"""
This module defines tensors with abstract index notation.
The abstract index notation has been first formalized by Penrose.
Tensor indices are formal objects, with a tensor type; there is no
notion of index range, it is only possible to assign the dimension,
used to trace the Kronecker delta; the dimension can be a Symbol.
The Einstein summation convention is used.
The covariant indices are indicated with a minus sign in front of the index.
For instance the tensor ``t = p(a)*A(b,c)*q(-c)`` has the index ``c``
contracted.
A tensor expression ``t`` can be called; called with its
indices in sorted order it is equal to itself:
in the above example ``t(a, b) == t``;
one can call ``t`` with different indices; ``t(c, d) == p(c)*A(d,a)*q(-a)``.
The contracted indices are dummy indices, internally they have no name,
the indices being represented by a graph-like structure.
Tensors are put in canonical form using ``canon_bp``, which uses
the Butler-Portugal algorithm for canonicalization using the monoterm
symmetries of the tensors.
If there is a (anti)symmetric metric, the indices can be raised and
lowered when the tensor is put in canonical form.
"""
from __future__ import print_function, division
from collections import defaultdict
from sympy import Matrix, Rational
from sympy.combinatorics.tensor_can import get_symmetric_group_sgs, \
bsgs_direct_product, canonicalize, riemann_bsgs
from sympy.core import Basic, sympify, Add, S
from sympy.core.compatibility import string_types, reduce, range
from sympy.core.containers import Tuple
from sympy.core.decorators import deprecated
from sympy.core.symbol import Symbol, symbols
from sympy.core.sympify import CantSympify
from sympy.external import import_module
from sympy.utilities.decorator import doctest_depends_on
from sympy.matrices import eye
class TIDS(CantSympify):
"""
Tensor-index data structure. This contains internal data structures about
components of a tensor expression, its free and dummy indices.
To create a ``TIDS`` object via the standard constructor, the required
arguments are
WARNING: this class is meant as an internal representation of tensor data
structures and should not be directly accessed by end users.
Parameters
==========
components : ``TensorHead`` objects representing the components of the tensor expression.
free : Free indices in their internal representation.
dum : Dummy indices in their internal representation.
Examples
========
>>> from sympy.tensor.tensor import TensorIndexType, tensor_indices, TIDS, tensorhead
>>> Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
>>> m0, m1, m2, m3 = tensor_indices('m0,m1,m2,m3', Lorentz)
>>> T = tensorhead('T', [Lorentz]*4, [[1]*4])
>>> TIDS([T], [(m0, 0, 0), (m3, 3, 0)], [(1, 2, 0, 0)])
TIDS([T(Lorentz,Lorentz,Lorentz,Lorentz)], [(m0, 0, 0), (m3, 3, 0)], [(1, 2, 0, 0)])
Notes
=====
In short, this has created the components, free and dummy indices for
the internal representation of a tensor T(m0, m1, -m1, m3).
Free indices are represented as a list of triplets. The elements of
each triplet identify a single free index and are
1. TensorIndex object
2. position inside the component
3. component number
Dummy indices are represented as a list of 4-plets. Each 4-plet stands
for couple for contracted indices, their original TensorIndex is not
stored as it is no longer required. The four elements of the 4-plet
are
1. position inside the component of the first index.
2. position inside the component of the second index.
3. component number of the first index.
4. component number of the second index.
"""
def __init__(self, components, free, dum):
self.components = components
self.free = free
self.dum = dum
self._ext_rank = len(self.free) + 2*len(self.dum)
self.dum.sort(key=lambda x: (x[2], x[0]))
def get_tensors(self):
"""
Get a list of ``Tensor`` objects having the same ``TIDS`` if multiplied
by one another.
"""
indices = self.get_indices()
components = self.components
tensors = [None for i in components] # pre-allocate list
ind_pos = 0
for i, component in enumerate(components):
prev_pos = ind_pos
ind_pos += component.rank
tensors[i] = Tensor(component, indices[prev_pos:ind_pos])
return tensors
def get_components_with_free_indices(self):
"""
Get a list of components with their associated indices.
Examples
========
>>> from sympy.tensor.tensor import TensorIndexType, tensor_indices, TIDS, tensorhead
>>> Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
>>> m0, m1, m2, m3 = tensor_indices('m0,m1,m2,m3', Lorentz)
>>> T = tensorhead('T', [Lorentz]*4, [[1]*4])
>>> A = tensorhead('A', [Lorentz], [[1]])
>>> t = TIDS.from_components_and_indices([T], [m0, m1, -m1, m3])
>>> t.get_components_with_free_indices()
[(T(Lorentz,Lorentz,Lorentz,Lorentz), [(m0, 0, 0), (m3, 3, 0)])]
>>> t2 = (A(m0)*A(-m0))._tids
>>> t2.get_components_with_free_indices()
[(A(Lorentz), []), (A(Lorentz), [])]
>>> t3 = (A(m0)*A(-m1)*A(-m0)*A(m1))._tids
>>> t3.get_components_with_free_indices()
[(A(Lorentz), []), (A(Lorentz), []), (A(Lorentz), []), (A(Lorentz), [])]
>>> t4 = (A(m0)*A(m1)*A(-m0))._tids
>>> t4.get_components_with_free_indices()
[(A(Lorentz), []), (A(Lorentz), [(m1, 0, 1)]), (A(Lorentz), [])]
>>> t5 = (A(m0)*A(m1)*A(m2))._tids
>>> t5.get_components_with_free_indices()
[(A(Lorentz), [(m0, 0, 0)]), (A(Lorentz), [(m1, 0, 1)]), (A(Lorentz), [(m2, 0, 2)])]
"""
components = self.components
ret_comp = []
free_counter = 0
if len(self.free) == 0:
return [(comp, []) for comp in components]
for i, comp in enumerate(components):
c_free = []
while free_counter < len(self.free):
if not self.free[free_counter][2] == i:
break
c_free.append(self.free[free_counter])
free_counter += 1
if free_counter >= len(self.free):
break
ret_comp.append((comp, c_free))
return ret_comp
@staticmethod
def from_components_and_indices(components, indices):
"""
Create a new ``TIDS`` object from ``components`` and ``indices``
``components`` ``TensorHead`` objects representing the components
of the tensor expression.
``indices`` ``TensorIndex`` objects, the indices. Contractions are
detected upon construction.
Examples
========
>>> from sympy.tensor.tensor import TensorIndexType, tensor_indices, TIDS, tensorhead
>>> Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
>>> m0, m1, m2, m3 = tensor_indices('m0,m1,m2,m3', Lorentz)
>>> T = tensorhead('T', [Lorentz]*4, [[1]*4])
>>> TIDS.from_components_and_indices([T], [m0, m1, -m1, m3])
TIDS([T(Lorentz,Lorentz,Lorentz,Lorentz)], [(m0, 0, 0), (m3, 3, 0)], [(1, 2, 0, 0)])
In case of many components the same indices have slightly different
indexes:
>>> A = tensorhead('A', [Lorentz], [[1]])
>>> TIDS.from_components_and_indices([A]*4, [m0, m1, -m1, m3])
TIDS([A(Lorentz), A(Lorentz), A(Lorentz), A(Lorentz)], [(m0, 0, 0), (m3, 0, 3)], [(0, 0, 1, 2)])
"""
tids = None
cur_pos = 0
for i in components:
tids_sing = TIDS([i], *TIDS.free_dum_from_indices(*indices[cur_pos:cur_pos+i.rank]))
if tids is None:
tids = tids_sing
else:
tids *= tids_sing
cur_pos += i.rank
if tids is None:
tids = TIDS([], [], [])
tids.free.sort(key=lambda x: x[0].name)
tids.dum.sort()
return tids
@deprecated(useinstead="get_indices")
def to_indices(self):
return self.get_indices()
@staticmethod
def free_dum_from_indices(*indices):
"""
Convert ``indices`` into ``free``, ``dum`` for single component tensor
``free`` list of tuples ``(index, pos, 0)``,
where ``pos`` is the position of index in
the list of indices formed by the component tensors
``dum`` list of tuples ``(pos_contr, pos_cov, 0, 0)``
Examples
========
>>> from sympy.tensor.tensor import TensorIndexType, tensor_indices, TIDS
>>> Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
>>> m0, m1, m2, m3 = tensor_indices('m0,m1,m2,m3', Lorentz)
>>> TIDS.free_dum_from_indices(m0, m1, -m1, m3)
([(m0, 0, 0), (m3, 3, 0)], [(1, 2, 0, 0)])
"""
n = len(indices)
if n == 1:
return [(indices[0], 0, 0)], []
# find the positions of the free indices and of the dummy indices
free = [True]*len(indices)
index_dict = {}
dum = []
for i, index in enumerate(indices):
name = index._name
typ = index._tensortype
contr = index._is_up
if (name, typ) in index_dict:
# found a pair of dummy indices
is_contr, pos = index_dict[(name, typ)]
# check consistency and update free
if is_contr:
if contr:
raise ValueError('two equal contravariant indices in slots %d and %d' %(pos, i))
else:
free[pos] = False
free[i] = False
else:
if contr:
free[pos] = False
free[i] = False
else:
raise ValueError('two equal covariant indices in slots %d and %d' %(pos, i))
if contr:
dum.append((i, pos, 0, 0))
else:
dum.append((pos, i, 0, 0))
else:
index_dict[(name, typ)] = index._is_up, i
free = [(index, i, 0) for i, index in enumerate(indices) if free[i]]
free.sort()
return free, dum
@staticmethod
def _check_matrix_indices(f_free, g_free, nc1):
