:
angles[idx] += 2 * mp.pi
while angles[idx] > 2 * mp.pi:
angles[idx] -= 2 * mp.pi
# Calculate the corresponding time for each angle and return
return [mp.nint((1 / ANGULAR_VELOCITY_EARTH) * angle) for angle in angles]
def rotate_3D_mp(x, y):
xnorm = mpm.norm(x)
ynorm = mpm.norm(y)
cos = mpm.fdot(x,y)/(xnorm*ynorm)
sin = mpm.sqrt(1.-cos**2)
K = (y.T*x-x.T*y)/(xnorm*ynorm*sin)
return mpm.eye(3) + sin*K + (1.-cos)*(K*K)
def run_gth_solve(A, verbose=0):
"""
gth_solve returns a stochastic vector x such that x A = 0
for an irreducible transition rate matrix A.
"""
if verbose > 1:
print("original matrix (gth_solve):\n", A)
x = mp.gth_solve(A)
if verbose > 1:
print("x\n", x)
eps = mp.exp(0.8 * mp.log(mp.eps)) # test_eigen.py
# x is a solution to x A = 0
err0 = mp.norm(x*A, p=1)
if verbose > 0:
print("|xA| (gth_solve):", err0)
assert err0 < eps
# x is a nonnegative vector
if verbose > 0:
print("min(x) (gth_solve):", min(x))
assert min(x) >= 0 - eps
# 1-norm of x is one
err1 = mp.fabs(mp.norm(x, p=1) - 1)
if verbose > 0:
print("||x| - 1| (gth_solve):", err1)
assert err1 < eps
def test_poincare_reflect0():
z = mpm.matrix([[0., 0., 0.5]])
x = mpm.matrix([[0.1, -0.2, 0.1]])
assert mpm.norm(poincare_reflect0(z, z)) < TOL
y = poincare_reflect0(z, x)
assert mpm.norm(poincare_reflect0(z, y) - x) < TOL
def test_poincare_reflect():
x = mpm.matrix([[0.3, -0.3, 0.0]])
a = mpm.matrix([[0.5, 0., 0.0]])
y = poincare_reflect(a, x)
R = mpm.norm(a)
r1 = mpm.norm(x - a)
r2 = mpm.norm(y - a)
assert R**2 - r1 * r2 < TOL
def rotate_mp(pts, x, y):
out = mpm.zeros(pts.rows, pts.cols)
v = x/mpm.norm(x)
cos = mpm.fdot(x,y) / (mpm.norm(x), mpm.norm(y))
sin = mpm.sqrt(1.-cos**2)
u = y - mpm.fdot(v, y) * v
mat = mpm.eye(x.cols) - u.T * u - v.T * v \
+ cos * u.T * u - sin * v.T * u + sin * u.T * v + cos * v.T * v
return mat
def sphericalknn(data, no_clusters):
H = len(data)
W = len(data[0])
#calculating the mean to remove the dc component
sum_sample = sum(data)
norm_sample = np.linalg.norm(sum_sample)
mean_sample = (sum_sample) / (norm_sample)
#calculating global mean to get centroids of the clusters
deviation = 0.01
mean_global = np.zeros([no_clusters, W])
for i in range(0, no_clusters):
random_sample = np.random.rand(1, W) - 0.5
random_norm = deviation * (np.random.rand())
random_sample2 = (random_norm *
random_sample) / np.linalg.norm(random_sample)
temp = mean_sample + random_sample2
mean_global[i, :] = temp / np.linalg.norm(temp)
#calculating mean from spherical kmeans
sum_sample3 = np.zeros([1, W])
difference = 1
epsilon = 0.01
number = 100
iteration = 0
while (difference > epsilon):
iteration = iteration + 1
number2 = number
#computing the nearest neighbour and assigning the points
#E Step in EM algorithm
mean_global2 = np.transpose(mean_global)
value = np.dot(data, mean_global2)
value_max = value.max(1)
clusters = np.argmax(value, axis=1)
#computing value of the function
number = sum(value_max)
#print(number)
#computing centroids for the clusters
#M step in EM algorithm
for i in range(0, no_clusters):
sum_sample3 = sum(data[np.where(clusters == i)])
if (mpmath.norm(sum_sample3) != 0):
temp2 = sum_sample3 / mpmath.norm(
sum_sample3) #np.linalg.norm(sum_sample3) #Check this
mean_global[i, :] = temp2
difference = abs(number - number2)
return clusters
def refine_model_mdnewton_mpmath(low_dimensional_model,
newton_steps=5,
expected_quality=(0.1, 0.01, 1e-3, 1e-5, 1e-8,
1e-12, 1e-18, 1e-24),
log10_stop_quality=-130,
still_good=lambda v70: True,
norm=1):
"""Refines a model using mpmath's MDNewton() solver.
For some solutions, this fails due to the critical point being not a 'generic'
one.
Args:
low_dimensional_model: A LowDimensionalModel.
still_good: f(v70) -> bool, determines if the solution is still good.
newton_steps: The number of multidimensional Newton steps to take.
expected_quality: List of expected quality thresholds for each step.
(Last entry implicitly repeats.)
log10_stop_quality: Stop early if this quality-threshold is reached.
norm: The `norm` parameter for mpmath.calculus.optimization.MDNewton().
Returns:
A refined LowDimensionalModel.
Raises:
ValueError, if the solution is no longer good.
