def test_jacobian(self):
C_0, C_1 = symbols('C_0 C_1')
state_vars = [C_0, C_1]
t = Symbol('t')
# linear compartmental matrix
# constant input
# The result is the compartmental matrix
input_fluxes = {0: 50, 1: 60}
internal_fluxes = {}
output_fluxes = {0: C_0, 1: 6 * C_1}
rm = SmoothReservoirModel(state_vars, t, input_fluxes, output_fluxes,
internal_fluxes)
self.assertEqual(rm.jacobian, -diag(1, 6))
# linear compartmental matrix
# but 'linear' state dependent input I=M*C+I0
# The result is J=B+M
sv = Matrix(state_vars)
B = -diag(1, 6)
M = Matrix([[1, 2], [3, 4]])
I0 = Matrix(2, 1, [50, 60])
I = M * sv + I0
rm = SmoothReservoirModel.from_B_u(sv, t, -diag(1, 6), I)
self.assertEqual(rm.jacobian, B + M)
# non linear compartmental matrix
# constant input
# The result is NOT the compartmental matrix B
state_vars = [C_0]
output_fluxes = {0: C_0**3}
#input_fluxes = {0:50,1:60}
input_fluxes = {}
rm = SmoothReservoirModel(state_vars, t, input_fluxes, output_fluxes,
internal_fluxes)
J = rm.jacobian
CM = Matrix([-3 * C_0**2])
# the next line breaks
self.assertEqual(J, CM)
# non linear compartmental matrix
# with (linear) external function in input
state_vars = [C_0]
output_fluxes = {0: C_0**3}
f_expr = Function('f')(t)
def f_func(t_val):
return t_val
func_set = {f_expr: f_func}
input_fluxes = {0: C_0 * f_expr}
#input_fluxes = {0:50,1:60}
internal_fluxes = {}
rm = SmoothReservoirModel(state_vars, t, input_fluxes, output_fluxes,
internal_fluxes)
J = rm.jacobian
CM = Matrix([-3 * C_0**2 + f_expr])
self.assertEqual(J, CM)
def sym():
# # Not necessary but gives nice-looking latex output
# # More info at: http://docs.sympy.org/latest/tutorial/printing.html
# sp.init_printing()
#
# sx, sy, rho = sp.symbols('sigma_x sigma_y rho')
# matrix = sp.Matrix([[sx ** 2, rho * sx * sy],
# [rho * sx * sy, sy ** 2]])
# print(sp.pretty(matrix.inv()))
# print(sp.pretty(sp.simplify(matrix.inv())))
print('lets go')
i = sp.symbols('i')
A = sp.Matrix(sp.symarray('a', (6, 3)))
sigmas = sp.symbols(' '.join(['s{}'.format(k) for k in range(6)]))
Sll = sp.diag(*sigmas)
N = A.T * Sll.inv() * A
# (sp.pretty(N))
A2 = A.copy()
Sll2 = sp.diag(*sigmas)
# Sll2[0, 0] = Sll2[0, 0] * (1 - i)
N2 = A.T * Sll2**-1 * A
# print(sp.pretty(N2))
Sxx = N.inv()
Sxx2 = N2.inv()
print('OK')
def __init__(self, scheme, output_txt=False):
# pylint: disable=unsubscriptable-object
self.nvtot = scheme.s.shape[0]
self.consm = list(scheme.consm.keys())
self.param = scheme.param
self.dim = scheme.dim
self.is_stable_l2 = True
self.output_txt = output_txt
if scheme.rel_vel is None:
jacobian = scheme.EQ.jacobian(self.consm)
else:
jacobian = (scheme.Tu * scheme.EQ).jacobian(self.consm)
relax_mat_m = sp.eye(self.nvtot) - sp.diag(*scheme.s)
relax_mat_m[:, :len(self.consm)] += sp.diag(*scheme.s) * jacobian
if scheme.rel_vel is not None:
relax_mat_m = scheme.Tmu * relax_mat_m * scheme.Tu
relax_mat_m = relax_mat_m.subs([
(i, j)
for i, j in zip([rel_ux, rel_uy, rel_uz], scheme.rel_vel)
])
self.relax_mat_f = scheme.invM * relax_mat_m * scheme.M
# alltogether(self.relax_mat_f)
velocities = sp.Matrix(scheme.stencil.get_all_velocities())
self.velocities = np.asarray(velocities).astype('float')
def get_matrix_of_converted_atoms(Nu, positions, pending_conversion, natural_influence, Omicron, D):
"""
:param Nu: A matrix with a shape=(<number of Matters in Universe>, <number of Atoms in Universe>) where
each Nu[i,j] stands for how many atoms of type j in matter of type i.
:type Nu: Matrix
:param ps: positions of matters
:type ps: [Matrix]
:return:
"""
x, y = symbols('x y')
number_of_matters, number_of_atoms = Nu.shape
M = zeros(0, number_of_atoms)
if number_of_matters != len(positions):
raise Exception("Parameters shapes mismatch.")
for (i, position) in enumerate(positions):
(a, b) = tuple(position)
K = get_conversion_ratio_matrix(pending_conversion, Nu[i, :])
M = M.col_join(((diag(*(ones(1, number_of_atoms)*diag(*K)*Omicron.transpose()))*D).transpose() *
natural_influence).transpose().subs({x: a, y: b}))
return M.evalf()
def __init__(self, scheme):
# pylint: disable=unsubscriptable-object
self.nvtot = scheme.s.shape[0]
self.consm = list(scheme.consm.keys())
self.param = scheme.param
self.dim = scheme.dim
self.is_stable_l2 = True
if scheme.rel_vel is None:
jacobian = scheme.EQ.jacobian(self.consm)
else:
jacobian = (scheme.Tu * scheme.EQ).jacobian(self.consm)
relax_mat_m = sp.eye(self.nvtot) - sp.diag(*scheme.s)
relax_mat_m[:, :len(self.consm)] += sp.diag(*scheme.s) * jacobian
if scheme.rel_vel is not None:
relax_mat_m = scheme.Tmu * relax_mat_m * scheme.Tu
relax_mat_m = relax_mat_m.subs(
[
(i, j) for i, j in zip(
[rel_ux, rel_uy, rel_uz], scheme.rel_vel
)
]
)
self.relax_mat_f = scheme.invM * relax_mat_m * scheme.M
# alltogether(self.relax_mat_f)
velocities = sp.Matrix(scheme.stencil.get_all_velocities())
self.velocities = np.asarray(velocities).astype('float')
def matrix_power(M, n):
P, jordan_cells = M.jordan_cells()
for j in jordan_cells:
jordan_cell_power(j, n)
a = P
b = sp.diag(*jordan_cells)
c = P.inv()
return P * sp.diag(*jordan_cells) * P.inv()
def get_refl_transform(nx, ny): nss = nx**2 + ny**2 v = Matrix([[nx, ny]]).T H2 = diag(1,1) - 2*v*v.T / nss H3 = diag(1,1,1) H3[0:2,0:2] = H2 return H3
def _compute_properties(self):
"""
Computes the macroscopic properties of the solution
"""
M_pmbr = Matrix([self.endmember_proportions]).T
self.site_species_proportions = (self.endmember_site_occupancies.T *
M_pmbr)
A = self.endmember_site_occupancies.T
M_alpha = Matrix([a for a in self.site_species_alphas])
self.endmember_alphas = simplify_matrix(np.array(A.T * M_alpha))
invalphap = Matrix([1. / a for a in self.endmember_alphas[:]])
B = diag(*M_alpha) * A * diag(*invalphap)
try:
B = simplify_matrix(np.array(B))
except TypeError:
pass
f = 2. / (np.einsum('i, j -> ij', self.site_species_alphas,
np.ones(self.n_site_species)) +
np.einsum('i, j -> ij', np.ones(self.n_site_species),
self.site_species_alphas))
Wmod = emult(self.site_species_interactions, simplify_matrix(f))
Q = B.T * Wmod * B
self.endmember_energies = Matrix(np.ones(self.n_mbrs))
self.endmember_interactions = Q[:, :]
for i in range(self.n_mbrs):
self.endmember_energies[i] = Q[i, i] * self.endmember_alphas[i]
for j in range(i, self.n_mbrs):
self.endmember_interactions[i, j] = (
(Q[i, j] + Q[j, i] - Q[i, i] - Q[j, j]) *
(self.endmember_alphas[i] + self.endmember_alphas[j]) / 2)
self.endmember_interactions[j, i] = 0
# Convert sympy objects to floats if possible
# also normalise solution to self.n_sites
normalise = symplify(self.n_sites)
self.endmember_alphas /= normalise
self.endmember_interactions *= normalise
self.endmember_energies *= normalise
self.endmember_alphas = vector_to_array(self.endmember_alphas)
self.endmember_interactions = matrix_to_array(
self.endmember_interactions)
self.site_species_proportions = vector_to_array(
self.site_species_proportions)
self.endmember_energies = vector_to_array(self.endmember_energies)
def test_replace_arrays_partial_derivative():
x, y, z, t = symbols("x y z t")
expr = PartialDerivative(A(i), A(j))
assert expr.replace_with_arrays({A(i): [x, y]}, [i, j]) == Array([[1, 0], [0, 1]])
expr = PartialDerivative(A(i), A(-i))
assert expr.replace_with_arrays({A(i): [x, y], L: diag(1, 1)}, []) == 2
assert expr.replace_with_arrays({A(i): [x, y], L: diag(1, -1)}, []) == 0
expr = PartialDerivative(A(-i), A(i))
assert expr.replace_with_arrays({A(i): [x, y], L: diag(1, 1)}, []) == 2
assert expr.replace_with_arrays({A(i): [x, y], L: diag(1, -1)}, []) == 0
def get_refl_transform(nx, ny):
if DEBUG:
print "Transforming normal:"
print nx, ny
nss = nx**2 + ny**2
v = Matrix([[nx, ny]]).T
H2 = diag(1,1) - 2*v*v.T / nss
H3 = diag(1,1,1)
H3[0:2,0:2] = H2
return H3
def test_replace_arrays_partial_derivative():
x, y, z, t = symbols("x y z t")
expr = PartialDerivative(A(i), A(j))