# This "private" method checks matrix indices.
# Matrix indices are special as there are only two, and observe
# anomalous substitution rules to determine contractions.
dum = []
# make sure that free indices appear in the same order as in their component:
f_free.sort(key=lambda x: (x[2], x[1]))
g_free.sort(key=lambda x: (x[2], x[1]))
matrix_indices_storage = {}
transform_right_to_left = {}
f_pop_pos = []
g_pop_pos = []
for free_pos, (ind, i, c) in enumerate(f_free):
index_type = ind._tensortype
if ind not in (index_type.auto_left, -index_type.auto_right):
continue
matrix_indices_storage[ind] = (free_pos, i, c)
for free_pos, (ind, i, c) in enumerate(g_free):
index_type = ind._tensortype
if ind not in (index_type.auto_left, -index_type.auto_right):
continue
if ind == index_type.auto_left:
if -index_type.auto_right in matrix_indices_storage:
other_pos, other_i, other_c = matrix_indices_storage.pop(-index_type.auto_right)
dum.append((other_i, i, other_c, c + nc1))
# mark to remove other_pos and free_pos from free:
g_pop_pos.append(free_pos)
f_pop_pos.append(other_pos)
continue
if ind in matrix_indices_storage:
other_pos, other_i, other_c = matrix_indices_storage.pop(ind)
dum.append((other_i, i, other_c, c + nc1))
# mark to remove other_pos and free_pos from free:
g_pop_pos.append(free_pos)
f_pop_pos.append(other_pos)
transform_right_to_left[-index_type.auto_right] = c
continue
if ind in transform_right_to_left:
other_c = transform_right_to_left.pop(ind)
if c == other_c:
g_free[free_pos] = (index_type.auto_left, i, c)
for i in reversed(sorted(f_pop_pos)):
f_free.pop(i)
for i in reversed(sorted(g_pop_pos)):
g_free.pop(i)
return dum
@staticmethod
def mul(f, g):
"""
The algorithms performing the multiplication of two ``TIDS`` instances.
In short, it forms a new ``TIDS`` object, joining components and indices,
checking that abstract indices are compatible, and possibly contracting
them.
Examples
========
>>> from sympy.tensor.tensor import TensorIndexType, tensor_indices, TIDS, tensorhead
>>> Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
>>> m0, m1, m2, m3 = tensor_indices('m0,m1,m2,m3', Lorentz)
>>> T = tensorhead('T', [Lorentz]*4, [[1]*4])
>>> A = tensorhead('A', [Lorentz], [[1]])
>>> tids_1 = TIDS.from_components_and_indices([T], [m0, m1, -m1, m3])
>>> tids_2 = TIDS.from_components_and_indices([A], [m2])
>>> tids_1 * tids_2
TIDS([T(Lorentz,Lorentz,Lorentz,Lorentz), A(Lorentz)],\
[(m0, 0, 0), (m3, 3, 0), (m2, 0, 1)], [(1, 2, 0, 0)])
In this case no contraction has been performed.
>>> tids_3 = TIDS.from_components_and_indices([A], [-m3])
>>> tids_1 * tids_3
TIDS([T(Lorentz,Lorentz,Lorentz,Lorentz), A(Lorentz)],\
[(m0, 0, 0)], [(1, 2, 0, 0), (3, 0, 0, 1)])
Free indices ``m3`` and ``-m3`` are identified as a contracted couple, and are
therefore transformed into dummy indices.
A wrong index construction (for example, trying to contract two
contravariant indices or using indices multiple times) would result in
an exception:
>>> tids_4 = TIDS.from_components_and_indices([A], [m3])
>>> # This raises an exception:
>>> # tids_1 * tids_4
"""
index_up = lambda u: u if u.is_up else -u
# lambda returns True is index is not a matrix index:
notmat = lambda i: i not in (i._tensortype.auto_left, -i._tensortype.auto_right)
f_free = f.free[:]
g_free = g.free[:]
nc1 = len(f.components)
dum = TIDS._check_matrix_indices(f_free, g_free, nc1)
# find out which free indices of f and g are contracted
free_dict1 = dict([(i if i.is_up else -i, (pos, cpos, i)) for i, pos, cpos in f_free])
free_dict2 = dict([(i if i.is_up else -i, (pos, cpos, i)) for i, pos, cpos in g_free])
free_names = set(free_dict1.keys()) & set(free_dict2.keys())
# find the new `free` and `dum`
dum2 = [(i1, i2, c1 + nc1, c2 + nc1) for i1, i2, c1, c2 in g.dum]
free1 = [(ind, i, c) for ind, i, c in f_free if index_up(ind) not in free_names]
free2 = [(ind, i, c + nc1) for ind, i, c in g_free if index_up(ind) not in free_names]
free = free1 + free2
dum.extend(f.dum + dum2)
for name in free_names:
ipos1, cpos1, ind1 = free_dict1[name]
ipos2, cpos2, ind2 = free_dict2[name]
cpos2 += nc1
if ind1._is_up == ind2._is_up:
raise ValueError('wrong index construction {0}'.format(ind1))
if ind1._is_up:
new_dummy = (ipos1, ipos2, cpos1, cpos2)
else:
new_dummy = (ipos2, ipos1, cpos2, cpos1)
dum.append(new_dummy)
return (f.components + g.components, free, dum)
def __mul__(self, other):
return TIDS(*self.mul(self, other))
def __str__(self):
return "TIDS({0}, {1}, {2})".format(self.components, self.free, self.dum)
def __repr__(self):
return self.__str__()
def sorted_components(self):
"""
Returns a ``TIDS`` with sorted components
The sorting is done taking into account the commutation group
of the component tensors.
"""
from sympy.combinatorics.permutations import _af_invert
cv = list(zip(self.components, range(len(self.components))))
sign = 1
n = len(cv) - 1
for i in range(n):
for j in range(n, i, -1):
c = cv[j-1][0].commutes_with(cv[j][0])
if c not in [0, 1]:
continue
if (cv[j-1][0]._types, cv[j-1][0]._name) > \
(cv[j][0]._types, cv[j][0]._name):
cv[j-1], cv[j] = cv[j], cv[j-1]
if c:
sign = -sign
# perm_inv[new_pos] = old_pos
components = [x[0] for x in cv]
perm_inv = [x[1] for x in cv]
perm = _af_invert(perm_inv)
free = [(ind, i, perm[c]) for ind, i, c in self.free]
free.sort()
dum = [(i1, i2, perm[c1], perm[c2]) for i1, i2, c1, c2 in self.dum]
dum.sort(key=lambda x: components[x[2]].index_types[x[0]])
return TIDS(components, free, dum), sign
def _get_sorted_free_indices_for_canon(self):
sorted_free = self.free[:]
sorted_free.sort(key=lambda x: x[0])
return sorted_free
def _get_sorted_dum_indices_for_canon(self):
return sorted(self.dum, key=lambda x: (x[2], x[0]))
def canon_args(self):
"""
Returns ``(g, dummies, msym, v)``, the entries of ``canonicalize``
see ``canonicalize`` in ``tensor_can.py``
"""
# to be called after sorted_components
from sympy.combinatorics.permutations import _af_new
# types = list(set(self._types))
# types.sort(key = lambda x: x._name)
n = self._ext_rank
g = [None]*n + [n, n+1]
pos = 0
vpos = []
components = self.components
for t in components:
vpos.append(pos)
pos += t._rank
# ordered indices: first the free indices, ordered by types
# then the dummy indices, ordered by types and contravariant before
# covariant
# g[position in tensor] = position in ordered indices
for i, (indx, ipos, cpos) in enumerate(self._get_sorted_free_indices_for_canon()):
pos = vpos[cpos] + ipos
g[pos] = i
pos = len(self.free)
j = len(self.free)
dummies = []
prev = None
a = []
msym = []
for ipos1, ipos2, cpos1, cpos2 in self._get_sorted_dum_indices_for_canon():
pos1 = vpos[cpos1] + ipos1
pos2 = vpos[cpos2] + ipos2
g[pos1] = j
g[pos2] = j + 1
j += 2
typ = components[cpos1].index_types[ipos1]
if typ != prev:
if a:
dummies.append(a)
a = [pos, pos + 1]
prev = typ
msym.append(typ.metric_antisym)
else:
a.extend([pos, pos + 1])
pos += 2
if a:
dummies.append(a)
numtyp = []
prev = None
for t in components:
if t == prev:
numtyp[-1][1] += 1
else:
prev = t
numtyp.append([prev, 1])
v = []
for h, n in numtyp:
if h._comm == 0 or h._comm == 1:
comm = h._comm
else:
comm = TensorManager.get_comm(h._comm, h._comm)
v.append((h._symmetry.base, h._symmetry.generators, n, comm))
return _af_new(g), dummies, msym, v
def perm2tensor(self, g, canon_bp=False):
"""
Returns a ``TIDS`` instance corresponding to the permutation ``g``
``g`` permutation corresponding to the tensor in the representation
used in canonicalization
``canon_bp`` if True, then ``g`` is the permutation
corresponding to the canonical form of the tensor
"""
vpos = []
components = self.components
pos = 0
for t in components:
vpos.append(pos)
pos += t._rank
sorted_free = [i[0] for i in self._get_sorted_free_indices_for_canon()]
nfree = len(sorted_free)
rank = self._ext_rank
dum = [[None]*4 for i in range((rank - nfree)//2)]
free = []
icomp = -1
for i in range(rank):
if i in vpos:
icomp += vpos.count(i)
pos0 = i
ipos = i - pos0
gi = g[i]
if gi < nfree:
ind = sorted_free[gi]
free.append((ind, ipos, icomp))
else:
j = gi - nfree
idum, cov = divmod(j, 2)
if cov:
dum[idum][1] = ipos
dum[idum][3] = icomp
else:
dum[idum][0] = ipos
dum[idum][2] = icomp
dum = [tuple(x) for x in dum]
return TIDS(components, free, dum)
def get_indices(self):
"""
Get a list of indices, creating new tensor indices to complete dummy indices.