"""
if len(low_dimensional_model.params) == 0:
return low_dimensional_model
def f_off(*model_vec):
v70 = numpy.dot(low_dimensional_model.v70_from_params,
numpy.array(model_vec, dtype=mpmath.mpf))
sinfo = scalar_sector_mpmath.mpmath_scalar_manifold_evaluator(v70)
return tuple(sinfo.grad_potential)
newton = mpmath.calculus.optimization.MDNewton(
mpmath.mp,
f_off,
tuple(low_dimensional_model.params),
verbose=1,
norm=lambda x: mpmath.norm(x, norm))
newton_iter = iter(newton)
for num_step in range(1, newton_steps + 1):
opt, stationarity = next(newton_iter)
opt_v70 = numpy.dot(low_dimensional_model.v70_from_params, opt)
if mpmath.log(stationarity, 10) <= log10_stop_quality:
break
expected_stationarity = expected_quality[min(
len(expected_quality) - 1, num_step)]
if stationarity > expected_stationarity:
raise ValueError('Stationarity does not improve as expected: '
'seen=%.6g, wanted=%.6g, step=%d' %
(stationarity, expected_stationarity, num_step))
if not still_good(opt_v70):
raise ValueError('Solution is no longer good.')
print('[MDNewton] Step %d: stat=%s' % (num_step, mpfmt(stationarity)))
return LowDimensionalModel(
v70_from_params=low_dimensional_model.v70_from_params,
params=numpy.array(opt, dtype=mpmath.mpf))
def poincare_dist0(x, c=1.0, precision=None):
''' Distance from 0 to x in the Poincare model with curvature -1/c'''
if precision is not None:
mpm.mp.dps = precision
x_norm = mpm.norm(x)
sqrt_c = mpm.sqrt(c)
return 2 / sqrt_c * mpm.atanh(x_norm / sqrt_c)
def are_nested(self, v1: NormalVector, r1: float, v2: NormalVector,
r2: float):
from numpy.linalg import norm
distance = norm(v1.get_coordinate() - v2.get_coordinate())
return max(r1, r2) >= distance + min(r1, r2)
def __init__(self, theta, phi, r=None, is_polar=True):
from mpmath import mpf, mp, norm, fabs, chop
if not r:
r = Point._scale
else:
r *= Point._scale
Point._can_change_the_scale = False
mp.dps = Point._precision
if not is_polar:
self.x = mpf(theta)
self.y = mpf(phi)
self.z = mpf(r)
self.norm = norm(self.get_coordinate(), 2)
self.set_polar_coordinate()
else:
self.norm = fabs(r)
self.theta = theta
self.phi = phi
self.set_coordinate()
# Setting float parameters
self.floatx = float(self.x)
self.floaty = float(self.y)
self.floatz = float(self.z)
def test_gth_solve_fp():
P = mp.fp.matrix([[-0.1, 0.075, 0.025],
[0.15, -0.2 , 0.05 ],
[0.25, 0.25 , -0.5 ]])
x_expected = mp.fp.matrix([[0.625, 0.3125, 0.0625]])
x = mp.fp.gth_solve(P)
eps = mp.exp(0.8 * mp.log(mp.eps)) # test_eigen.py
err0 = mp.norm(x-x_expected, p=1)
assert err0 < eps
def test_stoch_eig_fp():
P = mp.fp.matrix([[0.9 , 0.075, 0.025],
[0.15, 0.8 , 0.05 ],
[0.25, 0.25 , 0.5 ]])
x_expected = mp.fp.matrix([[0.625, 0.3125, 0.0625]])
x = mp.fp.stoch_eig(P)
eps = mp.exp(0.8 * mp.log(mp.eps)) # test_eigen.py
err0 = mp.norm(x-x_expected, p=1)
assert err0 < eps
def newtonMethod(VF, X_start):
X = X_start
Y = getYacobyMatrix(VF, X)
Y_1 = Y**-1
while mp.norm(Y_1 * VF(X), 2) > eps:
Y = getYacobyMatrix(VF, X)
Y_1 = Y**-1
X = X - (Y_1 * VF(X))
return X
def has_intersection(self, v1: NormalVector, r1: float, v2: NormalVector,
r2: float):
# v1 and v2 are two distinict points on unit upper semisphere.It returns true if two balls B[v1, r1] and
# B[v2, r2] has intersection on unit sphere and in this case answers will be returned as
# up_intersect and down_intersect relative to Z - axis. It is O(1)
from numpy import cross, dot
from numpy.linalg import norm
from math import sqrt
t1 = r_to_dot(r1)
t2 = r_to_dot(r2)
u1 = v1.get_coordinate()
u2 = v2.get_coordinate()
t = dot(u1, u2)
dot_equation_solution_on_u1u2 = ((t1 - t * t2) /
(1 - t**2)) * u1 + ((t2 - t * t1) /
(1 - t**2)) * u2
norm_on_u1u2 = norm(dot_equation_solution_on_u1u2)
if norm_on_u1u2 > 1:
return False, None, None
perp = cross(u1, u2)
perp /= norm(perp)
if perp[2] < 0:
perp *= -1
perp_projection = sqrt(1 - norm_on_u1u2**2) * perp
up = dot_equation_solution_on_u1u2 + perp_projection
down = dot_equation_solution_on_u1u2 - perp_projection
up_intersection = NormalVector(up[0], up[1], up[2], is_polar=False)
down_intersection = NormalVector(down[0],
down[1],
down[2],
is_polar=False)
return True, up_intersection, down_intersection
def visibility(self, posix_time):
"""Calculate the visibility function of the satellite and the site at a given time.
Args:
posix_time (float): The time to evaluate the visibility function at
Returns:
The value of the visibility function evaluated at the provided time.
Note:
This function assumes the FOV of the sensors on the satellite are 180 degrees
"""
# Since most helper functions don't play well with mpmath floats we have to perform a lossy
# conversion.
posix_time = float(posix_time)
site_pos = np.array(self.site_ecef) * mp.mpf(1.0)
site_normal_pos = site_pos / mp.norm(site_pos)
sat_pos = self.sat_irp.interpolate(posix_time)[0]
sat_site = np.subtract(sat_pos, site_pos)
return mp.mpf(mp.fdot(sat_site, site_normal_pos) / mp.norm(sat_site))
def coherent_state(self, alpha, dtype=complex):
from ..statevector import StateVector as sv
x = numpy.empty(self.N1, dtype=dtype)
x[0] = dtype("1")
for n in range(1, self.N1):
x[n] = x[n-1]*alpha/(n)**(0.5)
if issubclass(dtype, complex):
a = numpy.exp(-numpy.abs(alpha)**2/2.)