assert expr.replace_with_arrays({A(i): [x, y]}, [i, j]) == Array([[1, 0],
[0, 1]])
expr = PartialDerivative(A(i), A(-i))
assert expr.replace_with_arrays({A(i): [x, y], L: diag(1, 1)}, []) == 2
assert expr.replace_with_arrays({A(i): [x, y], L: diag(1, -1)}, []) == 0
expr = PartialDerivative(A(-i), A(i))
assert expr.replace_with_arrays({A(i): [x, y], L: diag(1, 1)}, []) == 2
assert expr.replace_with_arrays({A(i): [x, y], L: diag(1, -1)}, []) == 0
def make_state_space_sym(num_signals, num_states, is_homo):
mu_rho_dict_sym = {}
state_state_sym_dict = {}
mu_names = ['mu_'+str(i) for i in range(num_states)]
rho_names = ['rho_'+str(i) for i in range(num_states)]
mu_rho_names = mu_names + rho_names
mu_names_sym = [sympy.Symbol(x) for x in mu_names]
rho_names_sym = [sympy.Symbol(x) for x in rho_names]
mu_rho_names_sym = [sympy.Symbol(x) for x in mu_rho_names]
mu_rho_dict_sym.update(dict(zip(mu_rho_names, mu_rho_names_sym)))
A_identity_sym = sympy.eye(num_states)
A_rhoblock_sym = sympy.diag(*rho_names_sym)
A_sym = sympy.diag(A_identity_sym, A_rhoblock_sym)
D_sym_mu_part = sympy.ones(num_signals, num_states)
D_sym_zeta_part = sympy.ones(num_signals, num_states)
D_sym = D_sym_mu_part.row_join(D_sym_zeta_part)
if is_homo:
sigmas_signal_names = [
'sigma_signal_'+str(i) for i in range(num_signals)]
sigmas_state_names = ['sigma_state_'+str(i) for i in range(num_states)]
sigmas_signal_sym = [sympy.Symbol(x) for x in sigmas_signal_names]
sigmas_state_sym = [sympy.Symbol(x) for x in sigmas_state_names]
C_nonsingularblock_sym = sympy.diag(*sigmas_state_sym)
G_nonsingularblock_sym = sympy.diag(*sigmas_signal_sym)
C_singularblock_sym = sympy.zeros(num_states, num_states)
G_singularblock_sym = sympy.zeros(num_signals, num_states)
G_sym = G_singularblock_sym.row_join(G_nonsingularblock_sym)
C_sym = sympy.diag(C_singularblock_sym, C_nonsingularblock_sym)
main_matrices_sym = {
'A_z': A_sym, 'C_z': C_sym, 'D_s': D_sym, 'G_s': G_sym}
sub_matrices_sym = {'A_z_stable': A_rhoblock_sym,
'C_z_nonsingular': C_nonsingularblock_sym,
'G_s_nonsingular': G_nonsingularblock_sym}
state_state_sym_dict.update(main_matrices_sym)
state_state_sym_dict.update(sub_matrices_sym)
return state_state_sym_dict
def hamilton(M):
text = "Metoda Cayleya-Hamiltona<br>"
eigen_values, eigen_vectors = eigen(M)
text += 'Eigenvalues: $' + sp.latex(eigen_values) + ' = ' + sp.latex(
eigen_values.evalf(5)) + '$<br>'
eigen_values = eigen_values.evalf(5)
x = sp.symbols(
str(['a' + str(i)
for i in range(len(eigen_values))])[1:-1].replace("'", ""))
if len(eigen_values) == 1:
x = [x]
A = sp.ones(len(x), 1)
for i in range(1, len(x)):
A = sp.Matrix(sp.BlockMatrix([A, sp.HadamardPower(eigen_values, i)]))
b = sp.Matrix(sp.HadamardPower(sp.E, eigen_values * t))
result = sp.Matrix(list(sp.linsolve((A, b), *x))[0])
text += "$"+sp.latex(b)+' = '+sp.latex(A)+'\\bullet'+sp.latex(sp.Matrix(x))+'\\implies' + \
sp.latex(sp.Matrix(x))+' = '+sp.latex(result) + "$<br>"
s = sp.diag(*tuple([1] * len(x))) * result[0, 0]
for i in range(1, len(x)):
s += M * result[i, 0]
text += "$e^{\\mathbb{A}t} = " + sp.latex(sp.simplify(s)) + "$<br><br>"
return s, text
def Ernst(B=symbols("B"), M=symbols("M")):
"""
Black holes in a magnetic universe.
J. Math. Phys., 17:54–56, 1976.
Frederick J. Ernst.
Parameters
----------
M : ~sympy.core.basic.Basic or int or float
Mass of the black hole. Defaults to ``M``.
B : ~sympy.core.basic.Basic or int or float
The magnetic field strength
Defaults to ``B``.
"""
coords = symbols("t r theta phi")
t, r, th, ph = coords
# Helper functions
lambd = 1 + ((B * r * sin(th))**2)
w = 1 - ((2 * M) / r)
# define the metric
metric = diag(
-1 * (lambd**2) * w,
(lambd**2) / w,
((r * lambd)**2),
(((r * sin(th)) / lambd)**2),
).tolist()
return MetricTensor(metric, coords, "ll")
def _calculate_local_remainder_inhomogeneous(self):
r"""Calculate the non-quadratic remainder matrix :math:`W(x) = V(x) - U(x)` of the
quadratic approximation matrix :math:`U(x)` of the potential's eigenvalue matrix
:math:`\Lambda(x)`. This function is used for the inhomogeneous case.
"""
if self._remainder_eigen_ih_s is not None:
# Calculation already done at some earlier time
return
else:
self._remainder_eigen_ih_s = []
self.calculate_eigenvalues()
self.calculate_jacobian()
self.calculate_hessian()
# Quadratic Taylor series for all eigenvalues
quadratics = []
for index, eigenvalue in enumerate(self._eigenvalues_s):
# Point q where the Taylor series is computed
# This is a column vector q = (q1, ... ,qD)
qs = [
sympy.Symbol("q" + str(i))
for i, v in enumerate(self._variables)
]
pairs = [(xi, qi) for xi, qi in zip(self._variables, qs)]
V = self._eigenvalues_s[index].subs(pairs)
J = self._jacobian_s[index].subs(pairs)
H = self._hessian_s[index].subs(pairs)
# Symbolic expression for the quadratic Taylor expansion term
xmq = sympy.Matrix([(xi - qi)
for xi, qi in zip(self._variables, qs)])
quadratic = sympy.Matrix(
[[V]]) + J.T * xmq + sympy.Rational(1, 2) * xmq.T * H * xmq
try:
quadratic = quadratic.applyfunc(sympy.simplify)
except:
pass
quadratics.append(quadratic[0, 0])
# Symbolic expression for the Taylor expansion remainder term
U = sympy.diag(*quadratics)
remainder = self._potential_s - U
# Symbolic simplification may fail
if self._try_simplify:
try:
remainder = remainder.applyfunc(sympy.simplify)
except:
pass
self._remainder_eigen_ih_s = remainder
# Construct functions to evaluate the approximation at point q at the given nodes
self._remainder_eigen_ih_n = tuple(
sympy.lambdify(list(self._variables) + qs, entry, "numpy")
for entry in remainder)
def get_submatrix(self, matrix):
r"""
Returns
=======
macaulay_submatrix: Matrix
The Macaulay denominator matrix. Columns that are non reduced are kept.
The row which contains one of the a_{i}s is dropped. a_{i}s
are the coefficients of x_i ^ {d_i}.
"""
reduced, non_reduced = self.get_reduced_nonreduced()
# if reduced == [], then det(matrix) should be 1
if reduced == []:
return diag([1])
# reduced != []
reduction_set = [v ** self.degrees[i] for i, v
in enumerate(self.variables)]
ais = list([self.polynomials[i].coeff(reduction_set[i])
for i in range(self.n)])
reduced_matrix = matrix[:, reduced]
keep = []
for row in range(reduced_matrix.rows):
check = [ai in reduced_matrix[row, :] for ai in ais]
if True not in check:
keep.append(row)
return matrix[keep, non_reduced]
def get_hermitian_base(dim):
""" get complete Hermitian basis with a certain dimension
return (list of Matrix): a list of Hermitian basis
dim (int): dimension of the Hermitian matrix
"""
if dim==2:
return pauli_matrices
elif dim==4:
return gamma_matrices
elif dim==8:
return pg_matrices
assert dim > 0
base = [sp.diag(*[0]*i,1,*[0]*(dim-i-1)) for i in range(dim)]
for row in range(dim):
for col in range(row+1, dim):
basis = [[0 for i in range(dim)] for j in range(dim)]
basis[row][col] = 1
basis = sp.Matrix(basis)
base.append(basis+basis.H)
basis = [[0 for i in range(dim)] for j in range(dim)]
basis[row][col] = sp.nsimplify("I")
basis = sp.Matrix(basis)
base.append(basis+basis.H)
return base
def get_submatrix(self, matrix):
r"""
Returns
=======
macaulay_submatrix: Matrix
The Macaulay denominator matrix. Columns that are non reduced are kept.
The row which contains one of the a_{i}s is dropped. a_{i}s
are the coefficients of x_i ^ {d_i}.
"""
reduced, non_reduced = self.get_reduced_nonreduced()
# if reduced == [], then det(matrix) should be 1
if reduced == []:
return diag([1])
# reduced != []
reduction_set = [
v**self.degrees[i] for i, v in enumerate(self.variables)
]
ais = list([
self.polynomials[i].coeff(reduction_set[i]) for i in range(self.n)
])
reduced_matrix = matrix[:, reduced]
keep = []
for row in range(reduced_matrix.rows):
check = [ai in reduced_matrix[row, :] for ai in ais]
if True not in check:
keep.append(row)
return matrix[keep, non_reduced]
def __construct_kinetics(self):
"""Construct model's dynamics (M, tau_c, tau_g).