"""
components = self.components
free = self.free
dum = self.dum
indices = [None]*self._ext_rank
start = 0
pos = 0
vpos = []
for t in components:
vpos.append(pos)
pos += t.rank
cdt = defaultdict(int)
# if the free indices have names with dummy_fmt, start with an
# index higher than those for the dummy indices
# to avoid name collisions
for indx, ipos, cpos in free:
if indx._name.split('_')[0] == indx._tensortype._dummy_fmt[:-3]:
cdt[indx._tensortype] = max(cdt[indx._tensortype], int(indx._name.split('_')[1]) + 1)
start = vpos[cpos]
indices[start + ipos] = indx
for ipos1, ipos2, cpos1, cpos2 in dum:
start1 = vpos[cpos1]
start2 = vpos[cpos2]
typ1 = components[cpos1].index_types[ipos1]
assert typ1 == components[cpos2].index_types[ipos2]
fmt = typ1._dummy_fmt
nd = cdt[typ1]
indices[start1 + ipos1] = TensorIndex(fmt % nd, typ1)
indices[start2 + ipos2] = TensorIndex(fmt % nd, typ1, False)
cdt[typ1] += 1
return indices
def contract_metric(self, g):
"""
Returns new TIDS and sign.
Sign is either 1 or -1, to correct the sign after metric contraction
(for spinor indices).
"""
components = self.components
antisym = g.index_types[0].metric_antisym
#if not any(x == g for x in components):
# return self
# list of positions of the metric ``g``
gpos = [i for i, x in enumerate(components) if x == g]
if not gpos:
return self, 1
sign = 1
dum = self.dum[:]
free = self.free[:]
elim = set()
for gposx in gpos:
if gposx in elim:
continue
free1 = [x for x in free if x[-1] == gposx]
dum1 = [x for x in dum if x[-2] == gposx or x[-1] == gposx]
if not dum1:
continue
elim.add(gposx)
if len(dum1) == 2:
if not antisym:
dum10, dum11 = dum1
if dum10[3] == gposx:
# the index with pos p0 and component c0 is contravariant
c0 = dum10[2]
p0 = dum10[0]
else:
# the index with pos p0 and component c0 is covariant
c0 = dum10[3]
p0 = dum10[1]
if dum11[3] == gposx:
# the index with pos p1 and component c1 is contravariant
c1 = dum11[2]
p1 = dum11[0]
else:
# the index with pos p1 and component c1 is covariant
c1 = dum11[3]
p1 = dum11[1]
dum.append((p0, p1, c0, c1))
else:
dum10, dum11 = dum1
# change the sign to bring the indices of the metric to contravariant
# form; change the sign if dum10 has the metric index in position 0
if dum10[3] == gposx:
# the index with pos p0 and component c0 is contravariant
c0 = dum10[2]
p0 = dum10[0]
if dum10[1] == 1:
sign = -sign
else:
# the index with pos p0 and component c0 is covariant
c0 = dum10[3]
p0 = dum10[1]
if dum10[0] == 0:
sign = -sign
if dum11[3] == gposx:
# the index with pos p1 and component c1 is contravariant
c1 = dum11[2]
p1 = dum11[0]
sign = -sign
else:
# the index with pos p1 and component c1 is covariant
c1 = dum11[3]
p1 = dum11[1]
dum.append((p0, p1, c0, c1))
elif len(dum1) == 1:
if not antisym:
dp0, dp1, dc0, dc1 = dum1[0]
if dc0 == dc1:
# g(i, -i)
typ = g.index_types[0]
if typ._dim is None:
raise ValueError('dimension not assigned')
sign = sign*typ._dim
else:
# g(i0, i1)*p(-i1)
if dc0 == gposx:
p1 = dp1
c1 = dc1
else:
p1 = dp0
c1 = dc0
ind, p, c = free1[0]
free.append((ind, p1, c1))
else:
dp0, dp1, dc0, dc1 = dum1[0]
if dc0 == dc1:
# g(i, -i)
typ = g.index_types[0]
if typ._dim is None:
raise ValueError('dimension not assigned')
sign = sign*typ._dim
if dp0 < dp1:
# g(i, -i) = -D with antisymmetric metric
sign = -sign
else:
# g(i0, i1)*p(-i1)
if dc0 == gposx:
p1 = dp1
c1 = dc1
if dp0 == 0:
sign = -sign
else:
p1 = dp0
c1 = dc0
ind, p, c = free1[0]
free.append((ind, p1, c1))
dum = [x for x in dum if x not in dum1]
free = [x for x in free if x not in free1]
shift = 0
shifts = [0]*len(components)
for i in range(len(components)):
if i in elim:
shift += 1
continue
shifts[i] = shift
free = [(ind, p, c - shifts[c]) for (ind, p, c) in free if c not in elim]
dum = [(p0, p1, c0 - shifts[c0], c1 - shifts[c1]) for i, (p0, p1, c0, c1) in enumerate(dum) if c0 not in elim and c1 not in elim]
components = [c for i, c in enumerate(components) if i not in elim]
tids = TIDS(components, free, dum)
return tids, sign
class _TensorDataLazyEvaluator(CantSympify):
"""
EXPERIMENTAL: do not rely on this class, it may change without deprecation
warnings in future versions of SymPy.
This object contains the logic to associate components data to a tensor
expression. Components data are set via the ``.data`` property of tensor
expressions, is stored inside this class as a mapping between the tensor
expression and the ``ndarray``.
Computations are executed lazily: whereas the tensor expressions can have
contractions, tensor products, and additions, components data are not
computed until they are accessed by reading the ``.data`` property
associated to the tensor expression.
"""
_substitutions_dict = dict()
_substitutions_dict_tensmul = dict()
def __getitem__(self, key):
dat = self._get(key)
if dat is None:
return None
numpy = import_module("numpy")
if not isinstance(dat, numpy.ndarray):
return dat
if dat.ndim == 0:
return dat[()]
elif dat.ndim == 1 and dat.size == 1:
return dat[0]
return dat
def _get(self, key):
"""
Retrieve ``data`` associated with ``key``.
This algorithm looks into ``self._substitutions_dict`` for all
``TensorHead`` in the ``TensExpr`` (or just ``TensorHead`` if key is a
TensorHead instance). It reconstructs the components data that the
tensor expression should have by performing on components data the
operations that correspond to the abstract tensor operations applied.
Metric tensor is handled in a different manner: it is pre-computed in
``self._substitutions_dict_tensmul``.
"""
if key in self._substitutions_dict:
return self._substitutions_dict[key]
if isinstance(key, TensorHead):
return None
if isinstance(key, Tensor):
# special case to handle metrics. Metric tensors cannot be
# constructed through contraction by the metric, their
# components show if they are a matrix or its inverse.
signature = tuple([i.is_up for i in key.get_indices()])
srch = (key.component,) + signature
if srch in self._substitutions_dict_tensmul:
return self._substitutions_dict_tensmul[srch]
return self.data_tensmul_from_tensorhead(key, key.component)
if isinstance(key, TensMul):
tensmul_list = key.split()
if len(tensmul_list) == 1 and len(tensmul_list[0].components) == 1:
# special case to handle metrics. Metric tensors cannot be
# constructed through contraction by the metric, their
# components show if they are a matrix or its inverse.
signature = tuple([i.is_up for i in tensmul_list[0].get_indices()])
srch = (tensmul_list[0].components[0],) + signature
if srch in self._substitutions_dict_tensmul:
return self._substitutions_dict_tensmul[srch]
data_list = [self.data_tensmul_from_tensorhead(i, i.components[0]) for i in tensmul_list]
if all([i is None for i in data_list]):
return None
if any([i is None for i in data_list]):
raise ValueError("Mixing tensors with associated components "\
"data with tensors without components data")
data_result, tensmul_result = self.data_product_tensors(data_list, tensmul_list)
return data_result
if isinstance(key, TensAdd):
sumvar = S.Zero
data_list = [i.data for i in key.args]
if all([i is None for i in data_list]):
return None
if any([i is None for i in data_list]):
raise ValueError("Mixing tensors with associated components "\
"data with tensors without components data")
for i in data_list:
sumvar += i
return sumvar
return None
def data_tensorhead_from_tensmul(self, data, tensmul, tensorhead):
"""
This method is used when assigning components data to a ``TensMul``
object, it converts components data to a fully contravariant ndarray,
which is then stored according to the ``TensorHead`` key.