else:
import mpmath
a = mpmath.exp(-mpmath.norm(alpha)**2/2)
x = a*x[self.N0:self.N1]
return sv(x, basis=self)
def cg(vec_x, mat_a, vec_b, rtol, atol, max_times):
dim = vec_x.rows
r = vec_b - mat_a * vec_x
p = r
init_norm_r = mpmath.norm(r)
old_norm_r = init_norm_r
# メインループ
for times in range(max_times):
ap = mat_a * p
alpha = (r.T * p)[0] / (p.T * ap)[0]
vec_x = vec_x + alpha * p
r = r - alpha * ap
new_norm_r = mpmath.norm(r)
beta = new_norm_r * new_norm_r / (old_norm_r * old_norm_r)
# 収束判定
print(times, mpmath.nstr(new_norm_r / init_norm_r))
if (new_norm_r <= (rtol * init_norm_r + atol)):
break
p = r + beta * p
old_norm_r = new_norm_r
return times, vec_x
def test_mev():
output("""\
reim:{$[0>type x;1 0*x;2=count x;x;'`]};
mc:{((x[0]*y 0)-x[1]*y 1;(x[0]*y 1)+x[1]*y 0)};
mmc:{((.qml.mm[x 0]y 0)-.qml.mm[x 1]y 1;(.qml.mm[x 0]y 1)+.qml.mm[x 1]y 0)};
mev_:{[b;x]
if[2<>count wv:.qml.mev x;'`length];
if[not all over prec>=abs
mmc[flip vc;flip(flip')(reim'')flip x]-
flip(w:reim'[wv 0])mc'vc:(flip')(reim'')(v:wv 1);'`check];
/ Normalize sign; LAPACK already normalized to real
v*:1-2*0>{x a?max a:abs x}each vc[;0];
(?'[prec>=abs w[;1];w[;0];w];?'[b;v;0n])};""")
for A in eigenvalue_subjects:
if A.rows <= 3:
V = []
for w, n, r in A.eigenvects():
w = sp.simplify(sp.expand_complex(w))
if len(r) == 1:
r = r[0]
r = sp.simplify(sp.expand_complex(r))
r = r.normalized() / sp.sign(max(r, key=abs))
r = sp.simplify(sp.expand_complex(r))
else:
r = None
V.extend([(w, r)] * n)
V.sort(key=lambda (x, _): (-abs(x), -sp.im(x)))
else:
Am = mp.matrix(A)
# extra precision for complex pairs to be equal in sort
with mp.extradps(mp.mp.dps):
W, R = mp.eig(Am)
V = []
for w, r in zip(W, (R.column(i) for i in range(R.cols))):
w = mp.chop(w)
with mp.extradps(mp.mp.dps):
_, S, _ = mp.svd(Am - w * mp.eye(A.rows))
if sum(x == 0 for x in mp.chop(S)) == 1:
# nullity 1, so normalized eigenvector is unique
r /= mp.norm(r) * mp.sign(max(r, key=abs))
r = mp.chop(r)
else:
r = None
V.append((w, r))
V.sort(key=lambda (x, _): (-abs(x), -x.imag))
W, R = zip(*V)
test("mev_[%sb" % "".join("0" if r is None else "1" for r in R),
A, (W, [r if r is None else list(r) for r in R]),
complex_pair=True)
def test_mev():
output("""\
reim:{$[0>type x;1 0*x;2=count x;x;'`]};
mc:{((x[0]*y 0)-x[1]*y 1;(x[0]*y 1)+x[1]*y 0)};
mmc:{((.qml.mm[x 0]y 0)-.qml.mm[x 1]y 1;(.qml.mm[x 0]y 1)+.qml.mm[x 1]y 0)};
mev_:{[b;x]
if[2<>count wv:.qml.mev x;'`length];
if[not all over prec>=abs
mmc[flip vc;flip(flip')(reim'')flip x]-
flip(w:reim'[wv 0])mc'vc:(flip')(reim'')(v:wv 1);'`check];
/ Normalize sign; LAPACK already normalized to real
v*:1-2*0>{x a?max a:abs x}each vc[;0];
(?'[prec>=abs w[;1];w[;0];w];?'[b;v;0n])};""")
for A in eigenvalue_subjects:
if A.rows <= 3:
V = []
for w, n, r in A.eigenvects():
w = sp.simplify(sp.expand_complex(w))
if len(r) == 1:
r = r[0]
r = sp.simplify(sp.expand_complex(r))
r = r.normalized() / sp.sign(max(r, key=abs))
r = sp.simplify(sp.expand_complex(r))
else:
r = None
V.extend([(w, r)]*n)
V.sort(key=lambda (x, _): (-abs(x), -sp.im(x)))
else:
Am = mp.matrix(A)
# extra precision for complex pairs to be equal in sort
with mp.extradps(mp.mp.dps):
W, R = mp.eig(Am)
V = []
for w, r in zip(W, (R.column(i) for i in range(R.cols))):
w = mp.chop(w)
with mp.extradps(mp.mp.dps):
_, S, _ = mp.svd(Am - w*mp.eye(A.rows))
if sum(x == 0 for x in mp.chop(S)) == 1:
# nullity 1, so normalized eigenvector is unique
r /= mp.norm(r) * mp.sign(max(r, key=abs))
r = mp.chop(r)
else:
r = None
V.append((w, r))
V.sort(key=lambda (x, _): (-abs(x), -x.imag))
W, R = zip(*V)
test("mev_[%sb" % "".join("0" if r is None else "1" for r in R), A,
(W, [r if r is None else list(r) for r in R]), complex_pair=True)
def poincare_dist(x, y, c=1.0, precision=None):
'''
The hyperbolic distance between points in the Poincare model with curvature -1/c
Args:
x, y: size 1xD mpmath matrix representing point in the D-dimensional ball |x| < 1
precision (int): bits of precision to use
Returns:
mpmath float object. Can be converted back to regular float
'''
if precision is not None:
mpm.mp.dps = precision
x2 = mpm.fdot(x, x)
y2 = mpm.fdot(y, y)
xy = mpm.fdot(x, y)
sqrt_c = mpm.sqrt(c)
denom = 1 - 2 * c * xy + c**2 * x2 * y2
norm = mpm.norm(
(-(1 - 2 * c * xy + c * y2) * x + (1. - c * x2) * y) / denom)
return 2 / sqrt_c * mpm.atanh(sqrt_c * norm)
def solve_acos_bsin(a, b, c):