"""
self.logger.debug('__construct_kinetics')
# generate the mass matrix [6 x 6] for each body
self.M = [
sp.diag(self.m[i], self.m[i], self.m[i], 0, 0, self.Iz[i])
for i in self.dim
]
# dummy 0 for 1-based indexing
self.M.insert(0, 0)
# map spatial to generalized inertia
self.M = [self.Jc[i].T * self.M[i] * self.Jc[i] for i in self.dim]
# sum the mass product of each body
self.M = reduce(lambda x, y: x + y, self.M)
self.M = sp.trigsimp(self.M)
# Coriolis matrix
self.C = sp.trigsimp(coriolis_matrix(self.M, self.Q(), self.QDot()))
# Coriolis forces
self.tau_c = sp.trigsimp(self.C * sp.Matrix(self.QDot()))
# potential energy due to gravity force
self.V = 0
for i in self.dim:
self.V = self.V + self.m[i] * self.g * self.xc[i][1]
self.tau_g = sp.Matrix([sp.diff(self.V, x) for x in self.Q()])
def Kerr(c=constants.c, sch=symbols("r_s"), a=symbols("a")):
"""
Kerr Metric in Boyer Lindquist coordinates.
Parameters
----------
c : ~sympy.core.basic.Basic or int or float
Any value to assign to speed of light. Defaults to ``c``.
sch : ~sympy.core.basic.Basic or int or float
Any value to assign to Schwarzschild Radius of the central object.
Defaults to ``r_s``.
a : ~sympy.core.basic.Basic or int or float
Spin factor of the heavy body. Usually, given by ``J/(Mc)``,
where ``J`` is the angular momentum.
Defaults to ``a``.
"""
coords = symbols("t r theta phi")
t, r, theta, phi = coords
Sigma = r**2 + (a**2 * cos(theta)**2)
Delta = r**2 - sch * r + a**2
c2 = c**2
metric = diag(
1 - (sch * r / Sigma),
-Sigma / (Delta * c2),
-Sigma / c2,
-((r**2 + a**2 + (sch * r * (a**2) * (sin(theta)**2) / Sigma)) *
(sin(theta)**2)) / c2,
).tolist()
metric[0][3] = metric[3][0] = sch * r * a * (sin(theta)**2) / (Sigma * c)
return MetricTensor(metric, coords, "ll")
def A(self, p):
nx = len(self.variables['x'])
nmode = nx // 2
sigma = p[:nmode]
omega = p[nmode:nx]
A_diag = ([[s, w], [-w, s]] for s, w in zip(sigma, omega))
return sympy.diag(*A_diag)
def _generate_symbolic_relaxation_matrix(self):
"""
This function replaces the numbers in the relaxation matrix with symbols in this case, and returns also
the subexpressions, that assign the number to the newly introduced symbol
"""
rr = [
self.relaxation_matrix[i, i]
for i in range(self.relaxation_matrix.rows)
]
unique_relaxation_rates = set()
subexpressions = {}
for relaxation_rate in rr:
if relaxation_rate not in unique_relaxation_rates:
relaxation_rate = sp.sympify(relaxation_rate)
# special treatment for zero, sp.Zero would be an integer ..
if isinstance(relaxation_rate, Zero):
relaxation_rate = 0.0
if not isinstance(relaxation_rate, sp.Symbol):
rt_symbol = sp.Symbol(f"rr_{len(subexpressions)}")
subexpressions[relaxation_rate] = rt_symbol
unique_relaxation_rates.add(relaxation_rate)
new_rr = [
subexpressions[sp.sympify(e)]
if sp.sympify(e) in subexpressions else e for e in rr
]
substitutions = [
Assignment(e[1], e[0]) for e in subexpressions.items()
]
return substitutions, sp.diag(*new_rr)
def JanisNewmanWinicour(
c=constants.c, G=constants.G, gam=symbols("gam"), M=symbols("M")
):
"""
Reality of the Schwarzschild singularity.
Phys. Rev. Lett., 20:878–880, 1968.
A. I. Janis, E. T. Newman, and J. Winicour.
Parameters
----------
M : ~sympy.core.basic.Basic or int or float
Mass parameter, this is used for defining the schwarzschild metric. Defaults to ``M``.
gam : ~sympy.core.basic.Basic or int or float
Parameter for scaling Schwarzschild radius, for gamma=1 this will return the Schwarzschild metric
Defaults to ``gam``.
"""
coords = symbols("t r theta phi")
t, r, th, ph = coords
# Helper functions
r_s = (2 * G * M) / (c ** 2)
alpha = 1 - (r_s / (gam * r))
# define the metric
metric = diag(
-1 * (alpha ** gam),
(alpha ** -gam) / (c ** 2),
(r ** 2) * (alpha ** (-gam + 1)),
(r ** 2) * (alpha ** (-gam + 1)) * (sin(th) ** 2),
).tolist()
return MetricTensor(metric, coords, "ll", name="JanisNewmanWinicourMetric")
def get_random_A(dim=3, valid=True, diagonal=True, ints=True, max_val=1e+3):
"""
:param dim: dimension of A
:param valid: True iff A's eigenvalues are all negative
:param diagonal: if True, A will be diagonal
:param ints: A will have only integer elements
:param max_val: max value of A if diagonal, else the square of the max value
:return: A
"""
max_val = int(max_val)
sgn = -1 if valid else 1
r = random.randint if ints else random.uniform
diag = set()
# generate n unique numbers
for x in itertools.takewhile(lambda x: len(diag) < dim, _grand(r, 1, max_val, int=ints)):
diag.add(sgn * x)
D = sp.diag(*diag)
if not diagonal:
M = sp.Matrix([[sp.nan]])
while any(x == sp.nan for x in M): # avoid nan (due to X)
X = sp.randMatrix(dim, dim, min=-max_val, max=max_val, seed=int(time.time() * 1000))
# eigenvals from D: det(XDX^-1 -tI) = det(X(D-tI)X^-1) = det(D-tI)
M = X * D * (X ** -1)
return M
else:
return D # diagonal matrix with -r values
def matrix_decomposition(A: sp.Matrix):
"""
Decompose a matrix
Parameters
----------
A : Matrix
3 x 3 matrix to decompose.
Returns
-------
Tuple[Matrix, Matrix, Matrix, Matrix]
Symmetric, anti-symmetric, hydrostatic, and deviatoric matrix parts.
A_sym, A_anti, A_hydro, A_deviator
"""
A_sym = sp.zeros(*A.shape)
A_anti = sp.zeros(*A.shape)
for i in range(3):
for j in range(3):
A_sym[i, j] = (A[i, j] + A[j, i]) / 2
A_anti[i, j] = (A[i, j] - A[j, i]) / 2
A_hydro = A.trace() / 3
A_hydro = sp.diag(*[A_hydro] * 3)
A_deviator = A_sym - A_hydro
return A_sym, A_anti, A_hydro, A_deviator
def _generate_schwarzschild():
coords = symbols("t r theta phi", real=True)
t, r, th, ph = coords
schw = diag(1 - 1 / r, -1 / (1 - 1 / r), -r ** 2, -r ** 2 * sin(th) ** 2)
g = Metric("g", coords, schw)
mu, nu = indices("mu nu", g)
return (coords, t, r, th, ph, schw, g, mu, nu)
def _generate_minkowski():
coords = symbols("t x y z", real=True)
t, x, y, z = coords
mink = diag(1, -1, -1, -1)
eta = Metric("eta", coords, mink)
mu, nu = indices("mu nu", eta)
return (coords, t, x, y, z, mink, eta, mu, nu)
def shift_matrices(matrix):
"""Create two matrices from input matrix
(append row of ones to the beginning of one and
the end of the other) in order to find the
change between rows (years)
Args:
matrix (Symbolic Matrix): any matrix
Returns:
shift_term (Symbolic Matrix): matrix with rows shifted down
long_term (Symbolic Matrix): matrix with row appended to end
(so that the dimensions will match
shift term)
TODO: Is a row of ones the best way to do this?
"""
print('matrix shape', matrix.shape)
n = matrix.shape[0] # Length
shift = -1
ones_V = [sp.Matrix([0, 1])] + [1] * (n-np.abs(shift) - 1)
subdiag = sp.diag(*ones_V)
subdiag = subdiag.col_insert(subdiag.shape[1],
sp.zeros(subdiag.shape[0], 1))
sp.pprint(subdiag)
shift_term = subdiag * matrix
return shift_term, matrix
def force_definite(Q):
eigvectors, eigvalues = Q.diagonalize()
#Q = eigvectors * eigvalues * eigvectors.inv()
#fixed_eigvalues = [eigvalues[i,i] if eigvalues[i,i] > 0 else 0 for i in range(Q.shape[0])] #make negative eigvalues zero
fixed_eigvalues = [eigvalues[i,i] - min(eigvalues) for i in range(Q.shape[0])] #increase all eigenvalues by same amount such that the smallest is zero.
fixed_eigvalues = sympy.diag(*fixed_eigvalues)
return eigvectors* fixed_eigvalues * eigvectors.inv()
def are_lc_equiv(g1, g2):
"""
Tests whether two graphs are equivalent up to local complementation.
If True, also returns every unitary such that |g2> = U|g1>.