"""
if data is None:
return None
return self._correct_signature_from_indices(
data,
tensmul.get_indices(),
tensmul.free,
tensmul.dum,
True)
def data_tensmul_from_tensorhead(self, tensmul, tensorhead):
"""
This method corrects the components data to the right signature
(covariant/contravariant) using the metric associated with each
``TensorIndexType``.
"""
if tensorhead.data is None:
return None
return self._correct_signature_from_indices(
tensorhead.data,
tensmul.get_indices(),
tensmul.free,
tensmul.dum)
def data_product_tensors(self, data_list, tensmul_list):
"""
Given a ``data_list``, list of ``ndarray``'s and a ``tensmul_list``,
list of ``TensMul`` instances, compute the resulting ``ndarray``,
after tensor products and contractions.
"""
def data_mul(f, g):
"""
Multiplies two ``ndarray`` objects, it first calls ``TIDS.mul``,
then checks which indices have been contracted, and finally
contraction operation on data, according to the contracted indices.
"""
data1, tensmul1 = f
data2, tensmul2 = g
components, free, dum = TIDS.mul(tensmul1, tensmul2)
data = _TensorDataLazyEvaluator._contract_ndarray(tensmul1.free, tensmul2.free, data1, data2)
# TODO: do this more efficiently... maybe by just passing an index list
# to .data_product_tensor(...)
return data, TensMul.from_TIDS(S.One, TIDS(components, free, dum))
return reduce(data_mul, zip(data_list, tensmul_list))
def _assign_data_to_tensor_expr(self, key, data):
if isinstance(key, TensAdd):
raise ValueError('cannot assign data to TensAdd')
# here it is assumed that `key` is a `TensMul` instance.
if len(key.components) != 1:
raise ValueError('cannot assign data to TensMul with multiple components')
tensorhead = key.components[0]
newdata = self.data_tensorhead_from_tensmul(data, key, tensorhead)
return tensorhead, newdata
def _check_permutations_on_data(self, tens, data):
import numpy
if isinstance(tens, TensorHead):
rank = tens.rank
generators = tens.symmetry.generators
elif isinstance(tens, Tensor):
rank = tens.rank
generators = tens.components[0].symmetry.generators
elif isinstance(tens, TensorIndexType):
rank = tens.metric.rank
generators = tens.metric.symmetry.generators
# Every generator is a permutation, check that by permuting the array
# by that permutation, the array will be the same, except for a
# possible sign change if the permutation admits it.
for gener in generators:
sign_change = +1 if (gener(rank) == rank) else -1
data_swapped = data
last_data = data
permute_axes = list(map(gener, list(range(rank))))
# the order of a permutation is the number of times to get the
# identity by applying that permutation.
for i in range(gener.order()-1):
data_swapped = numpy.transpose(data_swapped, permute_axes)
# if any value in the difference array is non-zero, raise an error:
if (last_data - sign_change*data_swapped).any():
raise ValueError("Component data symmetry structure error")
last_data = data_swapped
def __setitem__(self, key, value):
"""
Set the components data of a tensor object/expression.
Components data are transformed to the all-contravariant form and stored
with the corresponding ``TensorHead`` object. If a ``TensorHead`` object
cannot be uniquely identified, it will raise an error.
"""
data = _TensorDataLazyEvaluator.parse_data(value)
self._check_permutations_on_data(key, data)
# TensorHead and TensorIndexType can be assigned data directly, while
# TensMul must first convert data to a fully contravariant form, and
# assign it to its corresponding TensorHead single component.
if not isinstance(key, (TensorHead, TensorIndexType)):
key, data = self._assign_data_to_tensor_expr(key, data)
if isinstance(key, TensorHead):
for dim, indextype in zip(data.shape, key.index_types):
if indextype.data is None:
raise ValueError("index type {} has no components data"\
" associated (needed to raise/lower index)".format(indextype))
if indextype.dim is None:
continue
if dim != indextype.dim:
raise ValueError("wrong dimension of ndarray")
self._substitutions_dict[key] = data
def __delitem__(self, key):
del self._substitutions_dict[key]
def __contains__(self, key):
return key in self._substitutions_dict
@staticmethod
def _contract_ndarray(free1, free2, ndarray1, ndarray2):
numpy = import_module('numpy')
def ikey(x):
return x[1:]
free1 = free1[:]
free2 = free2[:]
free1.sort(key=ikey)
free2.sort(key=ikey)
self_free = [_[0] for _ in free1]
axes1 = []
axes2 = []
for jpos, jindex in enumerate(free2):
if -jindex[0] in self_free:
nidx = self_free.index(-jindex[0])
else:
continue
axes1.append(nidx)
axes2.append(jpos)
contracted_ndarray = numpy.tensordot(
ndarray1,
ndarray2,
(axes1, axes2)
)
return contracted_ndarray
@staticmethod
def add_tensor_mul(prod, f, g):
def mul_function():
return _TensorDataLazyEvaluator._contract_ndarray(f.free, g.free, f.data, g.data)
_TensorDataLazyEvaluator._substitutions_dict[prod] = mul_function()
@staticmethod
def add_tensor_add(addition, f, g):
def add_function():
return f.data + g.data
_TensorDataLazyEvaluator._substitutions_dict[addition] = add_function()
def add_metric_data(self, metric, data):
"""
Assign data to the ``metric`` tensor. The metric tensor behaves in an
anomalous way when raising and lowering indices.
A fully covariant metric is the inverse transpose of the fully
contravariant metric (it is meant matrix inverse). If the metric is
symmetric, the transpose is not necessary and mixed
covariant/contravariant metrics are Kronecker deltas.
"""
# hard assignment, data should not be added to `TensorHead` for metric:
# the problem with `TensorHead` is that the metric is anomalous, i.e.
# raising and lowering the index means considering the metric or its
# inverse, this is not the case for other tensors.
self._substitutions_dict_tensmul[metric, True, True] = data
inverse_transpose = self.inverse_transpose_matrix(data)
# in symmetric spaces, the traspose is the same as the original matrix,
# the full covariant metric tensor is the inverse transpose, so this
# code will be able to handle non-symmetric metrics.
self._substitutions_dict_tensmul[metric, False, False] = inverse_transpose
# now mixed cases, these are identical to the unit matrix if the metric
# is symmetric.
m = Matrix(data)
invt = Matrix(inverse_transpose)
self._substitutions_dict_tensmul[metric, True, False] = m * invt
self._substitutions_dict_tensmul[metric, False, True] = invt * m
@staticmethod
def _flip_index_by_metric(data, metric, pos):
numpy = import_module('numpy')
data = numpy.tensordot(
metric,
data,
(1, pos))
return numpy.rollaxis(data, 0, pos+1)
@staticmethod
def inverse_matrix(ndarray):
m = Matrix(ndarray).inv()
return _TensorDataLazyEvaluator.parse_data(m)
@staticmethod
def inverse_transpose_matrix(ndarray):
m = Matrix(ndarray).inv().T
return _TensorDataLazyEvaluator.parse_data(m)
@staticmethod
def _correct_signature_from_indices(data, indices, free, dum, inverse=False):
"""
Utility function to correct the values inside the components data
ndarray according to whether indices are covariant or contravariant.
It uses the metric matrix to lower values of covariant indices.
"""
numpy = import_module('numpy')
# change the ndarray values according covariantness/contravariantness of the indices
# use the metric
for i, indx in enumerate(indices):
if not indx.is_up and not inverse:
data = _TensorDataLazyEvaluator._flip_index_by_metric(data, indx._tensortype.data, i)
elif not indx.is_up and inverse:
data = _TensorDataLazyEvaluator._flip_index_by_metric(
data,
_TensorDataLazyEvaluator.inverse_matrix(indx._tensortype.data),
i
)
if len(dum) > 0:
### perform contractions ###
axes1 = []
axes2 = []
for i, indx1 in enumerate(indices):
try:
nd = indices[:i].index(-indx1)
except ValueError:
continue
axes1.append(nd)
axes2.append(i)
for ax1, ax2 in zip(axes1, axes2):
data = numpy.trace(data, axis1=ax1, axis2=ax2)
return data
@staticmethod
def _sort_data_axes(old, new):
numpy = import_module('numpy')
new_data = old.data.copy()
old_free = [i[0] for i in old.free]
new_free = [i[0] for i in new.free]
for i in range(len(new_free)):
for j in range(i, len(old_free)):
if old_free[j] == new_free[i]:
old_free[i], old_free[j] = old_free[j], old_free[i]
new_data = numpy.swapaxes(new_data, i, j)
break
return new_data
@staticmethod
def add_rearrange_tensmul_parts(new_tensmul, old_tensmul):
def sorted_compo():
return _TensorDataLazyEvaluator._sort_data_axes(old_tensmul, new_tensmul)
_TensorDataLazyEvaluator._substitutions_dict[new_tensmul] = sorted_compo()
@staticmethod
@doctest_depends_on(modules=('numpy',))
def parse_data(data):
"""
Transform ``data`` to a numpy ndarray. The parameter ``data`` may
contain data in various formats, e.g. nested lists, sympy ``Matrix``,
and so on.