# It solves the equation $ a cos(theta) + b sin(theta) = c $ for theta as the unknown.
# This has either two or no answer. It is needed for the get_covered_phi function.
# It always returns the larger answer first.
from numpy.linalg import norm
from math import acos
Norm = norm((a, b))
if abs(c) > Norm:
return None, None
dot = c / Norm
dtheta = acos(dot)
if b >= 0:
angle = acos(a / Norm)
else:
angle = -acos(a / Norm)
upans = angle + dtheta
downans = angle - dtheta
return upans, downans
dtheta_values.append(
d_theta(angle_v, paramv['k'], paramv['R'], paramv['alpha'], An))
beamshape = [
20 * mpmath.log10(abs(each / dzero_value)) for each in dtheta_values
]
beamshape_df = pd.DataFrame(data={
'deg': np.degrees(np.float32(angles)),
'd0dthet': np.float32(beamshape)
})
error = beamshape - ka5['relonaxis_db']
mean_error = np.float(np.mean(np.abs(error)))
max_error = np.max(np.float32(np.abs(error)))
matrixsolve_error = mpmath.norm(mpmath.residual(Mmn, An, bm))
#print(f'Decimal precision: {decimalprec} \n Ntrend: {Ntrend}')
print(f'Matrix solve error : {matrixsolve_error }')
print(f'Mean abs error: {mean_error}, max error: {max_error}\n')
# # %%
# an_candidate = mpmath.lu_solve(Mmn,bm)
# res_mat = mpmath.residual(Mmn, an_candidate, bm)
# res = mpmath.norm(res_mat)
# print(f'matrix error is: {np.float64(res)}')
# %%
# an_candidate, res = mpmath.qr_solve(Mmn,bm)
# print(res)
# #%%
def run(self,**kwargs):
"""Run the microkinetic model. If recalculate is True then
data which is re-loaded will be used as an initial guess; otherwise it
will be assumed to be correct."""
for key in kwargs:
setattr(self,key,kwargs[key])
#set numerical representation
if self.numerical_representation == 'mpmath':
import mpmath as mp
mp.mp.dps = self.decimal_precision + 25 #pad decimal precision
self._math = mp
self._log_imports += "\nfrom mpmath import mpf \n\n"
self._kB = mp.mpf('8.617332478e-5') #eV/K (from NIST)
self._h = mp.mpf('4.135667516e-15') #eV s (from NIST)
self._mpfloat = mp.mpf
self._matrix = mp.matrix
self._Axb_solver = mp.lu_solve
self._math.infnorm = lambda x: mp.norm(x,'inf')
elif self.numerical_representation in ['numpy','python']:
self._log_imports += "\nfrom numpy import matrix \n\n"
self._math = np
self._math.infnorm = lambda x: np.linalg.norm(x,np.inf)
self._mpfloat = float
def matrixT(*args,**kwargs):
array = np.array(args[0])
while 1 in array.shape:
sum_idx = array.shape.index(1)
array = array.sum(sum_idx)
return array.T
self._matrix = matrixT
def Axb_solver(A,b):
try:
return np.linalg.solve(A,b.T)
except np.linalg.linalg.LinAlgError:
raise ZeroDivisionError
self._Axb_solver = Axb_solver
if self.decimal_precision > 15:
print(
'Warning: Max precision with numpy/python is 16 digits')
self.decimal_precision = 15
else:
raise AttributeError(
'Numerical representation must be mpmath, numpy, or python.')
#set up interaction model
if self.adsorbate_interaction_model == 'first_order':
interaction_model = catmap.thermodynamics.FirstOrderInteractions(self)
response_func = interaction_model.interaction_response_function
if not callable(response_func):
int_function = getattr(interaction_model,
response_func+'_response')
interaction_model.interaction_response_function = int_function
self.thermodynamics.__dict__['adsorbate_interactions'] = interaction_model
elif self.adsorbate_interaction_model == 'second_order':
interaction_model = catmap.thermodynamics.SecondOrderInteractions(self)
response_func = interaction_model.interaction_response_function
if not callable(response_func):
int_function = getattr(interaction_model,
response_func+'_response')
interaction_model.interaction_response_function = int_function
self.thermodynamics.__dict__['adsorbate_interactions'] = interaction_model
elif self.adsorbate_interaction_model in ['ideal',None]:
self.thermodynamics.adsorbate_interactions = None
else:
raise AttributeError(
'Invalid adsorbate_interaction_model specified.')
self.compatibility_check()
#determine whether or not to (re-)solve model
has_all = True
for v in self.output_variables:
if not hasattr(self,v+'_map'):
has_all = False
if not hasattr(self,'stdout'):
#any re-loaded model will have stdout
has_all = False
if self._solved == self._token():
#Do not solve the same model twice
has_all = True
elif has_all and not self.recalculate:
#All data has been loaded and no verification => solved.
self._solved = self._token()
self.log('input_success')
else:
#Make "volcano plot"
if hasattr(self,'descriptor_ranges') and hasattr(self,'resolution'):
self.descriptor_space_analysis()
#Get rates at single points
if hasattr(self,'descriptors'):
self.single_point_analysis(self.descriptors)
#Get rates at multiple points
if hasattr(self,'descriptor_values'):
self.multi_point_analysis()
#Save long attrs in data_file
for attr in dir(self):
if (not attr.startswith('_') and
not callable(getattr(self,attr)) and
attr not in self._classes):
if (len(repr(getattr(self,attr))) >
self._max_log_line_length):
#line is too long for logfile -> put into pickle
self._pickle_attrs.append(attr)
pickled_data = {}
for attr in self._pickle_attrs:
pickled_data[attr] = getattr(self,attr)
pickle.dump(pickled_data,open(self.data_file,'w'))
#Make logfile
log_txt = self._log_imports
log_txt += self._header(exclude_outputs=self._pickle_attrs)
self._stdout = '\n'.join(self._log_lines)
log_txt += 'stdout = ' + '"'*3+self._stdout+'"'*3
#this construction means that self.stdout will only be set
#for models which have been re-loaded.
