def __init__(self, dimension, ncomponents, eps):
r"""Initialize a new homogeneous Hagedorn wavepacket.
:param dimension: The space dimension :math:`D` the packet has.
:param ncomponents: The number :math:`N` of components the packet has.
:param eps: The semi-classical scaling parameter :math:`\varepsilon` of the basis functions.
:return: An instance of :py:class:`HagedornWavepacket`.
"""
self._dimension = dimension
self._number_components = ncomponents
self._eps = eps
# The basis shapes K_i
self._basis_shapes = []
# The coefficients c^i
self._coefficients = []
for d in xrange(self._number_components):
# Default basis shapes for all components
bs = HyperCubicShape( self._dimension*[1] )
self._basis_shapes.append(bs)
# A Gaussian
self._coefficients.append(zeros((bs.get_basis_size(),1), dtype=complexfloating))
# Cache basis sizes
self._basis_sizes = [ bs.get_basis_size() for bs in self._basis_shapes ]
# Default parameters of harmonic oscillator eigenstates
q = zeros((self._dimension, 1))
p = zeros((self._dimension, 1))
Q = eye(self._dimension)
P = 1.0j * eye(self._dimension)
S = 0.0
# The parameter set Pi
self._Pis = [q, p, Q, P, S]
# No quadrature set
self._QE = None
# Function for taking continuous roots
self._sqrt = ContinuousSqrt()
class HagedornWavepacket(HagedornWavepacketBase):
r"""This class represents homogeneous vector valued Hagedorn wavepackets
:math:`\Psi` with :math:`N` components in :math:`D` space dimensions.
"""
def __init__(self, dimension, ncomponents, eps):
r"""Initialize a new homogeneous Hagedorn wavepacket.
:param dimension: The space dimension :math:`D` the packet has.
:param ncomponents: The number :math:`N` of components the packet has.
:param eps: The semi-classical scaling parameter :math:`\varepsilon` of the basis functions.
:return: An instance of :py:class:`HagedornWavepacket`.
"""
self._dimension = dimension
self._number_components = ncomponents
self._eps = eps
# The basis shapes K_i
self._basis_shapes = []
# The coefficients c^i
self._coefficients = []
for d in xrange(self._number_components):
# Default basis shapes for all components
bs = HyperCubicShape( self._dimension*[1] )
self._basis_shapes.append(bs)
# A Gaussian
self._coefficients.append(zeros((bs.get_basis_size(),1), dtype=complexfloating))
# Cache basis sizes
self._basis_sizes = [ bs.get_basis_size() for bs in self._basis_shapes ]
# Default parameters of harmonic oscillator eigenstates
q = zeros((self._dimension, 1))
p = zeros((self._dimension, 1))
Q = eye(self._dimension)
P = 1.0j * eye(self._dimension)
S = 0.0
# The parameter set Pi
self._Pis = [q, p, Q, P, S]
# No quadrature set
self._QE = None
# Function for taking continuous roots
self._sqrt = ContinuousSqrt()
def __str__(self):
r""":return: A string describing the Hagedorn wavepacket :math:`\Psi`.
"""
s = ("Homogeneous Hagedorn wavepacket with "+str(self._number_components)
+" component(s) in "+str(self._dimension)+" space dimension(s)\n")
return s
def get_description(self):
r"""Return a description of this wavepacket object.
A description is a ``dict`` containing all key-value pairs
necessary to reconstruct the current instance. A description
never contains any data.
"""
d = {}
d["type"] = "HagedornWavepacket"
d["dimension"] = self._dimension
d["ncomponents"] = self._number_components
d["eps"] = self._eps
if self._QE is not None:
d["quadrature"] = self._QE.get_description()
return d
def clone(self, keepid=False):
# Parameters of this packet
params = self.get_description()
# Create a new Packet
# TODO: Consider using the block factory
other = HagedornWavepacket(params["dimension"],
params["ncomponents"],
params["eps"])
# If we wish to keep the packet ID
if keepid is True:
other.set_id(self.get_id())
# And copy over all (private) data
# Basis shapes are immutable, no issues with sharing same instance
other.set_basis_shape(self.get_basis_shape())
other.set_parameters(self.get_parameters())
other.set_coefficients(self.get_coefficients())
# Quadratures are immutable, no issues with sharing same instance
other.set_quadrature(self.get_quadrature())
# The complex root cache
other._sqrt = self._sqrt.clone()
return other
def get_parameters(self, component=None, aslist=False):
r"""Get the Hagedorn parameter set :math:`\Pi` of the wavepacket :math:`\Psi`.
:param component: Dummy parameter for API compatibility with the inhomogeneous packets.
:param aslist: Return a list of :math:`N` parameter tuples. This is for API compatibility
with inhomogeneous packets.
:return: The Hagedorn parameter set :math:`\Pi = (q, p, Q, P, S)` in this order.
"""
if aslist is True:
return self._number_components * [ self._Pis ]
return self._Pis[:]
def set_parameters(self, Pi, component=None):
r"""Set the Hagedorn parameters :math:`\Pi` of the wavepacket :math:`\Psi`.