Examples
========
>>> from sympy.tensor.tensor import _TensorDataLazyEvaluator
>>> _TensorDataLazyEvaluator.parse_data([1, 3, -6, 12])
[1 3 -6 12]
>>> _TensorDataLazyEvaluator.parse_data([[1, 2], [4, 7]])
[[1 2]
[4 7]]
"""
numpy = import_module('numpy')
if (numpy is not None) and (not isinstance(data, numpy.ndarray)):
if len(data) == 2 and hasattr(data[0], '__call__'):
def fromfunction_sympify(*x):
return sympify(data[0](*x))
data = numpy.fromfunction(fromfunction_sympify, data[1])
else:
vsympify = numpy.vectorize(sympify)
data = vsympify(numpy.array(data))
return data
_tensor_data_substitution_dict = _TensorDataLazyEvaluator()
[docs]class _TensorManager(object):
"""
Class to manage tensor properties.
Notes
=====
Tensors belong to tensor commutation groups; each group has a label
``comm``; there are predefined labels:
``0`` tensors commuting with any other tensor
``1`` tensors anticommuting among themselves
``2`` tensors not commuting, apart with those with ``comm=0``
Other groups can be defined using ``set_comm``; tensors in those
groups commute with those with ``comm=0``; by default they
do not commute with any other group.
"""
def __init__(self):
self._comm_init()
def _comm_init(self):
self._comm = [{} for i in range(3)]
for i in range(3):
self._comm[0][i] = 0
self._comm[i][0] = 0
self._comm[1][1] = 1
self._comm[2][1] = None
self._comm[1][2] = None
self._comm_symbols2i = {0:0, 1:1, 2:2}
self._comm_i2symbol = {0:0, 1:1, 2:2}
@property
def comm(self):
return self._comm
[docs] def comm_symbols2i(self, i):
"""
get the commutation group number corresponding to ``i``
``i`` can be a symbol or a number or a string
If ``i`` is not already defined its commutation group number
is set.
"""
if i not in self._comm_symbols2i:
n = len(self._comm)
self._comm.append({})
self._comm[n][0] = 0
self._comm[0][n] = 0
self._comm_symbols2i[i] = n
self._comm_i2symbol[n] = i
return n
return self._comm_symbols2i[i]
[docs] def comm_i2symbol(self, i):
"""
Returns the symbol corresponding to the commutation group number.
"""
return self._comm_i2symbol[i]
[docs] def set_comm(self, i, j, c):
"""
set the commutation parameter ``c`` for commutation groups ``i, j``
Parameters
==========
i, j : symbols representing commutation groups
c : group commutation number
Notes
=====
``i, j`` can be symbols, strings or numbers,
apart from ``0, 1`` and ``2`` which are reserved respectively
for commuting, anticommuting tensors and tensors not commuting
with any other group apart with the commuting tensors.
For the remaining cases, use this method to set the commutation rules;
by default ``c=None``.
The group commutation number ``c`` is assigned in correspondence
to the group commutation symbols; it can be
0 commuting
1 anticommuting
None no commutation property
Examples
========
``G`` and ``GH`` do not commute with themselves and commute with
each other; A is commuting.
>>> from sympy.tensor.tensor import TensorIndexType, tensor_indices, tensorhead, TensorManager
>>> Lorentz = TensorIndexType('Lorentz')
>>> i0,i1,i2,i3,i4 = tensor_indices('i0:5', Lorentz)
>>> A = tensorhead('A', [Lorentz], [[1]])
>>> G = tensorhead('G', [Lorentz], [[1]], 'Gcomm')
>>> GH = tensorhead('GH', [Lorentz], [[1]], 'GHcomm')
>>> TensorManager.set_comm('Gcomm', 'GHcomm', 0)
>>> (GH(i1)*G(i0)).canon_bp()
G(i0)*GH(i1)
>>> (G(i1)*G(i0)).canon_bp()
G(i1)*G(i0)
>>> (G(i1)*A(i0)).canon_bp()
A(i0)*G(i1)
"""
if c not in (0, 1, None):
raise ValueError('`c` can assume only the values 0, 1 or None')
if i not in self._comm_symbols2i:
n = len(self._comm)
self._comm.append({})
self._comm[n][0] = 0
self._comm[0][n] = 0
self._comm_symbols2i[i] = n
self._comm_i2symbol[n] = i
if j not in self._comm_symbols2i:
n = len(self._comm)
self._comm.append({})
self._comm[0][n] = 0
self._comm[n][0] = 0
self._comm_symbols2i[j] = n
self._comm_i2symbol[n] = j
ni = self._comm_symbols2i[i]
nj = self._comm_symbols2i[j]
self._comm[ni][nj] = c
self._comm[nj][ni] = c
[docs] def set_comms(self, *args):
"""
set the commutation group numbers ``c`` for symbols ``i, j``
Parameters
==========
args : sequence of ``(i, j, c)``
"""
for i, j, c in args:
self.set_comm(i, j, c)
[docs] def get_comm(self, i, j):
"""
Return the commutation parameter for commutation group numbers ``i, j``
see ``_TensorManager.set_comm``
"""
return self._comm[i].get(j, 0 if i == 0 or j == 0 else None)
[docs] def clear(self):
"""
Clear the TensorManager.
"""
self._comm_init()
TensorManager = _TensorManager()
@doctest_depends_on(modules=('numpy',))
[docs]class TensorIndexType(Basic):
"""
A TensorIndexType is characterized by its name and its metric.
Parameters
==========
name : name of the tensor type
metric : metric symmetry or metric object or ``None``
dim : dimension, it can be a symbol or an integer or ``None``
eps_dim : dimension of the epsilon tensor
dummy_fmt : name of the head of dummy indices
Attributes
==========
``name``
``metric_name`` : it is 'metric' or metric.name
``metric_antisym``
``metric`` : the metric tensor
``delta`` : ``Kronecker delta``
``epsilon`` : the ``Levi-Civita epsilon`` tensor
``dim``
``dim_eps``
``dummy_fmt``
``data`` : a property to add ``ndarray`` values, to work in a specified basis.
Notes
=====
The ``metric`` parameter can be:
``metric = False`` symmetric metric (in Riemannian geometry)
``metric = True`` antisymmetric metric (for spinor calculus)
``metric = None`` there is no metric
``metric`` can be an object having ``name`` and ``antisym`` attributes.
If there is a metric the metric is used to raise and lower indices.
In the case of antisymmetric metric, the following raising and
lowering conventions will be adopted:
``psi(a) = g(a, b)*psi(-b); chi(-a) = chi(b)*g(-b, -a)``
``g(-a, b) = delta(-a, b); g(b, -a) = -delta(a, -b)``
where ``delta(-a, b) = delta(b, -a)`` is the ``Kronecker delta``
(see ``TensorIndex`` for the conventions on indices).
If there is no metric it is not possible to raise or lower indices;
e.g. the index of the defining representation of ``SU(N)``
is 'covariant' and the conjugate representation is
'contravariant'; for ``N > 2`` they are linearly independent.
``eps_dim`` is by default equal to ``dim``, if the latter is an integer;
else it can be assigned (for use in naive dimensional regularization);
if ``eps_dim`` is not an integer ``epsilon`` is ``None``.
Examples
========
>>> from sympy.tensor.tensor import TensorIndexType
>>> Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
>>> Lorentz.metric
metric(Lorentz,Lorentz)
Examples with metric components data added, this means it is working on a
fixed basis:
>>> Lorentz.data = [1, -1, -1, -1]
>>> Lorentz
TensorIndexType(Lorentz, 0)
>>> Lorentz.data
[[1 0 0 0]
[0 -1 0 0]
[0 0 -1 0]
[0 0 0 -1]]
"""
def __new__(cls, name, metric=False, dim=None, eps_dim=None,
dummy_fmt=None):
if isinstance(name, string_types):
name = Symbol(name)
obj = Basic.__new__(cls, name, S.One if metric else S.Zero)
obj._name = str(name)
if not dummy_fmt:
obj._dummy_fmt = '%s_%%d' % obj.name
else:
obj._dummy_fmt = '%s_%%d' % dummy_fmt
if metric is None:
obj.metric_antisym = None
obj.metric = None
else:
if metric in (True, False, 0, 1):
metric_name = 'metric'
obj.metric_antisym = metric
else:
metric_name = metric.name
obj.metric_antisym = metric.antisym
sym2 = TensorSymmetry(get_symmetric_group_sgs(2, obj.metric_antisym))
S2 = TensorType([obj]*2, sym2)
obj.metric = S2(metric_name)
obj.metric._matrix_behavior = True
obj._dim = dim
obj._delta = obj.get_kronecker_delta()
obj._eps_dim = eps_dim if eps_dim else dim
obj._epsilon = obj.get_epsilon()
obj._autogenerated = []
return obj
@property
def auto_right(self):
if not hasattr(self, '_auto_right'):
self._auto_right = TensorIndex("auto_right", self)
return self._auto_right
@property
def auto_left(self):
if not hasattr(self, '_auto_left'):
self._auto_left = TensorIndex("auto_left", self)
return self._auto_left
@property
def auto_index(self):
if not hasattr(self, '_auto_index'):
self._auto_index = TensorIndex("auto_index", self)
return self._auto_index
@property
def data(self):
return _tensor_data_substitution_dict[self]
@data.setter
def data(self, data):
# This assignment is a bit controversial, should metric components be assigned
# to the metric only or also to the TensorIndexType object? The advantage here
# is the ability to assign a 1D array and transform it to a 2D diagonal array.
numpy = import_module('numpy')
data = _TensorDataLazyEvaluator.parse_data(data)
if data.ndim > 2:
raise ValueError("data have to be of rank 1 (diagonal metric) or 2.")