self._solved = self._token()
if hasattr(self,'log_file'):
logfile = self.log_file
else:
name,suffix = self.setup_file.rsplit('.',1)
if suffix != 'log':
suffix = 'log'
else:
suffix = 'out'
logfile = '.'.join([name,suffix])
f = open(logfile,'w')
f.write(log_txt)
f.close()
if getattr(self,'create_standalone',None):
self.make_standalone()
from pylab import * import mpmath mpmath.mp.dps = 50 # higher precision for demonstration #Nv=100000 Nv=100000 a = [mpmath.sin(2.*mpmath.pi*n/Nv) for n in range(Nv)] b = array(a) print b.dot(b) - Nv/2 b32= array(a,dtype=float32) b64= array(a,dtype=float64) b128= array(a,dtype=float128) print b32.dot(b32) - Nv/2. print b64.dot(b64) - Nv/2. print b128.dot(b128) - Nv/2. print print mpmath.norm(b) - mpmath.sqrt(Nv/2) print norm(b32) - sqrt(Nv/2) print norm(b64) - sqrt(Nv/2) print norm(b128) -sqrt(Nv/2)
],
'agm': [
'primitive',
[lambda x, y: mp.agm(x) if y is None else mp.agm(x, y[0]), None]
],
#
'matrix': [
'primitive',
[
lambda x, y: mp.matrix(x) if isa(x, Vector) else mp.matrix(x)
if y is None else mp.matrix(x, y[0]), None
]
],
'matrix-ref': ['primitive', [lambda x, y: x[y[0], y[1], [0]], None]],
'matrix-set': [
'primitive',
[lambda x, y: matrix_set(x, y[0], y[1][0], y[1][1][0]), None]
],
'zeros': ['primitive', [lambda x, y: mp.zeros(x), None]],
'ones': ['primitive', [lambda x, y: mp.ones(x), None]],
'eye': ['primitive', [lambda x, y: mp.eye(x), None]],
'diag': ['primitive', [lambda x, y: mp.diag(x), None]],
'randmatrix': ['primitive', [lambda x, y: mp.randmatrix(x), None]],
'matrix-inv': ['primitive', [lambda x, y: x**(-1), None]],
'norm': [
'primitive',
[lambda x, y: mp.norm(x) if y is None else mp.norm(x, y[0]), None]
],
'mnorm': ['primitive', [lambda x, y: mp.mnorm(x, y[0]), None]],
}
mat_a = mpmath.zeros(dim, dim)
for i in range(dim):
for j in range(dim):
mat_a[i, j] = mpmath.mpf(dim - max(i, j))
mat_a = mat_a.T * mat_a
# print(mat_a)
# x = [1 2 ... dim]
vec_true_x = mpmath.matrix([i for i in range(1, dim + 1)])
# nprint(vec_true_x)
# b = A * x
vec_b = mat_a * vec_true_x
# CG法実行
# vec_x := 0
vec_x = mpmath.zeros(dim, 1)
start_time1 = time.time()
iterative_times, vec_x = cg(vec_x, mat_a, vec_b, 1.0e-20, 0.0, dim * 10)
time1 = time.time() - start_time1
relerr = mpmath.norm(vec_x - vec_true_x) / mpmath.norm(vec_true_x)
print('CG: iteration, time = ', iterative_times, time1, ', relerr(vec_x) = ',
relerr)
# -------------------------------------
# Copyright (c) 2021 Tomonori Kouya
# All rights reserved.
# -------------------------------------
def make_polar_coordinate(self,
n,
is_twisted=True,
precision=30,
log_scale=0):
# This function distribute points with polar coordinate on sphere. This algorithm is O (n^2).
# It also returns a phi above which we know north pole is a local maximum.
from mpmath import mp, sin, cos, sqrt, pi, floor, mpf, asin
from numpy import array, cross
from numpy.linalg import norm
from os import remove
mp.dps = precision
remove(self.points_address)
if is_twisted:
name = "twisted_polar_coordinates_" + str(n)
self.point_set = PointSet(name=name, address=self.output_folder)
else:
name = "polar_coordinates_" + str(n)
self.point_set = PointSet(name=name, address=self.output_folder)
self.points_address = self.point_set.point_file_address
if not float(Point._log_scale) == log_scale:
Point.change_log_scale(log_scale)
print("Point scale is", Point.get_scale())
if not Point._precision == precision:
Point.change_precision(precision)
phis = [0] * (n - 1)
m = [0] * (n - 1)
for j in range(n - 1):
phis[j] = ((pi * (j + 1)) / n) - (pi / 2)
m[j] = int(floor(mpf(".5") + sqrt(3) * n * cos(phis[j])))
for j in range(n - 1):
for i in range(m[j]):
if is_twisted:
if j + 1 < n / 2:
shift = (2 * pi * (j + 1)) / (n * m[j])
else:
shift = (2 * pi * (1 - mpf(j + 1) / mpf(n))) / m[j]
else:
shift = 0
theta = ((2 * pi * i) / m[j]) + shift
new_point = Point(theta=theta, phi=phis[j])
self.point_set.add_point(new_point)
north_pole = Point(theta=0, phi=pi / 2)
south_pole = Point(theta=0, phi=-pi / 2)
self.point_set.add_point(north_pole)
self.point_set.add_point(south_pole)
self.min_discrepancy_distance = 1 / self.point_set.size
# In below we get the confidence phi around the north pole
highest_phi = -1
for i in range(int(n / 2) - 1, n - 2):
phi1 = phis[i]
phi2 = phis[i + 1]
v1 = array([mpf(0), cos(phi1), mpf(0)])
v2 = array([2 * cos(phi1), mpf(0), sin(phi2) - sin(phi1)])
normal_vector = cross(v1, v2)
normal_vector /= norm(normal_vector)
highest_phi = max(highest_phi, asin(abs(normal_vector[2])))
return float(highest_phi)
db3 scaling coefficients
Wavelet Analysis: The Scalable Structure of Information (2002)
Howard L. Resnikoff, Raymond O.Jr. Wells
Table 10.3 page 263
'''
db3 = [( _1 + sqrt10 + mp.sqrt(_5 + _2*sqrt10))/_16,
( _5 + sqrt10 + _3*mp.sqrt(_5 + _2*sqrt10))/_16,
(_10 - _2*sqrt10 + _2*mp.sqrt(_5 + _2*sqrt10))/_16,
(_10 - _2*sqrt10 - _2*mp.sqrt(_5 + _2*sqrt10))/_16,
( _5 + sqrt10 - _3*mp.sqrt(_5 + _2*sqrt10))/_16,
( _1 + sqrt10 - mp.sqrt(_5 + _2*sqrt10))/_16]
db3 = np.array(db3)
D6 = db3
for a in [D2, D4, D6]:
norm = mpmath.norm(a)
assert mpmath.almosteq(norm, mp.sqrt(_2))
def wavelet_coefficients(a):
# TODO: is this just true for Daubechies wavelets?
N = len(a)
return np.array([(-1)**k*a[N-1-k] for k in range(N)])
# maps from genus (i.e. vanishing moments) to the scaling coefficients
daubechies_wavelets = {
1: D2,
2: D4,
3: D6,
}
def calc_ellipsoid_axes(coords, uvals, cell, probability=0.5, longest=True):
"""
This method calculates the principal axes of an ellipsoid as list of two
fractional coordinate triples.