"""
# Gets adjacency matrices and returns false if differing bases
am1, k1 = get_adjacency_matrix(g1)
am2, k2 = get_adjacency_matrix(g2)
dim1, dim2 = len(k1), len(k2)
if k1 != k2 or am1.shape != (dim1, dim1) or am2.shape != (dim2, dim2):
return False, None
# Defines binary matrices
Id = sp.eye(dim1)
S1 = sp.Matrix(am1).col_join(Id)
S2 = sp.Matrix(am2).col_join(Id)
# Defines symbolic variables
A = sp.symbols('a:' + str(dim1), bool=True)
B = sp.symbols('b:' + str(dim1), bool=True)
C = sp.symbols('c:' + str(dim1), bool=True)
D = sp.symbols('d:' + str(dim1), bool=True)
# Defines solution matrix basis
abcd = flatten(list(zip(A, B, C, D)))
no_vars = len(abcd)
no_qubits = no_vars // 4
# Creates symbolic binary matrix
A, B, C, D = sp.diag(*A), sp.diag(*B), sp.diag(*C), sp.diag(*D)
Q = A.row_join(B).col_join(C.row_join(D))
P = sp.zeros(dim1).row_join(Id).col_join(Id.row_join(sp.zeros(dim1)))
# Constructs matrix to solve
X = [i for i in S1.T * Q.T * P * S2]
X = np.array([[x.coeff(v) for v in abcd] for x in X], dtype=int)
# Removes any duplicated and all-zero rows
X = np.unique(X, axis=0)
X = X[~(X == 0).all(1)]
# Finds the solutions (the nullspace of X)
V = list(GF2nullspace(X))
if len(V) > 4:
V = [(v1 + v2) % 2 for v1, v2 in it.combinations(V, 2)]
else:
V = [sum(vs) % 2 for vs in powerset(V)]
V = [np.reshape(v, (no_qubits, 4)) for v in V]
V = [v for v in V if all((a * d + b * c) % 2 == 1 for a, b, c, d in v)]
if V:
V = [[bin2gate[tuple(r)] for r in v] for v in V]
return True, V
else:
return False, None
def __create_symbolic_jacobian_matrix(self):
self.__mu_vec = [
sympy.symbols('v_' + str(i + 1)) for i in self.__edges_true_outflow
]
diag_mu = sympy.diag(*self.__mu_vec)
self.__symbolic_J = self.__S_to * diag_mu * self.__Y_r.T
def _init_matrixes(self):
self.X = sm.zeros(rows=self.n_e, cols=1)
for edge_index, (edge, edge_info) in enumerate(sorted(self.edges.items(),
key=lambda edge_sort_key: int(edge_sort_key[0][0]))):
edge_index = list(self.edges()).index(edge)
self.X[edge_index] = edge_info['variable_x']
self.A = nx.incidence_matrix(self, oriented=True).todense()
self.Q_symbols = sm.Matrix(deepcopy(self.Q))
self.Q = self.A * self.X
self.AF = deepcopy(self.A)
self.AL = deepcopy(self.A)
for row in range(self.AF.shape[0]):
for col in range(self.AF.shape[1]):
if self.AF[row, col] == -1:
self.AF[row, col] = 0
if self.AL[row, col] == 1:
self.AL[row, col] = 0
self.DF = sm.zeros(len(self.edges()), 1)
self.DL = sm.zeros(len(self.edges()), 1)
for equation_index, (edge, edge_info) in enumerate(sorted(self.edges.items(),
key=lambda edge_sort_key: edge_sort_key[0][0])):
u = edge[0]
v = edge[1]
p_s = self.nodes[u]['variable_p']
p_f = self.nodes[v]['variable_p']
a = edge_info['A']
eq = sm.sqrt((p_s ** 2 - p_f ** 2) / a)
self.DF[equation_index] = sm.diff(eq, p_s)
self.DL[equation_index] = -sm.diff(eq, p_f)
self.DF = sm.diag(*self.DF)
self.DL = sm.diag(*self.DL)
self.P = sm.Matrix(self.P)
self.Q = sm.Matrix(self.Q)
self.dP = sm.Matrix(self.dP)
self.dQ = sm.Matrix(self.dQ)
def diag(x):
def m(i, j):
return x[i, i]
try:
a = Matrix(x.cols, 1, m)
except:
a = sympy.diag(*x)
return a
def _sympy_diag(args,sargs):
vecmat = args[0]
if vecmat.returnType().type in ['N','V']:
return sympy.diag(*sargs[0])
elif vecmat.returnType().type == 'M':
raise NotImplementedError("TODO: Get diagonal of matrix in Sympy")
else:
raise ValueError("Unknown return type from symbolic Expression")
def __add__(core1, core2):
"""
Add core1 and core2 and return a new core.
Every vectors are concatenated and structure matrices with same labels
are diagonally stacked into a big (square) structure matrix.
"""
assert set(core1.symbs_names) == set(core2.symbs_names)
core = Core(label=core1.label)
# Concatenate lists of symbols
for name in core1.symbs_names:
attr1 = getattr(core1, name)
attr2 = getattr(core2, name)
core.setsymb(name, attr1 + attr2)
for vari in core.dims.names:
for varj in core.dims.names:
Mij1 = getattr(core1, 'M'+vari+varj)()
Mij2 = getattr(core2, 'M'+vari+varj)()
Mij = types.matrix_types[0](sympy.diag(Mij1, Mij2))
if all(dim > 0 for dim in Mij.shape):
set_func = getattr(core, 'set_M'+vari+varj)
set_func(Mij)
# Concatenate lists of symbols
for name in core1.symbs_names:
attr1 = getattr(core1, name)
attr2 = getattr(core2, name)
core.setsymb(name, attr1 + attr2)
# Update subs disctionary
core.subs = {}
core.subs.update(core1.subs)
core.subs.update(core2.subs)
# Update observers dictionary
core.observers.update(core1.observers)
core.observers.update(core2.observers)
# Set Hamiltonian expression
core.setexpr('H', core1.H + core2.H)
# Concatenate lists of expressions
core.setexpr('z', core1.z + core2.z)
core.connectors = core1.connectors + core2.connectors
core.force_wnl = core1.force_wnl + core2.force_wnl
for vari in core.dims.names:
for varj in core.dims.names:
Mij = getattr(core, 'M'+vari+varj)()
if all(dim > 0 for dim in Mij.shape):
set_func = getattr(core, 'set_M'+vari+varj)
set_func(Mij)
return core
def __add__(core1, core2):
"""
Add core1 and core2 and return a new core.
Every vectors are concatenated and structure matrices with same labels
are diagonally stacked into a big (square) structure matrix.
"""
assert set(core1.symbs_names) == set(core2.symbs_names)
core = Core(label=core1.label)
# Concatenate lists of symbols
for name in core1.symbs_names:
attr1 = getattr(core1, name)
attr2 = getattr(core2, name)
core.setsymb(name, attr1 + attr2)
for vari in core.dims.names:
for varj in core.dims.names:
Mij1 = getattr(core1, 'M' + vari + varj)()
Mij2 = getattr(core2, 'M' + vari + varj)()
Mij = types.matrix_types[0](sympy.diag(Mij1, Mij2))
if all(dim > 0 for dim in Mij.shape):
set_func = getattr(core, 'set_M' + vari + varj)
set_func(Mij)
# Concatenate lists of symbols
for name in core1.symbs_names:
attr1 = getattr(core1, name)
attr2 = getattr(core2, name)
core.setsymb(name, attr1 + attr2)
# Update subs disctionary
core.subs = {}
core.subs.update(core1.subs)
core.subs.update(core2.subs)
# Update observers dictionary
core.observers.update(core1.observers)
core.observers.update(core2.observers)
# Set Hamiltonian expression
core.setexpr('H', core1.H + core2.H)
# Concatenate lists of expressions
core.setexpr('z', core1.z + core2.z)
core.connectors = core1.connectors + core2.connectors
core.force_wnl = core1.force_wnl + core2.force_wnl
for vari in core.dims.names:
for varj in core.dims.names:
Mij = getattr(core, 'M' + vari + varj)()
if all(dim > 0 for dim in Mij.shape):
set_func = getattr(core, 'set_M' + vari + varj)
set_func(Mij)
return core
def Lambda(self):
"""Diagonal matrix of eigenvalues."""
# Perhaps faster to use diagonalize
# E, L = self.A.diagonalize()
# return L
e = self.eigenvalues
return sMatrix(sym.diag(*e))
def force_definite(Q):
eigvectors, eigvalues = Q.diagonalize()
#Q = eigvectors * eigvalues * eigvectors.inv()
#fixed_eigvalues = [eigvalues[i,i] if eigvalues[i,i] > 0 else 0 for i in range(Q.shape[0])] #make negative eigvalues zero
fixed_eigvalues = [
eigvalues[i, i] - min(eigvalues) for i in range(Q.shape[0])
] #increase all eigenvalues by same amount such that the smallest is zero.
fixed_eigvalues = sympy.diag(*fixed_eigvalues)
return eigvectors * fixed_eigvalues * eigvectors.inv()
def Lambda(self):
"""Diagonal matrix of eigenvalues."""
# Perhaps faster to use diagonalize
# E, L = self.A.diagonalize()
# return L
e = self.eigenvalues
return sMatrix(sym.diag(*e))
def bravyi_haah_gmat(k):
"""Tri-orthogonal matrix from Bravyi-Haah
"""
bigl = np.block([[LMAT] for _ in range(k)]+ [[S1MAT]])
bigm = sympy.diag(*[MMAT for _ in range(k)])
bigs2 = np.block([S2MAT for _ in range(k)])
gmatr = np.block([[bigm], [bigs2]])
gmat = np.block([bigl, gmatr])
return sympy.Matrix(gmat)
def _calculate_local_remainder_component(self, diagonal_component=None):
r"""Calculate the non-quadratic remainder :math:`W(x) = V(x) - U(x)` of the quadratic
Taylor approximation :math:`U(x)` of the potential's eigenvalue :math:`\lambda_i(x)`.
Note that this function is idempotent.
:param diagonal_component: Specifies the index :math:`i` of the eigenvalue :math:`\lambda_i`
that gets expanded into a Taylor series :math:`u_i`.