if data.ndim == 1:
if self.dim is not None:
nda_dim = data.shape[0]
if nda_dim != self.dim:
raise ValueError("Dimension mismatch")
dim = data.shape[0]
newndarray = numpy.zeros((dim, dim), dtype=object)
for i, val in enumerate(data):
newndarray[i, i] = val
data = newndarray
dim1, dim2 = data.shape
if dim1 != dim2:
raise ValueError("Non-square matrix tensor.")
if self.dim is not None:
if self.dim != dim1:
raise ValueError("Dimension mismatch")
_tensor_data_substitution_dict[self] = data
_tensor_data_substitution_dict.add_metric_data(self.metric, data)
delta = self.get_kronecker_delta()
i1 = TensorIndex('i1', self)
i2 = TensorIndex('i2', self)
delta(i1, -i2).data = _TensorDataLazyEvaluator.parse_data(eye(dim1))
@data.deleter
def data(self):
if self in _tensor_data_substitution_dict:
del _tensor_data_substitution_dict[self]
if self.metric in _tensor_data_substitution_dict:
del _tensor_data_substitution_dict[self.metric]
@property
def name(self):
return self._name
@property
def dim(self):
return self._dim
@property
def delta(self):
return self._delta
@property
def eps_dim(self):
return self._eps_dim
@property
def epsilon(self):
return self._epsilon
@property
def dummy_fmt(self):
return self._dummy_fmt
def get_kronecker_delta(self):
sym2 = TensorSymmetry(get_symmetric_group_sgs(2))
S2 = TensorType([self]*2, sym2)
delta = S2('KD')
delta._matrix_behavior = True
return delta
def get_epsilon(self):
if not isinstance(self._eps_dim, int):
return None
sym = TensorSymmetry(get_symmetric_group_sgs(self._eps_dim, 1))
Sdim = TensorType([self]*self._eps_dim, sym)
epsilon = Sdim('Eps')
return epsilon
def __lt__(self, other):
return self.name < other.name
def __str__(self):
return self.name
__repr__ = __str__
def _components_data_full_destroy(self):
"""
EXPERIMENTAL: do not rely on this API method.
This destroys components data associated to the ``TensorIndexType``, if
any, specifically:
* metric tensor data
* Kronecker tensor data
"""
if self in _tensor_data_substitution_dict:
del _tensor_data_substitution_dict[self]
def delete_tensmul_data(key):
if key in _tensor_data_substitution_dict._substitutions_dict_tensmul:
del _tensor_data_substitution_dict._substitutions_dict_tensmul[key]
# delete metric data:
delete_tensmul_data((self.metric, True, True))
delete_tensmul_data((self.metric, True, False))
delete_tensmul_data((self.metric, False, True))
delete_tensmul_data((self.metric, False, False))
# delete delta tensor data:
delta = self.get_kronecker_delta()
if delta in _tensor_data_substitution_dict:
del _tensor_data_substitution_dict[delta]
@doctest_depends_on(modules=('numpy',))
[docs]class TensorIndex(Basic):
"""
Represents an abstract tensor index.
Parameters
==========
name : name of the index, or ``True`` if you want it to be automatically assigned
tensortype : ``TensorIndexType`` of the index
is_up : flag for contravariant index
Attributes
==========
``name``
``tensortype``
``is_up``
Notes
=====
Tensor indices are contracted with the Einstein summation convention.
An index can be in contravariant or in covariant form; in the latter
case it is represented prepending a ``-`` to the index name.
Dummy indices have a name with head given by ``tensortype._dummy_fmt``
Examples
========
>>> from sympy.tensor.tensor import TensorIndexType, TensorIndex, TensorSymmetry, TensorType, get_symmetric_group_sgs
>>> Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
>>> i = TensorIndex('i', Lorentz); i
i
>>> sym1 = TensorSymmetry(*get_symmetric_group_sgs(1))
>>> S1 = TensorType([Lorentz], sym1)
>>> A, B = S1('A,B')
>>> A(i)*B(-i)
A(L_0)*B(-L_0)
If you want the index name to be automatically assigned, just put ``True``
in the ``name`` field, it will be generated using the reserved character
``_`` in front of its name, in order to avoid conflicts with possible
existing indices:
>>> i0 = TensorIndex(True, Lorentz)
>>> i0
_i0
>>> i1 = TensorIndex(True, Lorentz)
>>> i1
_i1
>>> A(i0)*B(-i1)
A(_i0)*B(-_i1)
>>> A(i0)*B(-i0)
A(L_0)*B(-L_0)
"""
def __new__(cls, name, tensortype, is_up=True):
if isinstance(name, string_types):
name_symbol = Symbol(name)
elif isinstance(name, Symbol):
name_symbol = name
elif name is True:
name = "_i{0}".format(len(tensortype._autogenerated))
name_symbol = Symbol(name)
tensortype._autogenerated.append(name_symbol)
else:
raise ValueError("invalid name")
obj = Basic.__new__(cls, name_symbol, tensortype, S.One if is_up else S.Zero)
obj._name = str(name)
obj._tensortype = tensortype
obj._is_up = is_up
return obj
@property
def name(self):
return self._name
@property
def tensortype(self):
return self._tensortype
@property
def is_up(self):
return self._is_up
def _print(self):
s = self._name
if not self._is_up:
s = '-%s' % s
return s
def __lt__(self, other):
return (self._tensortype, self._name) < (other._tensortype, other._name)
def __neg__(self):
t1 = TensorIndex(self._name, self._tensortype,
(not self._is_up))
return t1
[docs]def tensor_indices(s, typ):
"""
Returns list of tensor indices given their names and their types
Parameters
==========
s : string of comma separated names of indices
typ : list of ``TensorIndexType`` of the indices
Examples
========
>>> from sympy.tensor.tensor import TensorIndexType, tensor_indices
>>> Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
>>> a, b, c, d = tensor_indices('a,b,c,d', Lorentz)
"""
if isinstance(s, str):
a = [x.name for x in symbols(s, seq=True)]
else:
raise ValueError('expecting a string')
tilist = [TensorIndex(i, typ) for i in a]
if len(tilist) == 1:
return tilist[0]
return tilist
@doctest_depends_on(modules=('numpy',))
[docs]class TensorSymmetry(Basic):
"""
Monoterm symmetry of a tensor
Parameters
==========
bsgs : tuple ``(base, sgs)`` BSGS of the symmetry of the tensor
Attributes
==========
``base`` : base of the BSGS
``generators`` : generators of the BSGS
``rank`` : rank of the tensor
Notes
=====
A tensor can have an arbitrary monoterm symmetry provided by its BSGS.
Multiterm symmetries, like the cyclic symmetry of the Riemann tensor,
are not covered.
See Also
========
sympy.combinatorics.tensor_can.get_symmetric_group_sgs
Examples
========
Define a symmetric tensor
>>> from sympy.tensor.tensor import TensorIndexType, tensor_indices, TensorSymmetry, TensorType, get_symmetric_group_sgs
>>> Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
>>> sym2 = TensorSymmetry(get_symmetric_group_sgs(2))
>>> S2 = TensorType([Lorentz]*2, sym2)
>>> V = S2('V')
"""
def __new__(cls, *args, **kw_args):
if len(args) == 1:
base, generators = args[0]
elif len(args) == 2:
base, generators = args
else:
raise TypeError("bsgs required, either two separate parameters or one tuple")
if not isinstance(base, Tuple):
base = Tuple(*base)
if not isinstance(generators, Tuple):
generators = Tuple(*generators)
obj = Basic.__new__(cls, base, generators, **kw_args)
return obj
@property
def base(self):
return self.args[0]
@property
def generators(self):
return self.args[1]
@property
def rank(self):
return self.args[1][0].size - 2
[docs]def tensorsymmetry(*args):
"""
Return a ``TensorSymmetry`` object.
One can represent a tensor with any monoterm slot symmetry group
using a BSGS.
``args`` can be a BSGS
``args[0]`` base
``args[1]`` sgs
Usually tensors are in (direct products of) representations
of the symmetric group;
``args`` can be a list of lists representing the shapes of Young tableaux
Notes
=====
For instance:
``[[1]]`` vector
``[[1]*n]`` symmetric tensor of rank ``n``
``[[n]]`` antisymmetric tensor of rank ``n``
``[[2, 2]]`` monoterm slot symmetry of the Riemann tensor
``[[1],[1]]`` vector*vector
``[[2],[1],[1]`` (antisymmetric tensor)*vector*vector
Notice that with the shape ``[2, 2]`` we associate only the monoterm
symmetries of the Riemann tensor; this is an abuse of notation,
since the shape ``[2, 2]`` corresponds usually to the irreducible
representation characterized by the monoterm symmetries and by the
cyclic symmetry.