Many thanks to R. W. Grosse-Kunstleve and P. D. Adams
for their great publication on the handling of atomic anisotropic displacement
parameters:
R. W. Grosse-Kunstleve, P. D. Adams, J Appl Crystallogr 2002, 35, 477–480.
F = ... * exp ( -2π²[ h²(a*)²U11 + k²(b*)²U22 + ... + 2hka*b*U12 ] )
SHELXL atom:
Name type x y z occ U11 U22 U33 U23 U13 U12
F3 4 0.210835 0.104067 0.437922 21.00000 0.07243 0.03058 =
0.03216 -0.01057 -0.01708 0.03014
>>> import mpmath as mpm
>>> cell = (10.5086, 20.9035, 20.5072, 90, 94.13, 90)
>>> coords = [0.210835, 0.104067, 0.437922]
>>> uvals = [0.07243, 0.03058, 0.03216, -0.01057, -0.01708, 0.03014]
>>> l = calc_ellipsoid_axes(coords, uvals, cell, longest=True)
>>> print(mpm.nstr(l))
[[0.24765096, 0.11383281, 0.43064756], [0.17401904, 0.09430119, 0.44519644]]
>>> calc_ellipsoid_axes(coords, uvals, cell, longest=False)
[[[0.24765096, 0.11383281, 0.43064756], [0.218406, 0.09626142, 0.43746127], [0.21924358, 0.10514684, 0.44886868]], [[0.17401904, 0.09430119, 0.44519644], [0.203264, 0.11187258, 0.43838273], [0.20242642, 0.10298716, 0.42697532]]]
>>> cell = (10.5086, 20.9035, 20.5072, 90, 94.13, 90)
>>> coords = [0.210835, 0.104067, 0.437922]
>>> uvals = [0.07243, -0.03058, 0.03216, -0.01057, -0.01708, 0.03014]
>>> calc_ellipsoid_axes(coords, uvals, cell, longest=True)
<BLANKLINE>
Ellipsoid is non positive definite!
<BLANKLINE>
False
>>> uvals = [0.07243, 0.03058, 0.03216, -0.01057, -0.01708]
>>> calc_ellipsoid_axes(coords, uvals, cell, longest=False)
Traceback (most recent call last):
...
Exception: 6 Uij values have to be supplied!
>>> cell = (10.5086, 20.9035, 90, 94.13, 90)
>>> coords = [0.210835, 0.104067, 0.437922]
>>> uvals = [0.07243, 0.03058, 0.03216, -0.01057, -0.01708, 0.03014]
>>> calc_ellipsoid_axes(coords, uvals, cell, longest=True)
Traceback (most recent call last):
...
Exception: cell needs six parameters!
:param coords: coordinates of the respective atom in fractional coordinates
:type coords: list
:param uvals: Uij valiues of the respective ellipsoid on fractional
basis like in cif and SHELXL format
:type uvals: list
:param cell: unit cell of the structure: a, b, c, alpha, beta, gamma
:type cell: list
:param probability: thermal probability of the ellipsoid
:type probability: float or int
:param longest: not always the length is important. make to False to
get all three coordiantes of the ellipsoid axes.
:type longest: boolean
"""
from misc import A
probability += 1
# Uij is symmetric:
if len(uvals) != 6:
raise Exception('6 Uij values have to be supplied!')
if len(cell) != 6:
raise Exception('cell needs six parameters!')
# orthogonalization matrix that transforms the fractional coordinates
# with respect to a crystallographic basis system to coordinates
# with respect to a Cartesian basis:
A = A(cell).orthogonal_matrix
Ucart = ufrac_to_ucart(A, cell, uvals)
# print(Ucart)
# E => eigenvalues, Q => eigenvectors:
E, Q = mpm.eig(Ucart)
# calculate vectors of ellipsoid axes
try:
sqrt(E[0])
sqrt(E[1])
sqrt(E[2])
except ValueError:
print('\nEllipsoid is non positive definite!\n')
return False
v1 = mpm.matrix([Q[0, 0], Q[1, 0], Q[2, 0]])
v2 = mpm.matrix([Q[0, 1], Q[1, 1], Q[2, 1]])
v3 = mpm.matrix([Q[0, 2], Q[1, 2], Q[2, 2]])
v1i = v1 * (-1)
v2i = v2 * (-1)
v3i = v3 * (-1)
# multiply probability (usually 50%)
e1 = sqrt(E[0]) * probability
e2 = sqrt(E[1]) * probability
e3 = sqrt(E[2]) * probability
# scale axis vectors to eigenvalues
v1, v2, v3, v1i, v2i, v3i = v1 * e1, v2 * e2, v3 * e3, v1i * e1, v2i * e2, v3i * e3
# find out which vector is the longest:
length = mpm.norm(v1)
v = 0
if mpm.norm(v2) > length:
length = mpm.norm(v2)
v = 1
elif mpm.norm(v3) > length:
length = mpm.norm(v3)
v = 2
# move vectors back to atomic position
atom = A * mpm.matrix(coords)
v1, v1i = v1 + atom, v1i + atom
v2, v2i = v2 + atom, v2i + atom
v3, v3i = v3 + atom, v3i + atom
# go back into fractional coordinates:
a1 = cart_to_frac(v1, cell)
a2 = cart_to_frac(v2, cell)
a3 = cart_to_frac(v3, cell)
a1i = cart_to_frac(v1i, cell)
a2i = cart_to_frac(v2i, cell)
a3i = cart_to_frac(v3i, cell)
allvec = [[a1, a2, a3], [a1i, a2i, a3i]]
if longest:
# only the longest vector
return [allvec[0][v], allvec[1][v]]
else:
# all vectors:
return allvec
def test_left_eigen_vec(self):
self.assertTrue(mp.norm(v - v*mp.mp.matrix(P), 'inf') <= TOL)
# chebyshev approximation
start_2 = time.time()
sv_vector = mp.matrix(12, 2)
A_abs = mp.matrix(12)
for i in range(12):
for j in range(12):
A_abs[i, j] = abs(A[i, j])
for i in range(12):
sv_vector[i, 0] = A_abs[i, i] - sum(A_abs[i, :])
sv_vector[i, 1] = A_abs[i, i] - sum(A_abs[:, i])
s_min = min(max(sv_vector[:, 0]), max(sv_vector[:, 1]))
s_max = mp.sqrt(mp.norm(A, p=1) * mp.norm(A, p=10))
# S = mp.svd_r(A, compute_uv=False)
# delta = max(s_max, abs(1 - s_max))
m = 2
dim = 12
delta = s_min ** 2/ (s_max ** 2 + s_min ** 2)
# delta_2 = min(S) ** 2/ (max(S) ** 2 + min(S) ** 2)
# s_min, s_max = min(S), max(S)
interval = [-1, 1]
func = lambda x : mp.log(1 - ((1 - 2 * delta) * x + 1) / 2)