"""
# Calculation already done at some earlier time?
if self._remainder_eigen_s.has_key(diagonal_component):
return
self.calculate_eigenvalues()
self.calculate_jacobian()
self.calculate_hessian()
# Point q where the taylor series is computed
# This is a column vector q = (q1, ... ,qD)
qs = [sympy.Symbol("q" + str(i)) for i in xrange(len(self._variables))]
pairs = [(xi, qi) for xi, qi in zip(self._variables, qs)]
V = self._eigenvalues_s[diagonal_component].subs(pairs)
J = self._jacobian_s[diagonal_component].subs(pairs)
H = self._hessian_s[diagonal_component].subs(pairs)
# Symbolic expression for the quadratic Taylor expansion term
xmq = sympy.Matrix([(xi - qi) for xi, qi in zip(self._variables, qs)])
quadratic = sympy.Matrix(
[[V]]) + J.T * xmq + sympy.Rational(1, 2) * xmq.T * H * xmq
# Symbolic simplification may fail
if self._try_simplify:
try:
quadratic = quadratic.applyfunc(sympy.simplify)
except:
pass
# Symbolic expression for the Taylor expansion remainder term
U = sympy.diag(*self._number_components * [quadratic[0, 0]])
remainder = self._potential_s - U
# Symbolic simplification may fail
if self._try_simplify:
try:
remainder = remainder.applyfunc(sympy.simplify)
except:
pass
self._remainder_eigen_s[diagonal_component] = remainder
# Construct functions to evaluate the approximation at point q at the given nodes
# The variable ordering in lambdify is [x1, ..., xD, q1, ...., qD]
self._remainder_eigen_n[diagonal_component] = tuple([
sympy.lambdify(list(self._variables) + qs, entry, "numpy")
for entry in remainder
])
def __init__(self, label, nodes, **kwargs):
pars = ['Is', 'betaR', 'betaF', 'Vt', 'mu', 'Rb', 'Rc', 'Re']
for par in pars:
assert par in kwargs.keys()
Is, betaR, betaF, Vt, mu, Rb, Rc, Re = symbols(pars)
# dissipation variable
wbjt = symbols(["w"+label+ind for ind in ['bc', 'be']])
# bjt dissipation funcion
coeffs = types.matrix_types[0]([[(betaR+1)/betaR, -1],
[-1, (betaF+1)/betaF]])
funcs = [Is*(sympy.exp(wbjt[0]/(mu*Vt))-1) + GMIN*wbjt[0],
Is*(sympy.exp(wbjt[1]/(mu*Vt))-1) + GMIN*wbjt[1]]
zbjt = coeffs*types.matrix_types[0](funcs)
# bjt edges data
data_bc = {'label': wbjt[0],
'type': 'dissipative',
'ctrl': 'e',
'z': {'e_ctrl': zbjt[0], 'f_ctrl': sympy.sympify(0)},
'link': None}
data_be = {'label': wbjt[1],
'type': 'dissipative',
'z': {'e_ctrl': zbjt[1], 'f_ctrl': sympy.sympify(0)},
'ctrl': 'e',
'link': None}
# connector resistances dissipative functions
wR = symbols(["w"+label+ind for ind in ['rb', 'rc', 're']])
Rmat = types.matrix_types[0](sympy.diag(Rb, Rc, Re))
zR = Rmat*types.matrix_types[0](wR)
# connector resistances edges data
data_rb = {'label': wR[0],
'z': {'e_ctrl': wR[0]/Rb, 'f_ctrl': Rb*wR[0]},
'type': 'dissipative',
'ctrl': '?',
'link': None}
data_rc = {'label': wR[1],
'z': {'e_ctrl': wR[1]/Rc, 'f_ctrl': Rc*wR[1]},
'type': 'dissipative',
'ctrl': '?',
'link': None}
data_re = {'label': wR[2],
'z': {'e_ctrl': wR[2]/Re, 'f_ctrl': Re*wR[2]},
'type': 'dissipative',
'ctrl': '?',
'link': None}
# edge
Nb, Nc, Ne = nodes
iNb, iNc, iNe = [str(el)+label for el in (Nb, Nc, Ne)]
edges = [(iNb, iNc, data_bc),
(iNb, iNe, data_be),
(Nb, iNb, data_rb),
(Nc, iNc, data_rc),
(Ne, iNe, data_re)]
# init component
DissipativeNonLinear.__init__(self, label, edges, wbjt + wR,
list(zbjt) + list(zR), **kwargs)
def _calculate_local_remainder_inhomogeneous(self):
r"""Calculate the non-quadratic remainder matrix :math:`W(x) = V(x) - U(x)` of the
quadratic approximation matrix :math:`U(x)` of the potential's eigenvalue matrix
:math:`\Lambda(x)`. This function is used for the inhomogeneous case.
"""
if self._remainder_eigen_ih_s is not None:
# Calculation already done at some earlier time
return
else:
self._remainder_eigen_ih_s = []
self.calculate_eigenvalues()
self.calculate_jacobian()
self.calculate_hessian()
# Quadratic taylor series for all eigenvalues
quadratics = []
for index, eigenvalue in enumerate(self._eigenvalues_s):
# Point q where the taylor series is computed
# This is a column vector q = (q1, ... ,qD)
qs = [ sympy.Symbol("q"+str(i)) for i,v in enumerate(self._variables) ]
pairs = [ (xi,qi) for xi,qi in zip(self._variables, qs) ]
V = self._eigenvalues_s[index].subs(pairs)
J = self._jacobian_s[index].subs(pairs)
H = self._hessian_s[index].subs(pairs)
# Symbolic expression for the quadratic Taylor expansion term
xmq = sympy.Matrix([ (xi-qi) for xi,qi in zip(self._variables, qs) ])
quadratic = sympy.Matrix([[V]]) + J.T*xmq + sympy.Rational(1,2)*xmq.T*H*xmq
try:
quadratic = quadratic.applyfunc(sympy.simplify)
except:
pass
quadratics.append(quadratic[0,0])
# Symbolic expression for the Taylor expansion remainder term
U = sympy.diag( *quadratics )
remainder = self._potential_s - U
# Symbolic simplification may fail
if self._try_simplify:
try:
remainder = remainder.applyfunc(sympy.simplify)
except:
pass
self._remainder_eigen_ih_s = remainder
# Construct functions to evaluate the approximation at point q at the given nodes
self._remainder_eigen_ih_n = tuple([
sympy.lambdify(list(self._variables) + qs, entry, "numpy") for entry in remainder ])
def _calculate_local_remainder_component(self, diagonal_component=None):
r"""Calculate the non-quadratic remainder :math:`W(x) = V(x) - U(x)` of the quadratic
Taylor approximation :math:`U(x)` of the potential's eigenvalue :math:`\lambda_i(x)`.
Note that this function is idempotent.
:param diagonal_component: Specifies the index :math:`i` of the eigenvalue :math:`\lambda_i`
that gets expanded into a Taylor series :math:`u_i`.
"""
# Calculation already done at some earlier time?
if self._remainder_eigen_s.has_key(diagonal_component):
return
self.calculate_eigenvalues()
self.calculate_jacobian()
self.calculate_hessian()
# Point q where the taylor series is computed
# This is a column vector q = (q1, ... ,qD)
qs = [ sympy.Symbol("q"+str(i)) for i in xrange(len(self._variables)) ]
pairs = [ (xi,qi) for xi,qi in zip(self._variables, qs) ]
V = self._eigenvalues_s[diagonal_component].subs(pairs)
J = self._jacobian_s[diagonal_component].subs(pairs)
H = self._hessian_s[diagonal_component].subs(pairs)
# Symbolic expression for the quadratic Taylor expansion term
xmq = sympy.Matrix([ (xi-qi) for xi,qi in zip(self._variables, qs) ])
quadratic = sympy.Matrix([[V]]) + J.T*xmq + sympy.Rational(1,2)*xmq.T*H*xmq
# Symbolic simplification may fail
if self._try_simplify:
try:
quadratic = quadratic.applyfunc(sympy.simplify)
except:
pass
# Symbolic expression for the Taylor expansion remainder term
U = sympy.diag( *self._number_components*[quadratic[0,0]] )
remainder = self._potential_s - U
# Symbolic simplification may fail
if self._try_simplify:
try:
remainder = remainder.applyfunc(sympy.simplify)
except:
pass
self._remainder_eigen_s[diagonal_component] = remainder
# Construct functions to evaluate the approximation at point q at the given nodes
# The variable ordering in lambdify is [x1, ..., xD, q1, ...., qD]
self._remainder_eigen_n[diagonal_component] = tuple([
sympy.lambdify(list(self._variables) + qs, entry, "numpy") for entry in remainder ])
def _age_vector_dens(u, B, Qt):
"""Return the (symbolic) probability density vector of the compartment ages.
Args:
u (SymPy dx1-matrix): external input vector
B (SymPy dxd-matrix): compartment matrix
Qt (SymPy dxd-matrix): Qt = :math:`e^{t\\,B}`
Returns:
SymPy dx1-matrix: probability density vector of the compartment ages
:math:`f_a(y) = (X^\\ast)^{-1}\\,e^{y\\,B}\\,u`
"""
xss = -(B**-1)*u
X = diag(*xss)
return (X**-1)*Qt*u
def connect(self):
"""
Effectively connect inputs and outputs defined in core.connectors.
See also
--------
See help of method :code:`core.add_connector` for details.
"""
all_alpha = list()
# recover connectors and sort cy and cu
for i, c in enumerate(self.connectors):
all_alpha.append(c['alpha'])
i_primal = getattr(self, 'cy').index(c['y'][0])
self.move_connector(i_primal, 2*i)
i_dual = getattr(self, 'cy').index(c['y'][1])
self.move_connector(i_dual, 2*i+1)
Mswitch_list = [alpha * sympy.Matrix([[0, -1],
[1, 0]])
for alpha in all_alpha]
Mswitch = types.matrix_types[0](sympy.diag(*Mswitch_list))
nxwy = self.dims.x() + self.dims.w() + self.dims.y()
# Gain matrix
G_connectors = self.M[:nxwy, nxwy:]
# Observation matrix
O_connectors = self.M[nxwy:, :nxwy]
N_connectors = types.matrix_types[0](sympy.eye(self.dims.cy()) -
self.Mcycy() * Mswitch)
try:
iN_connectors = inverse(N_connectors, dosimplify=False)
# Interconnection Matrix due to the connectors
M_connectors = G_connectors*Mswitch*iN_connectors*O_connectors
# Store new structure
self.M = types.matrix_types[0](self.M[:nxwy, :nxwy] + M_connectors)
# clean
setattr(self, 'cy', list())
setattr(self, 'cu', list())
setattr(self, 'connectors', list())
except ValueError:
raise Exception('Can not resolve the connection.\n\nABORD')
def absorbing_jump_chain(self):
"""Return the absorbing jump chain as a discrete-time Markov chain.