Examples
========
Symmetric tensor using a Young tableau
>>> from sympy.tensor.tensor import TensorIndexType, TensorType, tensorsymmetry
>>> Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
>>> sym2 = tensorsymmetry([1, 1])
>>> S2 = TensorType([Lorentz]*2, sym2)
>>> V = S2('V')
Symmetric tensor using a ``BSGS`` (base, strong generator set)
>>> from sympy.tensor.tensor import TensorSymmetry, get_symmetric_group_sgs
>>> sym2 = tensorsymmetry(*get_symmetric_group_sgs(2))
>>> S2 = TensorType([Lorentz]*2, sym2)
>>> V = S2('V')
"""
from sympy.combinatorics import Permutation
def tableau2bsgs(a):
if len(a) == 1:
# antisymmetric vector
n = a[0]
bsgs = get_symmetric_group_sgs(n, 1)
else:
if all(x == 1 for x in a):
# symmetric vector
n = len(a)
bsgs = get_symmetric_group_sgs(n)
elif a == [2, 2]:
bsgs = riemann_bsgs
else:
raise NotImplementedError
return bsgs
if not args:
return TensorSymmetry(Tuple(), Tuple(Permutation(1)))
if len(args) == 2 and isinstance(args[1][0], Permutation):
return TensorSymmetry(args)
base, sgs = tableau2bsgs(args[0])
for a in args[1:]:
basex, sgsx = tableau2bsgs(a)
base, sgs = bsgs_direct_product(base, sgs, basex, sgsx)
return TensorSymmetry(Tuple(base, sgs))
@doctest_depends_on(modules=('numpy',))
[docs]class TensorType(Basic):
"""
Class of tensor types.
Parameters
==========
index_types : list of ``TensorIndexType`` of the tensor indices
symmetry : ``TensorSymmetry`` of the tensor
Attributes
==========
``index_types``
``symmetry``
``types`` : list of ``TensorIndexType`` without repetitions
Examples
========
Define a symmetric tensor
>>> from sympy.tensor.tensor import TensorIndexType, tensorsymmetry, TensorType
>>> Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
>>> sym2 = tensorsymmetry([1, 1])
>>> S2 = TensorType([Lorentz]*2, sym2)
>>> V = S2('V')
"""
is_commutative = False
def __new__(cls, index_types, symmetry, **kw_args):
assert symmetry.rank == len(index_types)
obj = Basic.__new__(cls, Tuple(*index_types), symmetry, **kw_args)
return obj
@property
def index_types(self):
return self.args[0]
@property
def symmetry(self):
return self.args[1]
@property
def types(self):
return sorted(set(self.index_types), key=lambda x: x.name)
def __str__(self):
return 'TensorType(%s)' % ([str(x) for x in self.index_types])
def __call__(self, s, comm=0, matrix_behavior=0):
"""
Return a TensorHead object or a list of TensorHead objects.
``s`` name or string of names
``comm``: commutation group number
see ``_TensorManager.set_comm``
Examples
========
Define symmetric tensors ``V``, ``W`` and ``G``, respectively
commuting, anticommuting and with no commutation symmetry
>>> from sympy.tensor.tensor import TensorIndexType, tensor_indices, tensorsymmetry, TensorType, canon_bp
>>> Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
>>> a, b = tensor_indices('a,b', Lorentz)
>>> sym2 = tensorsymmetry([1]*2)
>>> S2 = TensorType([Lorentz]*2, sym2)
>>> V = S2('V')
>>> W = S2('W', 1)
>>> G = S2('G', 2)
>>> canon_bp(V(a, b)*V(-b, -a))
V(L_0, L_1)*V(-L_0, -L_1)
>>> canon_bp(W(a, b)*W(-b, -a))
0
"""
if isinstance(s, str):
names = [x.name for x in symbols(s, seq=True)]
else:
raise ValueError('expecting a string')
if len(names) == 1:
return TensorHead(names[0], self, comm, matrix_behavior=matrix_behavior)
else:
return [TensorHead(name, self, comm, matrix_behavior=matrix_behavior) for name in names]
def tensorhead(name, typ, sym, comm=0, matrix_behavior=0):
"""
Function generating tensorhead(s).
Parameters
==========
name : name or sequence of names (as in ``symbol``)
typ : index types
sym : same as ``*args`` in ``tensorsymmetry``
comm : commutation group number
see ``_TensorManager.set_comm``
Examples
========
>>> from sympy.tensor.tensor import TensorIndexType, tensor_indices, tensorhead
>>> Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
>>> a, b = tensor_indices('a,b', Lorentz)
>>> A = tensorhead('A', [Lorentz]*2, [[1]*2])
>>> A(a, -b)
A(a, -b)
"""
sym = tensorsymmetry(*sym)
S = TensorType(typ, sym)
th = S(name, comm, matrix_behavior=matrix_behavior)
return th
@doctest_depends_on(modules=('numpy',))
[docs]class TensorHead(Basic):
r"""
Tensor head of the tensor
Parameters
==========
name : name of the tensor
typ : list of TensorIndexType
comm : commutation group number
Attributes
==========
``name``
``index_types``
``rank``
``types`` : equal to ``typ.types``
``symmetry`` : equal to ``typ.symmetry``
``comm`` : commutation group
Notes
=====
A ``TensorHead`` belongs to a commutation group, defined by a
symbol on number ``comm`` (see ``_TensorManager.set_comm``);
tensors in a commutation group have the same commutation properties;
by default ``comm`` is ``0``, the group of the commuting tensors.
Examples
========
>>> from sympy.tensor.tensor import TensorIndexType, tensorhead, TensorType
>>> Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
>>> A = tensorhead('A', [Lorentz, Lorentz], [[1],[1]])
Examples with ndarray values, the components data assigned to the
``TensorHead`` object are assumed to be in a fully-contravariant
representation. In case it is necessary to assign components data which
represents the values of a non-fully covariant tensor, see the other
examples.
>>> from sympy.tensor.tensor import tensor_indices, tensorhead
>>> Lorentz.data = [1, -1, -1, -1]
>>> i0, i1 = tensor_indices('i0:2', Lorentz)
>>> A.data = [[j+2*i for j in range(4)] for i in range(4)]
in order to retrieve data, it is also necessary to specify abstract indices
enclosed by round brackets, then numerical indices inside square brackets.
>>> A(i0, i1)[0, 0]
0
>>> A(i0, i1)[2, 3] == 3+2*2
True
Notice that square brackets create a valued tensor expression instance:
>>> A(i0, i1)
A(i0, i1)
To view the data, just type:
>>> A.data
[[0 1 2 3]
[2 3 4 5]
[4 5 6 7]
[6 7 8 9]]
Turning to a tensor expression, covariant indices get the corresponding
components data corrected by the metric:
>>> A(i0, -i1).data
[[0 -1 -2 -3]
[2 -3 -4 -5]
[4 -5 -6 -7]
[6 -7 -8 -9]]
>>> A(-i0, -i1).data
[[0 -1 -2 -3]
[-2 3 4 5]
[-4 5 6 7]
[-6 7 8 9]]
while if all indices are contravariant, the ``ndarray`` remains the same
>>> A(i0, i1).data
[[0 1 2 3]
[2 3 4 5]
[4 5 6 7]
[6 7 8 9]]
When all indices are contracted and components data are added to the tensor,
accessing the data will return a scalar, no numpy object. In fact, numpy
ndarrays are dropped to scalars if they contain only one element.
>>> A(i0, -i0)
A(L_0, -L_0)
>>> A(i0, -i0).data
-18
It is also possible to assign components data to an indexed tensor, i.e. a
tensor with specified covariant and contravariant components. In this
example, the covariant components data of the Electromagnetic tensor are
injected into `A`:
>>> from sympy import symbols
>>> Ex, Ey, Ez, Bx, By, Bz = symbols('E_x E_y E_z B_x B_y B_z')
>>> c = symbols('c', positive=True)
Let's define `F`, an antisymmetric tensor, we have to assign an
antisymmetric matrix to it, because `[[2]]` stands for the Young tableau
representation of an antisymmetric set of two elements:
>>> F = tensorhead('A', [Lorentz, Lorentz], [[2]])
>>> F(-i0, -i1).data = [
... [0, Ex/c, Ey/c, Ez/c],
... [-Ex/c, 0, -Bz, By],
... [-Ey/c, Bz, 0, -Bx],
... [-Ez/c, -By, Bx, 0]]
Now it is possible to retrieve the contravariant form of the Electromagnetic
tensor:
>>> F(i0, i1).data
[[0 -E_x/c -E_y/c -E_z/c]
[E_x/c 0 -B_z B_y]
[E_y/c B_z 0 -B_x]
[E_z/c -B_y B_x 0]]
and the mixed contravariant-covariant form:
>>> F(i0, -i1).data
[[0 E_x/c E_y/c E_z/c]
[E_x/c 0 B_z -B_y]
[E_y/c -B_z 0 B_x]
[E_z/c B_y -B_x 0]]
To convert the numpy's ndarray to a sympy matrix, just cast:
>>> from sympy import Matrix
>>> Matrix(F.data)
Matrix([
[ 0, -E_x/c, -E_y/c, -E_z/c],
[E_x/c, 0, -B_z, B_y],
[E_y/c, B_z, 0, -B_x],
[E_z/c, -B_y, B_x, 0]])
Still notice, in this last example, that accessing components data from a
tensor without specifying the indices is equivalent to assume that all
indices are contravariant.