:param Pi: The Hagedorn parameter set :math:`\Pi = (q, p, Q, P, S)` in this order.
:param component: Dummy parameter for API compatibility with the inhomogeneous packets.
"""
self._Pis = [ atleast_2d(array(item, dtype=complexfloating)) for item in Pi ]
def evaluate_basis_at(self, grid, component, prefactor=False):
r"""Evaluate the basis functions :math:`\phi_k` recursively at the given nodes :math:`\gamma`.
:param grid: The grid :math:\Gamma` containing the nodes :math:`\gamma`.
:type grid: A class having a :py:meth:`get_nodes(...)` method.
:param component: The index :math:`i` of a single component :math:`\Phi_i` to evaluate.
We need this to choose the correct basis shape.
:param prefactor: Whether to include a factor of :math:`\frac{1}{\sqrt{\det(Q)}}`.
:type prefactor: bool, default is ``False``.
:return: A two-dimensional ndarray :math:`H` of shape :math:`(|\mathcal{K}_i|, |\Gamma|)` where
the entry :math:`H[\mu(k), i]` is the value of :math:`\phi_k(\gamma_i)`.
"""
D = self._dimension
bas = self._basis_shapes[component]
bs = self._basis_sizes[component]
# TODO: Consider putting this into the Grid class as 2nd level API
# Allow ndarrays for the 'grid' argument
if isinstance(grid, Grid):
# The overall number of nodes
nn = grid.get_number_nodes(overall=True)
# The grid nodes
nodes = grid.get_nodes()
else:
# The overall number of nodes
nn = prod(grid.shape[1:])
# The grid nodes
nodes = grid
# Allocate the storage array
phi = zeros((bs, nn), dtype=complexfloating)
# Precompute some constants
q, p, Q, P, S = self._Pis
Qinv = inv(Q)
Qbar = conj(Q)
QQ = dot(Qinv, Qbar)
# Compute the ground state phi_0 via direct evaluation
mu0 = bas[tuple(D*[0])]
phi[mu0,:] = self._evaluate_phi0(self._Pis, nodes, prefactor=False)
# Compute all higher order states phi_k via recursion
for d in xrange(D):
# Iterator for all valid index vectors k
indices = bas.get_node_iterator(mode="chain", direction=d)
for k in indices:
# Current index vector
ki = vstack(k)
# Access predecessors
phim = zeros((D, nn), dtype=complexfloating)
for j, kpj in bas.get_neighbours(k, selection="backward"):
mukpj = bas[kpj]
phim[j,:] = phi[mukpj,:]
# Compute 3-term recursion
p1 = (nodes - q) * phi[bas[k],:]
p2 = sqrt(ki) * phim
t1 = sqrt(2.0/self._eps**2) * dot(Qinv[d,:], p1)
t2 = dot(QQ[d,:], p2)
# Find multi-index where to store the result
kped = bas.get_neighbours(k, selection="forward", direction=d)
# Did we find this k?
if len(kped) > 0:
kped = kped[0]
# Store computed value
phi[bas[kped[1]],:] = (t1 - t2) / sqrt(ki[d] + 1.0)
if prefactor is True:
phi = phi / self._sqrt(det(Q))
return phi
def evaluate_at(self, grid, component=None, prefactor=False):
r"""Evaluate the Hagedorn wavepacket :math:`\Psi` at the given nodes :math:`\gamma`.
:param grid: The grid :math:\Gamma` containing the nodes :math:`\gamma`.
:type grid: A class having a :py:meth:`get_nodes(...)` method.
:param component: The index :math:`i` of a single component :math:`\Phi_i` to evaluate.
(Defaults to ``None`` for evaluating all components.)
:param prefactor: Whether to include a factor of :math:`\frac{1}{\sqrt{\det(Q)}}`.
:type prefactor: bool, default is ``False``.
:return: A list of arrays or a single array containing the values of the :math:`\Phi_i` at the nodes :math:`\gamma`.
"""
# The global phase part
phase = exp(1.0j * self._Pis[4] / self._eps**2)
if component is not None:
basis = self.evaluate_basis_at(grid, component, prefactor=prefactor)
values = phase * sum(self._coefficients[component] * basis, axis=0)
else:
values = []
for component in xrange(self._number_components):
# Note: This is very inefficient! We may evaluate the same basis functions multiple
# times. But as long as we don't know that the basis shapes are true subsets
# of the largest one, we can not evaluate just all functions in this
# maximal set.
# TODO: Find more efficient way to do this
basis = self.evaluate_basis_at(grid, component, prefactor=prefactor)
values.append( phase * sum(self._coefficients[component] * basis, axis=0) )
return values
class HagedornWavepacket(HagedornWavepacketBase):
r"""This class represents homogeneous vector valued Hagedorn wavepackets
:math:`\Psi` with :math:`N` components in :math:`D` space dimensions.
"""
def __init__(self, dimension, ncomponents, eps):
r"""Initialize a new homogeneous Hagedorn wavepacket.
:param dimension: The space dimension :math:`D` the packet has.