The generator of the absorbing chain is just given by :math:`B`, which
allows the computation of the transition probability matrix :math:`P`
from :math:`B=(P-I)\\,D` with :math:`D` being the diagonal matrix with
diagonal entries taken from :math:`-B`.
Returns:
:class:`~.DTMC.DTMC`: :class:`DTMC` (beta, P)
"""
# B = (P - I) * D
d = self.B.rows
D = diag(*[-self.B[j,j] for j in range(d)])
P = self.B * D**(-1) + eye(d)
return DTMC(self.beta, P)
def _age_vector_nth_moment(u, B, n):
"""Return the (symbolic) vector of nth moments of the compartment ages.
Args:
u (SymPy dx1-matrix): external input vector
B (SymPy dxd-matrix): compartment matrix
n (positive int): order of the moment
Returns:
SymPy dx1-matrix: vector of nth moments of the compartment ages
:math:`\\mathbb{E}[a^n]=(-1)^n\\,n!\\,X^\\ast)^{-1}\\,B^{-n}\\,x^\\ast`
See Also:
:func:`_age_vector_exp`: Return the (symbolic) vector of expected values
of the compartment ages.
"""
xss = -(B**-1)*u
X = diag(*xss)
return (-1)**n*factorial(n)*(X**-1)*(B**-n)*xss
def _age_vector_cum_dist(u, B, Qt):
"""Return the (symbolic) cumulative distribution function vector of the
compartment ages.
Args:
u (SymPy dx1-matrix): external input vector
B (SymPy dxd-matrix): compartment matrix
Qt (SymPy dxd-matrix): Qt = :math:`e^{t\\,B}`
Returns:
SymPy dx1-matrix: cumulative distribution function vector of the
compartment ages
:math:`f_a(y)=(X^\\ast)^{-1}\\,B^{-1}\\,(e^{y\\,B}-I)\\,u`
"""
d = B.rows
xss = -(B**-1)*u
X = diag(*xss)
return (X**-1)*(B**-1)*(Qt-eye(d))*u
def compute_vect2_from_pixel(self):
# Parameters related to Mirror2:
self.k2, self.c2, self.d = symbols("k_2, c_2, d", real=True, positive=True)
self.u2, self.v2 = symbols("u_2, v_2", real=True, nonnegative=True)
# Viewpoint (focus) position vector:
self.f2x, self.f2y, self.f2z = symbols("x_f_2, y_f_2, z_f_2", real=True)
# (Coaxial alignment assumption)
# where
self.f2x = 0
self.f2y = 0
self.f2z = self.d - self.c2
self.f2 = Matrix([self.f2x, self.f2y, self.f2z])
# Reflex mirror's normal vector wrt [C]
self.n_ref = Matrix([0, 0, -1])
# Virtual focus
self.f2virt = Matrix([self.f2x, self.f2y, self.d]) # where f2virtz = d
# Pixel vector
self.m2h = Matrix([self.u2, self.v2, 1])
# Point in the normalized projection plane
self.q2 = self.Kc_inv * self.m2h
# Planar Reflection Matrix (Change of coordinates)
self.M_ref = (eye(3) + 2 * diag(*self.n_ref)).row_join(self.f2virt)
# Change coordinates from C' to C (according to explanation in journal paper)
self.q2v = self.M_ref * self.q2.col_join(eye(1))
# Point in mirror wrt C
self.t2 = self.c2 / (self.k2 - self.q2.norm() * sqrt(self.k2 * (self.k2 - 2)))
self.p2 = self.f2virt + self.t2 * (self.q2v - self.f2virt)
self.p2h = self.p2.col_join(eye(1))
# Transform matrix from C to F1 frame
self.T2_CtoF2 = eye(3).row_join(-self.f2)
# Direction vector
self.d2_vect = self.T2_CtoF2 * self.p2h
return self.d2_vect
def __init__(self, equations, u_x_names=[], u_y_names=[], u_z_names=[],
x_names=[], w_names=[], param_names=[], block_indices={},
par_to_values_dict={}, fwd_shift_idx=[], aux_eqs=[],
vars_initvalues_dict={}, u_trans_dict={}, ss_sol_dict={}):
print u_x_names
print u_y_names
print u_z_names
print x_names
print w_names
ModelBase.__init__(self, equations, x_names, w_names, param_names,
par_to_values_dict=par_to_values_dict,
vars_initvalues_dict=vars_initvalues_dict,
compute_ss=False)
u_x_tp1_sym_d = make_basic_sym_dic(u_x_names, 'tp1')
u_x_t_sym_d = make_basic_sym_dic(u_x_names, 't')
u_x_tm1_sym_d = make_basic_sym_dic(u_x_names, 'tm1')
u_y_tp1_sym_d = make_basic_sym_dic(u_y_names, 'tp1')
u_y_t_sym_d = make_basic_sym_dic(u_y_names, 't')
u_z_tp1_sym_d = make_basic_sym_dic(u_z_names, 'tp1')
u_z_t_sym_d = make_basic_sym_dic(u_z_names, 't')
u_w_tp1_sym_d = make_basic_sym_dic(w_names, 'tp1')
u_w_t_sym_d = make_basic_sym_dic(w_names, 't')
self.u_x_tp1_sym = list(sympy.ordered(u_x_tp1_sym_d.values()))
self.u_x_t_sym = list(sympy.ordered(u_x_t_sym_d.values()))
self.u_x_tm1_sym = list(sympy.ordered(u_x_tm1_sym_d.values()))
self.u_y_tp1_sym = list(sympy.ordered(u_y_tp1_sym_d.values()))
self.u_y_t_sym = list(sympy.ordered(u_y_t_sym_d.values()))
self.u_z_tp1_sym = list(sympy.ordered(u_z_tp1_sym_d.values()))
self.u_z_t_sym = list(sympy.ordered(u_z_t_sym_d.values()))
self.u_w_tp1_sym = list(sympy.ordered(u_w_tp1_sym_d.values()))
self.u_w_t_sym = list(sympy.ordered(u_w_t_sym_d.values()))
if fwd_shift_idx != []:
equations = self.shift_eqns_fwd(equations, fwd_shift_idx)
new_eqs = [x.subs(u_trans_dict) for x in equations]
new_eqs.extend(aux_eqs)
equations = new_eqs
if ss_sol_dict == {}:
eqns_no_param = self.make_ss_version_of_eqs(equations)
self.ss_solutions_dict = get_sstate_sol_dict_from_sympy_eqs(
eqns_no_param, self.x_in_ss_sym,
vars_initvalues_dict=vars_initvalues_dict)
self.ss_residuals = [x.subs(self.ss_solutions_dict) for x in
eqns_no_param]
self.block_indices = block_indices
# print block_indices
non_expec_idx = block_indices['non_expectational_block']
expec_idx = block_indices['expectational_block']
z_idx = block_indices['z_block']
if x_names == []:
x_names = u_x_names + u_y_names + u_z_names
self.u_param_sym = self.param_sym_d.values()
self.u_param_sym = list(sympy.ordered(self.u_param_sym))
self.eqns = equations
print "self.eqns"
print self.eqns
if isinstance(non_expec_idx, int):
self.eqns_non_expec = [equations[non_expec_idx]]
else:
self.eqns_non_expec = [equations[i] for i in non_expec_idx]
if isinstance(expec_idx, int):
self.eqns_expec = [equations[expec_idx]]
else:
self.eqns_expec = [equations[i] for i in expec_idx]
if isinstance(z_idx, int):
self.eqns_z = [equations[z_idx]]
else:
self.eqns_z = [equations[i] for i in z_idx]
print "\nself.eqns_non_expec"
print self.eqns_non_expec
print "\nself.eqns_expec"
print self.eqns_expec
print "\nself.eqns_z"
print self.eqns_z
self.x_to_devss_dict = make_devss_subs_dict(self.x_names,
self.x_dates)
self.eqns_expec_devss = [x.subs(self.x_to_devss_dict) for x
in self.eqns_expec]
self.eqns_non_expec_devss = [x.subs(self.x_to_devss_dict) for x in
self.eqns_non_expec]
self.eqns_z_devss = [x.subs(self.x_to_devss_dict) for x in
self.eqns_z]
print "\nself.eqns_non_expec_devss"
print self.eqns_non_expec_devss
print "\nself.eqns_expec_devss"
print self.eqns_expec_devss
print "\nself.eqns_z_devss"
print self.eqns_z_devss
self.jacobians_unev, self.jacobians_unev_ss = self.jacobians_sym_uh()
self.uA_sym = self.jacobians_unev[0]
self.uB_sym = self.jacobians_unev[1]
self.uC_sym = self.jacobians_unev[2]
self.uD_sym = self.jacobians_unev[3]
self.uF_sym = self.jacobians_unev[4]
self.uG_sym = self.jacobians_unev[5]
self.uH_sym = self.jacobians_unev[6]
self.uJ_sym = self.jacobians_unev[7]
self.uK_sym = self.jacobians_unev[8]
self.uL_sym = self.jacobians_unev[9]