coeffs = mp.chebyfit(func, interval, N=m)
# B = mp.eye(12) - A
B = 1/(s_max ** 2 + s_min ** 2) * A.T * A
def Han_algorithm(B, m, dim):
"""
def run(self, **kwargs):
"""Run the microkinetic model. If recalculate is True then
data which is re-loaded will be used as an initial guess; otherwise it
will be assumed to be correct.
:param recalculate: If True solve model again using previous results as initial guess
:type recalculate: bool
"""
for key in kwargs:
setattr(self, key, kwargs[key])
#ensure resolution has the proper dimensions
if not hasattr(self.resolution, '__iter__'):
self.resolution = [self.resolution] * len(self.descriptor_names)
#set numerical representation
if self.numerical_representation == 'mpmath':
import mpmath as mp
mp.mp.dps = self.decimal_precision + 25 #pad decimal precision
self._math = mp
self._log_imports += "\nfrom mpmath import mpf \n\n"
self._kB = mp.mpf('8.617332478e-5') #eV/K (from NIST)
self._h = mp.mpf('4.135667516e-15') #eV s (from NIST)
self._mpfloat = mp.mpf
self._matrix = mp.matrix
self._Axb_solver = mp.lu_solve
self._math.infnorm = lambda x: mp.norm(x, 'inf')
elif self.numerical_representation in ['numpy', 'python']:
self._log_imports += "\nfrom numpy import matrix \n\n"
self._math = np
self._math.infnorm = lambda x: np.linalg.norm(x, np.inf)
self._mpfloat = float
def matrixT(*args, **kwargs):
array = np.array(args[0])
while 1 in array.shape:
sum_idx = array.shape.index(1)
array = array.sum(sum_idx)
return array.T
self._matrix = matrixT
def Axb_solver(A, b):
try:
return np.linalg.solve(A, b.T)
except np.linalg.linalg.LinAlgError:
raise ZeroDivisionError
self._Axb_solver = Axb_solver
if self.decimal_precision > 15:
print('Warning: Max precision with numpy/python is 16 digits')
self.decimal_precision = 15
else:
raise AttributeError(
'Numerical representation must be mpmath, numpy, or python.')
#set up interaction model
if self.adsorbate_interaction_model == 'first_order':
interaction_model = catmap.thermodynamics.FirstOrderInteractions(
self)
interaction_model.get_interaction_info()
response_func = interaction_model.interaction_response_function
if not callable(response_func):
int_function = getattr(interaction_model,
response_func + '_response')
interaction_model.interaction_response_function = int_function
self.thermodynamics.__dict__[
'adsorbate_interactions'] = interaction_model
elif self.adsorbate_interaction_model == 'second_order':
interaction_model = catmap.thermodynamics.SecondOrderInteractions(
self)
interaction_model.get_interaction_info()
response_func = interaction_model.interaction_response_function
if not callable(response_func):
int_function = getattr(interaction_model,
response_func + '_response')
interaction_model.interaction_response_function = int_function
self.thermodynamics.__dict__[
'adsorbate_interactions'] = interaction_model
elif self.adsorbate_interaction_model in ['ideal', None]:
self.thermodynamics.adsorbate_interactions = None
else:
raise AttributeError(
'Invalid adsorbate_interaction_model specified.')
self.compatibility_check()
#determine whether or not to (re-)solve model
has_all = True
for v in self.output_variables:
if not hasattr(self, v + '_map'):
has_all = False
if not hasattr(self, 'stdout'):
#any re-loaded model will have stdout
has_all = False
if self._solved == self._token() and not getattr(
self, 'recalculate', None):
#Do not solve the same model twice
has_all = True
elif has_all and not getattr(self, 'recalculate', None):
#All data has been loaded and no verification => solved.
self._solved = self._token()
self.log('input_success')
else:
ran_dsa = False #When no map exists, run descriptor space analysis first
if not getattr(self, 'coverage_map', None):
#Make "volcano plot"
if getattr(self, 'descriptor_ranges', None) and getattr(
self, 'resolution', None):
self.descriptor_space_analysis()
ran_dsa = True
#Get rates at single points
if getattr(self, 'descriptors', None):
self.single_point_analysis(self.descriptors)
#Get rates at multiple points
if getattr(self, 'descriptor_values', None):
self.multi_point_analysis()
#If a map exists, run descriptor space analysis last (so that single-point guesses are
#not discarded)
if not ran_dsa and getattr(self, 'descriptor_ranges',
None) and getattr(
self, 'resolution', None):
self.descriptor_space_analysis()
#Save long attrs in data_file
for attr in dir(self):
if (not attr.startswith('_')
and not callable(getattr(self, attr))
and attr not in self._classes):
if (len(repr(getattr(self, attr))) >
self._max_log_line_length):
#line is too long for logfile -> put into pickle
self._pickle_attrs.append(attr)
pickled_data = {}
for attr in self._pickle_attrs:
pickled_data[attr] = getattr(self, attr)
pickle.dump(pickled_data, open(self.data_file, 'w'))
#Make logfile
log_txt = self._log_imports
log_txt += self._header(exclude_outputs=self._pickle_attrs)
self._stdout = '\n'.join(self._log_lines)
log_txt += 'stdout = ' + '"' * 3 + self._stdout + '"' * 3
#this construction means that self.stdout will only be set
#for models which have been re-loaded.