:param ncomponents: The number :math:`N` of components the packet has.
:param eps: The semi-classical scaling parameter :math:`\varepsilon` of the basis functions.
:return: An instance of :py:class:`HagedornWavepacket`.
"""
self._dimension = dimension
self._number_components = ncomponents
self._eps = eps
# The basis shapes K_i
self._basis_shapes = []
# The coefficients c^i
self._coefficients = []
for d in xrange(self._number_components):
# Default basis shapes for all components
bs = HyperCubicShape( self._dimension*[1] )
self._basis_shapes.append(bs)
# A Gaussian
self._coefficients.append(zeros((bs.get_basis_size(),1), dtype=complexfloating))
# Cache basis sizes
self._basis_sizes = [ bs.get_basis_size() for bs in self._basis_shapes ]
# Default parameters of harmonic oscillator eigenstates
q = zeros((self._dimension, 1), dtype=complexfloating)
p = zeros((self._dimension, 1), dtype=complexfloating)
Q = eye(self._dimension, dtype=complexfloating)
P = 1.0j * eye(self._dimension, dtype=complexfloating)
S = zeros((1, 1), dtype=complexfloating)
# The parameter set Pi
self._Pis = [q, p, Q, P, S]
# No inner product set
self._IP = None
# Function for taking continuous roots
self._sqrt = ContinuousSqrt(reference=angle(det(Q)))
def __str__(self):
r""":return: A string describing the Hagedorn wavepacket :math:`\Psi`.
"""
s = ("Homogeneous Hagedorn wavepacket with "+str(self._number_components)
+" component(s) in "+str(self._dimension)+" space dimension(s)\n")
return s
def _get_sqrt(self, component):
r"""Compatibility method
"""
return self._sqrt
def get_description(self):
r"""Return a description of this wavepacket object.
A description is a ``dict`` containing all key-value pairs
necessary to reconstruct the current instance. A description
never contains any data.
"""
d = {}
d["type"] = "HagedornWavepacket"
d["dimension"] = self._dimension
d["ncomponents"] = self._number_components
d["eps"] = self._eps
if self._IP is not None:
d["innerproduct"] = self._IP.get_description()
return d
def clone(self, keepid=False):
# Parameters of this packet
params = self.get_description()
# Create a new Packet
# TODO: Consider using the block factory
other = HagedornWavepacket(params["dimension"],
params["ncomponents"],
params["eps"])
# If we wish to keep the packet ID
if keepid is True:
other.set_id(self.get_id())
# And copy over all (private) data
# Basis shapes are immutable, no issues with sharing same instance
other.set_basis_shapes(self.get_basis_shapes())
other.set_parameters(self.get_parameters())
other.set_coefficients(self.get_coefficients())
# Innerproducts are stateless and finally immutable,
# no issues with sharing same instance
other.set_innerproduct(self.get_innerproduct())
# The complex root cache
other._sqrt = self._sqrt.clone()
return other
def get_parameters(self, component=None, aslist=False, key=("q","p","Q","P","S")):
r"""Get the Hagedorn parameter set :math:`\Pi` of the wavepacket :math:`\Psi`.
:param component: Dummy parameter for API compatibility with the inhomogeneous packets.
:param aslist: Return a list of :math:`N` parameter tuples. This is for API compatibility
with inhomogeneous packets.
:return: The Hagedorn parameter set :math:`\Pi = (q, p, Q, P, S)` in this order.
"""
Pilist = []
for k in key:
if k == "q":
Pilist.append(self._Pis[0])
elif k == "p":
Pilist.append(self._Pis[1])
elif k == "Q":
Pilist.append(self._Pis[2])
elif k == "P":
Pilist.append(self._Pis[3])
elif k == "S":
Pilist.append(self._Pis[4])
elif k == "adQ":
Pilist.append(array(self._get_sqrt(component).get(), dtype=complexfloating))
else:
raise KeyError("Invalid parameter key: "+str(key))
if aslist is True:
return self._number_components * [ Pilist ]
return Pilist
def set_parameters(self, Pi, component=None, key=("q","p","Q","P","S")):
r"""Set the Hagedorn parameters :math:`\Pi` of the wavepacket :math:`\Psi`.
:param Pi: The Hagedorn parameter set :math:`\Pi = (q, p, Q, P, S)` in this order.
:param component: Dummy parameter for API compatibility with the inhomogeneous packets.
"""
for k, item in zip(key, Pi):
if k == "q":
self._Pis[0] = atleast_2d(array(item, dtype=complexfloating))
elif k == "p":
self._Pis[1] = atleast_2d(array(item, dtype=complexfloating))
elif k == "Q":
self._Pis[2] = atleast_2d(array(item, dtype=complexfloating))
elif k == "P":
self._Pis[3] = atleast_2d(array(item, dtype=complexfloating))
elif k == "S":
self._Pis[4] = atleast_2d(array(item, dtype=complexfloating))
elif k == "adQ":
self._get_sqrt(component).set(squeeze(item))
else:
raise KeyError("Invalid parameter key: "+str(key))