self.uM_sym = self.jacobians_unev[10]
self.uN_sym = self.jacobians_unev[11]
self.uA_sym_ss = self.jacobians_unev_ss[0]
self.uB_sym_ss = self.jacobians_unev_ss[1]
self.uC_sym_ss = self.jacobians_unev_ss[2]
self.uD_sym_ss = self.jacobians_unev_ss[3]
self.uF_sym_ss = self.jacobians_unev_ss[4]
self.uG_sym_ss = self.jacobians_unev_ss[5]
self.uH_sym_ss = self.jacobians_unev_ss[6]
self.uJ_sym_ss = self.jacobians_unev_ss[7]
self.uK_sym_ss = self.jacobians_unev_ss[8]
self.uL_sym_ss = self.jacobians_unev_ss[9]
self.uM_sym_ss = self.jacobians_unev_ss[10]
self.uN_sym_ss = self.jacobians_unev_ss[11]
args = self.x_in_ss_sym + self.u_param_sym
self.jac_ss_funcs = sympy.lambdify(args, self.jacobians_unev_ss)
x_ss_num = [x.subs(self.ss_solutions_dict) for x in self.x_in_ss_sym]
par_ss_num = [x.subs(self.par_to_values_dict) for x in self.param_sym]
x_par_ss_num = x_ss_num + par_ss_num
self.jac_ss_num = [matrix2numpyfloat(x) for x
in self.jac_ss_funcs(*x_par_ss_num)]
# # print '\nself.jac_ss_num'
# print self.jac_ss_num
self.uA_num_ss = self.jac_ss_num[0]
self.uB_num_ss = self.jac_ss_num[1]
self.uC_num_ss = self.jac_ss_num[2]
self.uD_num_ss = self.jac_ss_num[3]
# print self.uD_sym
# print self.uD_sym_ss
# print self.uD_num_ss
self.uF_num_ss = self.jac_ss_num[4]
self.uG_num_ss = self.jac_ss_num[5]
self.uH_num_ss = self.jac_ss_num[6]
self.uJ_num_ss = self.jac_ss_num[7]
self.uK_num_ss = self.jac_ss_num[8]
self.uL_num_ss = self.jac_ss_num[9]
self.uM_num_ss = self.jac_ss_num[10]
self.uN_num_ss = self.jac_ss_num[11]
self.u_x_ss_sym = [x.subs(self.normal_and_0_to_ss) for x in self.u_x_t_sym]
self.u_y_ss_sym = [x.subs(self.normal_and_0_to_ss) for x in self.u_y_t_sym]
self.u_z_ss_sym = [x.subs(self.normal_and_0_to_ss) for x in self.u_z_t_sym]
self.u_x_ss_num = [x.subs(self.ss_solutions_dict) for x in self.u_x_ss_sym]
self.u_y_ss_num = [x.subs(self.ss_solutions_dict) for x in self.u_y_ss_sym]
self.u_z_ss_num = [x.subs(self.ss_solutions_dict) for x in self.u_z_ss_sym]
# self.u_z_ss_num = [0.03, 0.03, 1]
# self.u_z_ss_num = [0.03, 1]
non_zero_z_idx = np.nonzero(self.u_z_ss_num)
z_for_diag = np.ones_like(self.u_z_ss_num)
z_for_diag[non_zero_z_idx] = self.u_z_ss_num
print "foooooo\n"
print self.u_z_ss_num
print non_zero_z_idx
print z_for_diag
print "moooooo\n"
self.di_u_x_ss_sym = sympy.diag(*self.u_x_ss_sym)
self.di_u_y_ss_sym = sympy.diag(*self.u_y_ss_sym)
self.di_u_z_ss_sym = sympy.diag(*self.u_z_ss_sym)
self.di_u_x_ss_num = np.diag(self.u_x_ss_num)
self.di_u_y_ss_num = np.diag(self.u_y_ss_num)
self.di_u_z_ss_num = np.diag(z_for_diag)
self.uA_num_ss_log = np.dot(self.uA_num_ss, self.di_u_x_ss_num)
self.uB_num_ss_log = np.dot(self.uB_num_ss, self.di_u_x_ss_num)
self.uC_num_ss_log = np.dot(self.uC_num_ss, self.di_u_y_ss_num)
self.uD_num_ss_log = np.dot(self.uD_num_ss, self.di_u_z_ss_num)
self.uF_num_ss_log = np.dot(self.uF_num_ss, self.di_u_x_ss_num)
self.uG_num_ss_log = np.dot(self.uG_num_ss, self.di_u_x_ss_num)
self.uH_num_ss_log = np.dot(self.uH_num_ss, self.di_u_x_ss_num)
self.uJ_num_ss_log = np.dot(self.uJ_num_ss, self.di_u_y_ss_num)
self.uK_num_ss_log = np.dot(self.uK_num_ss, self.di_u_y_ss_num)
self.uL_num_ss_log = np.dot(self.uL_num_ss, self.di_u_z_ss_num)
self.uM_num_ss_log = np.dot(self.uM_num_ss, self.di_u_z_ss_num)
self.uN_num_ss_log = np.dot(self.uN_num_ss, self.di_u_z_ss_num)
Omicron = Matrix([[1, 0, 0, 0],
[0, 1, 0, 0],
[0, 0, 0, 1],
[0, 1, 0, 0],
[0, 0, 0, 1]])
D = Matrix([[0, 1, 0, 0],
[0, 0, 1, 0],
[1, 0, 0, 0],
[0, 0, 0, 1],
[0, 0, 0, 1]])
# Multiplicative component of natural law
multiplicative_components = [0, 0, 0, 0, 0]
Upsilon = diag(*multiplicative_components)
# Additive component of natural law
additive_components = [0, 1, 0, 0, 0]
S = diag(*additive_components)
# Accelerator
Alpha = Matrix([[1, 0, 0, 0, 0, 0],
[0, 0, 1, 0, 0, 0],
[0, 0, 0, 0, 1, 0],
[0, 0, 0, 0, 0, 1],
[0, 0, 0, 1, 0, 0]])
number_of_matters = 3
number_of_atoms = 4
def bend(pts, facets):
edges = {}
for f in range(0, len(facets)):
edges[f] = []
vertices = facets[f]
for v in range(0, len(vertices) - 1):
edges[f].append([vertices[v], vertices[v + 1]])
edges[f].append([vertices[len(vertices) - 1], vertices[0]])
if DEBUG:
print "Edges:"
print edges
def edges_eq(e1, e2):
if e1 == e2 or (e1[0] == e2[1] and e1[1] == e2[0]):
return True
return False
def common_edge(f1, f2):
for e1 in edges[f1]:
for e2 in edges[f2]:
if edges_eq(e1, e2):
return e1
return None
neighbours = {}
for f1 in range(0, len(facets)):
neighbours[f1] = []
for f2 in range(0, len(facets)):
if f2 == f1:
continue
if common_edge(f1, f2):
neighbours[f1].append(f2)
def transform_point(ptr, T):
p = pts[ptr]
v = Matrix([[p.x],
[p.y],
[1]])
u = T * v
return Point(u[0,0], u[1,0])
def transform_facet(f, T):
new_facet = []
for p in facets[f]:
new_facet.append(transform_point(p, T))
return new_facet
if DEBUG:
print "Neighbours:"
print neighbours
to_process = []
for f in neighbours[0]:
to_process.append([f, common_edge(0, f), diag(1, 1, 1)])
processed = {0: diag(1, 1, 1)}
while len(to_process) > 0:
if DRAW:
draw_side_by_side([[pts[p] for p in f] for f in facets], [transform_facet(f, processed[f]) for f in processed])
#draw_side_by_side([], [transform_facet(f, processed[f]) for f in processed])
curr = to_process.pop(0)
curr_n = neighbours[curr[0]]
curr_t = curr[2]
curr_e = curr[1]
curr_t = get_shift_transform(-pts[curr_e[0]].x, -pts[curr_e[0]].y) * curr_t
curr_t = get_refl_transform(
-(pts[curr_e[1]].y - pts[curr_e[0]].y),
(pts[curr_e[1]].x - pts[curr_e[0]].x)
) * curr_t
curr_t = get_shift_transform(pts[curr_e[0]].x, pts[curr_e[0]].y) * curr_t
if DEBUG:
print "Curr:"
print curr[0]
print facets[curr[0]]
print "Common edge with parent:"
print curr[1]
print "Parent matrix:"
print curr[2]
print "Curr matrix:"
print curr_t
print "Neighbours:"
print curr_n
for f in curr_n:
if f not in processed:
to_process.append([f, common_edge(curr[0], f), curr_t])
processed[curr[0]] = curr_t
if DEBUG:
print "Processed:"
print processed
# if DRAW:
draw_side_by_side([[pts[p] for p in f] for f in facets], [transform_facet(f, processed[f]) for f in processed])
solution = {}
for f in range(0, len(facets)):
for p in facets[f]:
solution[p] = transform_point(p, processed[f])
print len(pts)
for p in pts:
print '{},{}'.format(p.x, p.y)
print len(facets)
for f in facets:
facet_str = str(len(f))
facet_str += ' ' + ' '.join(map(str, f))
# for p in f:
# facet_str += str(p) + ","
# facet_str = facet_str[:-1]
print facet_str
alpha, beta = get_random_angles()
for p in solution:
x0, y0 = solution[p].x, solution[p].y
x = alpha * x0 + beta * y0 - 1
y = -beta * x0 + alpha * y0 - 2
print "{},{}".format(x, y)
from metric import Metric
import sympy as sp
x=sp.symbols(['z','t','x','y'])
gs=[sp.Symbol('g'+str(i))(x[0]) for i in x]
M=Metric(x,sp.diag(*gs))