self._solved = self._token()
if hasattr(self, 'log_file'):
logfile = self.log_file
else:
name, suffix = self.setup_file.rsplit('.', 1)
if suffix != 'log':
suffix = 'log'
else:
suffix = 'out'
logfile = '.'.join([name, suffix])
f = open(logfile, 'w')
f.write(log_txt)
f.close()
if getattr(self, 'create_standalone', None):
self.make_standalone()
def sarkar_embedding(tree, root, **kwargs):
'''
Embed a tree in the Poincare disc using Sarkar's algorithm
from "Low Distortion Delaunay Embedding of Trees in Hyperbolic Plane.
Args:
tree (networkx.Graph) : The tree represented with int node labels.
Weighted trees should have the edge attribute "weight"
root (int): The node to use as the root of the embedding
Keyword Args:
weighted (bool): True if the tree is weighted (default True)
tau (float): the scaling factor for distances.
By default it is calculated based on statistics of the tree.
epsilon (float): parameter >0 controlling distortion bound (default 0.1).
precision (int): number of bits of precision to use.
By default it is calculated based on tau and epsilon.
Returns:
size N x 2 mpmath.matrix containing the coordinates of embedded nodes
'''
eps = kwargs.get("epsilon",0.1)
weighted = kwargs.get("weighted", True)
tau = kwargs.get("tau")
max_deg = max(tree.degree)[1]
if tau is None:
tau = (1+eps)/eps * mpm.log(2*max_deg/ mpm.pi)
prc = kwargs.get("precision")
if prc is None:
prc = _embedding_precision(tree,root,eps)
mpm.mp.dps = prc
n = tree.order()
emb = mpm.zeros(n,2)
place = []
# place the children of root
for i, v in enumerate(tree[root]):
if weighted:
r = mpm.tanh( tau*tree[root][v]["weight"])
else:
r = mpm.tanh(tau)
theta = 2*i*mpm.pi / tree.degree[root]
emb[v,0] = r*mpm.cos(theta)
emb[v,1] = r*mpm.sin(theta)
place.append((root,v))
# TODO parallelize this
while place:
u, v = place.pop() # u is the parent of v
p, x = emb[u,:], emb[v,:]
rp = poincare_reflect0(x, p, precision=prc)
arg = mpm.acos(rp[0]/mpm.norm(rp))
if rp[1] < 0:
arg = 2*mpm.pi - arg
theta = 2*mpm.pi / tree.degree[v]
i=0
for w in tree[v]:
if w == u: continue
i+=1
if weighted:
r = mpm.tanh(tau*tree[v][w]["weight"])
else:
r = mpm.tanh(tau)
w_emb = r * mpm.matrix([mpm.cos(arg+theta*i),mpm.sin(arg+theta*i)]).T
w_emb = poincare_reflect0(x, w_emb, precision=prc)
emb[w,:] = w_emb
place.append((v,w))
return emb
def characterize_solution(ss, _matrix):
residues = residues_calc(ss, _matrix)
sae_residues = norm(residues, 1)
sse_residues = norm(residues, 2)
return sae_residues, sse_residues
def lyapunov_exponent(mathcalA, varphi, periodic_points, alg='basic', norm=2):
k = max([len(word) for word in periodic_points])
if alg == 'basic':
approx_basic = []
for n in range(1, k + 1):
integral = sum([
mpmath.log(mpmath.norm(cocycle(word, mathcalA), p=norm)) *
weight(word, varphi) for word in periodic_points
if len(word) == n
])
normalization = sum([
weight(word, varphi) for word in periodic_points
if len(word) == n
])
approx_basic.append(integral / (n * normalization))
return approx_basic
elif alg == 'pollicott':
#Compute the operator trace for each periodic point
op_trace = {
word: operator_trace(cocycle(word, mathcalA), 0)[0]
for word in periodic_points
}
op_trace_der = {
word: operator_trace(cocycle(word, mathcalA), 0)[1]
for word in periodic_points
}
#Compute traces for products of transfer operator put in dictionary indexed by power
trace = [
sum([(op_trace[word] * weight(word, varphi))
for word in periodic_points if len(word) == n])
for n in range(1, k + 1)
]
trace_der = [
sum([(op_trace_der[word] * weight(word, varphi))
for word in periodic_points if len(word) == n])
for n in range(1, k + 1)
]
coefficients = [mpmath.mpf(1)]
coefficients_der = [mpmath.mpf(0)]
for n in range(1, k + 1):
M = mpmath.matrix(n)
Der_M = mpmath.matrix(n)
for i in range(0, n):
for j in range(0, n):
if j > i + 1:
M[i, j] = 0
Der_M[i, j] = 0
elif j == i + 1:
M[i, j] = n - j
Der_M[i, j] = 0
else:
M[i, j] = trace[i - j]
Der_M[i, j] = trace_der[i - j]
coefficients.append((((-1)**n) / mpmath.fac(n)) * mpmath.det(M))
if n == 1:
coefficients_der.append(
(((-1)**n) / mpmath.fac(n)) * trace_der[0])
else:
#Use Jacobi's formula to compute derivative of coefficients
coefficients_der.append(
(((-1)**n) / mpmath.fac(n)) * trace_of(adj(M) * Der_M))
approximation = []
for n in range(1, k + 1):
approximation.append(
sum([coefficients_der[m] for m in range(1, n + 1)]) /
sum([m * coefficients[m] for m in range(1, n + 1)]))
return approximation
else:
return "Choices of algorithm are 'basic' and 'pollicott'"