A=[sp.Symbol(t)(M.x[0]) for t in ['Az','At','Ax','Ay']]
der=M.covariantVectorDerivative(A)
assert(M.g.is_diagonal())
sp.pprint(sp.simplify(sum(der[i][i]*M.ginv[i,i] for i in range(len(M.x)))))
m20, m21, m22, m23, m30, m31, m32, m33):
return mat4x4(m00, m01, m02, m03, m10, m11, m12, m13,
m20, m21, m22, m23, m30, m31, m32, m33).transpose()
if __name__ == '__main__':
print('行列の構成子のテスト')
print('mat4x4(1, 2, ..., 16):\n{0}'.format(mat4x4(*range(1, 17))))
print('tmat4x4(1, 2, ..., 16):\n{0}'.format(tmat4x4(*range(1, 17))))
# ## 拡大縮小行列: $\mathrm {Scale}(s_x, s_y, s_z)$
# In[7]:
ScaleSymbols = sp.var('s_x s_y s_z')
Symbolic['Scale'] = sp.diag(s_x, s_y, s_z, 1)
scale = Symbolic['scale'] = Symbolic['M32'](ScaleSymbols, Symbolic['Scale'])
if __name__ == '__main__':
md('拡大縮小変換行列$\mathrm {Scale}(s_x, s_y, s_z)$に同次座標を乗ずると，',
'その座標の$X$-, $Y$-, $Z$-成分をそれぞれ$s_x, s_y, s_z$倍した同次座標を与えます．')
p = Vec3(x, y, z)
md('$$\mathrm {Scale}(s_x, s_y, s_z)', Homogeneous(p), '=',
Symbolic['Scale'], Homogeneous(p), '=',
(Symbolic['Scale'] * Homogeneous(p)), '$$')
# ## 回転変換: Rotate_
# In[8]:
AdSfr=Metric(x, sp.diag(1/f,-f,x[0]**2,x[0]**2))
AdSfr.f=f
#used by mcgreevy, hereby shown to be equivalent
xn=[sp.Symbol('z',positive=True)]+sp.symbols(['t','x1','x2'])
AdSfz=changeOfVars(AdSfr,xn,[1/xn[0]]+xn[1:])
AdSfz.f=sp.Symbol('f')(1/xn[0])
#scwarzchild f
w=sp.Wild('w')
AdSBHz=Metric(AdSfz.x, AdSfz.g.applyfunc(lambda i:i.replace(sp.Symbol('f')(w),w**2-1/w) ), name='AdSBH')
"""
zh, L = sp.symbols(["zh", "L"], positive=True)
x = [sp.Symbol("z", positive=True)] + sp.symbols(["t", "x1", "x2"])
f = 1 - (x[0] / zh) ** (len(x) - 1)
AdSBHz = Metric(x, sp.diag(1 / f, -f, 1, 1) * (L / x[0]) ** 2, "AdSBHz")
AdSBHz.zh, AdSBHz.L = (zh, L)
zh, L = sp.symbols(["zh", "L"], positive=True)
x = [sp.Symbol("z", positive=True)] + sp.symbols(["tau", "x1", "x2"])
f = 1 - (x[0] / zh) ** (len(x) - 1)
AdSBHzE = Metric(x, sp.diag(1 / f, f, 1, 1) * (L / x[0]) ** 2, "AdSBHzE")
AdSBHzE.zh, AdSBHzE.L = (zh, L)
f = sp.Symbol("f")(x[0])
AdSBHf = Metric(x, sp.diag(1 / f, -f, 1, 1) * (L / x[0]) ** 2)
AdSBHf.L = L
AdSBHf.f = f
if __name__ == "__main__":
# sp.pprint(AdSfr.R.subs(AdSfr.f,AdSfr.r**2-1/AdSfr.r).doit().ratsimp())
def test_H2():
TP = sympy.diffgeom.TensorProduct
R2 = sympy.diffgeom.rn.R2
y = R2.y
dy = R2.dy
dx = R2.dx
g = (TP(dx, dx) + TP(dy, dy))*y**(-2)
automat = twoform_to_matrix(g)
mat = diag(y**(-2), y**(-2))
assert mat == automat
gamma1 = metric_to_Christoffel_1st(g)
assert gamma1[0, 0, 0] == 0
assert gamma1[0, 0, 1] == -y**(-3)
assert gamma1[0, 1, 0] == -y**(-3)
assert gamma1[0, 1, 1] == 0
assert gamma1[1, 1, 1] == -y**(-3)
assert gamma1[1, 1, 0] == 0
assert gamma1[1, 0, 1] == 0
assert gamma1[1, 0, 0] == y**(-3)
gamma2 = metric_to_Christoffel_2nd(g)
assert gamma2[0, 0, 0] == 0
assert gamma2[0, 0, 1] == -y**(-1)
assert gamma2[0, 1, 0] == -y**(-1)
assert gamma2[0, 1, 1] == 0
assert gamma2[1, 1, 1] == -y**(-1)
assert gamma2[1, 1, 0] == 0
assert gamma2[1, 0, 1] == 0
assert gamma2[1, 0, 0] == y**(-1)
Rm = metric_to_Riemann_components(g)
assert Rm[0, 0, 0, 0] == 0
assert Rm[0, 0, 0, 1] == 0
assert Rm[0, 0, 1, 0] == 0
assert Rm[0, 0, 1, 1] == 0
assert Rm[0, 1, 0, 0] == 0
assert Rm[0, 1, 0, 1] == -y**(-2)
assert Rm[0, 1, 1, 0] == y**(-2)
assert Rm[0, 1, 1, 1] == 0
assert Rm[1, 0, 0, 0] == 0
assert Rm[1, 0, 0, 1] == y**(-2)
assert Rm[1, 0, 1, 0] == -y**(-2)
assert Rm[1, 0, 1, 1] == 0
assert Rm[1, 1, 0, 0] == 0
assert Rm[1, 1, 0, 1] == 0
assert Rm[1, 1, 1, 0] == 0
assert Rm[1, 1, 1, 1] == 0
Ric = metric_to_Ricci_components(g)
assert Ric[0, 0] == -y**(-2)
assert Ric[0, 1] == 0
assert Ric[1, 0] == 0
assert Ric[0, 0] == -y**(-2)
assert Ric == ImmutableDenseNDimArray([-y**(-2), 0, 0, -y**(-2)], (2, 2))
## scalar curvature is -2
#TODO - it would be nice to have index contraction built-in
R = (Ric[0, 0] + Ric[1, 1])*y**2
assert R == -2
## Gauss curvature is -1
assert R/2 == -1
Omicron = Matrix([[1, 0, 0, 0],
[0, 1, 0, 0],
[0, 0, 0, 1],
[0, 1, 0, 0],
[0, 0, 0, 1]])
D = Matrix([[0, 1, 0, 0],
[0, 0, 1, 0],
[1, 0, 0, 0],
[0, 0, 0, 1],
[0, 0, 0, 1]])
# Multiplicative component of natural law
multiplicative_components = [0, 0, 0, 0, 0]
Upsilon = diag(*multiplicative_components)
# Additive component of natural law
additive_components = [0, 1, 0, 0, 0]
S = diag(*additive_components)
# Accelerator
Alpha = Matrix([[1, 0, 0, 0, 0, 0],
[0, 0, 1, 0, 0, 0],
[0, 0, 0, 0, 1, 0],
[0, 0, 0, 0, 0, 1],
[0, 0, 0, 1, 0, 0]])
p = [[1, 4], [6, -1], [1, -6]]
def __init__(self, scheme):
# TODO: add source terms
t, x, y, z = sp.symbols("t x y z", type='real')
consm = list(scheme.consm.keys())
nconsm = len(scheme.consm)
self.consm = sp.Matrix(consm)
self.dim = scheme.dim
space = [x, y, z]
LA = scheme.symb_la
if not LA:
LA = scheme.la
func = []
for i in range(nconsm):
func.append(sp.Function('f{}'.format(i))(t, x, y, z)) #pylint: disable=not-callable
func = sp.Matrix(func)
sublist = [(i, j) for i, j in zip(consm, func)]
sublist_inv = [(j, i) for i, j in zip(consm, func)]
eq_func = sp.Matrix(scheme.EQ[nconsm:]).subs(sublist)
s = sp.Matrix(scheme.s[nconsm:])
all_vel = scheme.stencil.get_all_velocities()
Lambda = []
for i in range(all_vel.shape[1]):
vd = LA*sp.diag(*all_vel[:, i])
Lambda.append(scheme.M*vd*scheme.invM)
phi1 = sp.zeros(s.shape[0], 1) #pylint: disable=unsubscriptable-object
sigma = sp.diag(*s).inv() - sp.eye(len(s))/2
gamma_1 = sp.zeros(nconsm, 1)
self.coeff_order1 = []
for dim, lambda_ in enumerate(Lambda):
A, B = sp.Matrix([lambda_[:nconsm, :nconsm]]), sp.Matrix([lambda_[:nconsm, nconsm:]])
C, D = sp.Matrix([lambda_[nconsm:, :nconsm]]), sp.Matrix([lambda_[nconsm:, nconsm:]])
self.coeff_order1.append(A*func + B*eq_func)
alltogether(self.coeff_order1[-1], nsimplify=True)
for i in range(nconsm):
gamma_1[i] += sp.Derivative(self.coeff_order1[-1][i], space[dim])
dummy = -C*func - D*eq_func
alltogether(dummy, nsimplify=True)
for i in range(dummy.shape[0]):
phi1[i] += sp.Derivative(dummy[i], space[dim])
self.coeff_order2 = [[sp.zeros(nconsm) for _ in range(scheme.dim)] for _ in range(scheme.dim)]
for dim, lambda_ in enumerate(Lambda):
A, B = sp.Matrix([lambda_[:nconsm, :nconsm]]), sp.Matrix([lambda_[:nconsm, nconsm:]])
meq = sp.Matrix(scheme.EQ[nconsm:])
jac = meq.jacobian(consm)
jac = jac.subs(sublist)
delta1 = jac*gamma_1
phi1_ = phi1 + delta1
sphi1 = B*sigma*phi1_
sphi1 = sphi1.doit()
alltogether(sphi1, nsimplify=True)
for i in range(scheme.dim):
for jc in range(nconsm):
for ic in range(nconsm):
self.coeff_order2[dim][i][ic, jc] += sphi1[ic].expand().coeff(sp.Derivative(func[jc], space[i]))
for ic, c in enumerate(self.coeff_order1):
self.coeff_order1[ic] = c.subs(sublist_inv)
for ic, c in enumerate(self.coeff_order2):
for jc, cc in enumerate(c):
self.coeff_order2[ic][jc] = cc.subs(sublist_inv)