It is also possible to store symbolic components data inside a tensor, for
example, define a four-momentum-like tensor:
>>> from sympy import symbols
>>> P = tensorhead('P', [Lorentz], [[1]])
>>> E, px, py, pz = symbols('E p_x p_y p_z', positive=True)
>>> P.data = [E, px, py, pz]
The contravariant and covariant components are, respectively:
>>> P(i0).data
[E p_x p_y p_z]
>>> P(-i0).data
[E -p_x -p_y -p_z]
The contraction of a 1-index tensor by itself is usually indicated by a
power by two:
>>> P(i0)**2
E**2 - p_x**2 - p_y**2 - p_z**2
As the power by two is clearly identical to `P_\mu P^\mu`, it is possible to
simply contract the ``TensorHead`` object, without specifying the indices
>>> P**2
E**2 - p_x**2 - p_y**2 - p_z**2
"""
is_commutative = False
def __new__(cls, name, typ, comm=0, matrix_behavior=0, **kw_args):
if isinstance(name, string_types):
name_symbol = Symbol(name)
elif isinstance(name, Symbol):
name_symbol = name
else:
raise ValueError("invalid name")
comm2i = TensorManager.comm_symbols2i(comm)
obj = Basic.__new__(cls, name_symbol, typ, **kw_args)
obj._matrix_behavior = matrix_behavior
obj._name = obj.args[0].name
obj._rank = len(obj.index_types)
obj._types = typ.types
obj._symmetry = typ.symmetry
obj._comm = comm2i
return obj
@property
def name(self):
return self._name
@property
def rank(self):
return self._rank
@property
def types(self):
return self._types[:]
@property
def symmetry(self):
return self._symmetry
@property
def typ(self):
return self.args[1]
@property
def comm(self):
return self._comm
@property
def index_types(self):
return self.args[1].index_types[:]
def __lt__(self, other):
return (self.name, self.index_types) < (other.name, other.index_types)
[docs] def commutes_with(self, other):
"""
Returns ``0`` if ``self`` and ``other`` commute, ``1`` if they anticommute.
Returns ``None`` if ``self`` and ``other`` neither commute nor anticommute.
"""
r = TensorManager.get_comm(self._comm, other._comm)
return r
def _print(self):
return '%s(%s)' %(self.name, ','.join([str(x) for x in self.index_types]))
def _check_auto_matrix_indices_in_call(self, *indices):
matrix_behavior_kinds = dict()
if len(indices) != len(self.index_types):
if not self._matrix_behavior:
raise ValueError('wrong number of indices')
# Take the last one or two missing
# indices as auto-matrix indices:
ldiff = len(self.index_types) - len(indices)
if ldiff > 2:
raise ValueError('wrong number of indices')
if ldiff == 2:
mat_ind = [len(indices), len(indices) + 1]
elif ldiff == 1:
mat_ind = [len(indices)]
not_equal = True
else:
not_equal = False
mat_ind = [i for i, e in enumerate(indices) if e is True]
if mat_ind:
not_equal = True
indices = tuple([_ for _ in indices if _ is not True])
for i, el in enumerate(indices):
if not isinstance(el, TensorIndex):
not_equal = True
break
if el._tensortype != self.index_types[i]:
not_equal = True
break
if not_equal:
for el in mat_ind:
eltyp = self.index_types[el]
if eltyp in matrix_behavior_kinds:
elind = -self.index_types[el].auto_right
matrix_behavior_kinds[eltyp].append(elind)
else:
elind = self.index_types[el].auto_left
matrix_behavior_kinds[eltyp] = [elind]
indices = indices[:el] + (elind,) + indices[el:]
return indices, matrix_behavior_kinds
def __call__(self, *indices, **kw_args):
"""
Returns a tensor with indices.
There is a special behavior in case of indices denoted by ``True``,
they are considered auto-matrix indices, their slots are automatically
filled, and confer to the tensor the behavior of a matrix or vector
upon multiplication with another tensor containing auto-matrix indices
of the same ``TensorIndexType``. This means indices get summed over the
same way as in matrix multiplication. For matrix behavior, define two
auto-matrix indices, for vector behavior define just one.
Examples
========
>>> from sympy.tensor.tensor import TensorIndexType, tensor_indices, tensorhead
>>> Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
>>> a, b = tensor_indices('a,b', Lorentz)
>>> A = tensorhead('A', [Lorentz]*2, [[1]*2])
>>> t = A(a, -b)
>>> t
A(a, -b)
To use the auto-matrix index behavior, just put a ``True`` on the
desired index position.
>>> r = A(True, True)
>>> r
A(auto_left, -auto_right)
Here ``auto_left`` and ``auto_right`` are automatically generated
tensor indices, they are only two for every ``TensorIndexType`` and
can be assigned to just one or two indices of a given type.
Auto-matrix indices can be assigned many times in a tensor, if indices
are of different ``TensorIndexType``
>>> Spinor = TensorIndexType('Spinor', dummy_fmt='S')
>>> B = tensorhead('B', [Lorentz, Lorentz, Spinor, Spinor], [[1]*4])
>>> s = B(True, True, True, True)
>>> s
B(auto_left, -auto_right, auto_left, -auto_right)
Here, ``auto_left`` and ``auto_right`` are repeated twice, but they are
not the same indices, as they refer to different ``TensorIndexType``s.
Auto-matrix indices are automatically contracted upon multiplication,
>>> r*s
A(auto_left, L_0)*B(-L_0, -auto_right, auto_left, -auto_right)
The multiplication algorithm has found an ``auto_right`` index in ``A``
and an ``auto_left`` index in ``B`` referring to the same
``TensorIndexType`` (``Lorentz``), so they have been contracted.
Auto-matrix indices can be accessed from the ``TensorIndexType``:
>>> Lorentz.auto_right
auto_right
>>> Lorentz.auto_left
auto_left
There is a special case, in which the ``True`` parameter is not needed
to declare an auto-matrix index, i.e. when the matrix behavior has been
declared upon ``TensorHead`` construction, in that case the last one or
two tensor indices may be omitted, so that they automatically become
auto-matrix indices:
>>> C = tensorhead('C', [Lorentz, Lorentz], [[1]*2], matrix_behavior=True)
>>> C()
C(auto_left, -auto_right)
"""
indices, matrix_behavior_kinds = self._check_auto_matrix_indices_in_call(*indices)
tensor = Tensor._new_with_dummy_replacement(self, indices, **kw_args)
return tensor
def __pow__(self, other):
if self.data is None:
raise ValueError("No power on abstract tensors.")
numpy = import_module('numpy')
metrics = [_.data for _ in self.args[1].args[0]]
marray = self.data
for metric in metrics:
marray = numpy.tensordot(marray, numpy.tensordot(metric, marray, (1, 0)), (0, 0))
pow2 = marray[()]
return pow2 ** (Rational(1, 2) * other)
@property
def data(self):
return _tensor_data_substitution_dict[self]
@data.setter
def data(self, data):
_tensor_data_substitution_dict[self] = data
@data.deleter
def data(self):
if self in _tensor_data_substitution_dict:
del _tensor_data_substitution_dict[self]
def __iter__(self):
return self.data.flatten().__iter__()
def _components_data_full_destroy(self):
"""
EXPERIMENTAL: do not rely on this API method.
Destroy components data associated to the ``TensorHead`` object, this
checks for attached components data, and destroys components data too.
"""
# do not garbage collect Kronecker tensor (it should be done by
# ``TensorIndexType`` garbage collection)
if self.name == "KD":
return
# the data attached to a tensor must be deleted only by the TensorHead
# destructor. If the TensorHead is deleted, it means that there are no
# more instances of that tensor anywhere.
if self in _tensor_data_substitution_dict:
del _tensor_data_substitution_dict[self]
@doctest_depends_on(modules=('numpy',))
[docs]class TensExpr(Basic):
"""
Abstract base class for tensor expressions
Notes
=====
A tensor expression is an expression formed by tensors;
currently the sums of tensors are distributed.
A ``TensExpr`` can be a ``TensAdd`` or a ``TensMul``.
``TensAdd`` objects are put in canonical form using the Butler-Portugal
algorithm for canonicalization under monoterm symmetries.
``TensMul`` objects are formed by products of component tensors,
and include a coefficient, which is a SymPy expression.
In the internal representation contracted indices are represented
by ``(ipos1, ipos2, icomp1, icomp2)``, where ``icomp1`` is the position
of the component tensor with contravariant index, ``ipos1`` is the
slot which the index occupies in that component tensor.
Contracted indices are therefore nameless in the internal representation.
"""
_op_priority = 11.0
is_commutative = False
def __neg__(self):
return self*S.NegativeOne
def __abs__(self):
raise NotImplementedError
def __add__(self, other):
raise NotImplementedError
def __radd__(self, other):
raise NotImplementedError
def __sub__(self, other):
raise NotImplementedError
def __rsub__(self, other):
raise NotImplementedError
def __mul__(self, other):
raise NotImplementedError
def __rmul__(self, other):
raise NotImplementedError
def __pow__(self, other):
if self.data is None:
raise ValueError("No power without ndarray data.")
numpy = import_module('numpy')
free = self.free
marray = self.data
for metric in free:
marray = numpy.tensordot(
marray,
numpy.tensordot(
metric[0]._tensortype.data,
marray,
(1, 0)
),
(0, 0)
)
pow2 = marray[()]
return pow2 ** (Rational(1, 2) * other)
def __rpow__(self, other):
raise NotImplementedError
def __div__(self, other):
raise NotImplementedError
def __rdiv__(self, other):
raise NotImplementedError()
__truediv__ = __div__
__rtruediv__ = __rdiv__
@doctest_depends_on(modules=('numpy',))
[docs] def get_matrix(self):
"""
Returns ndarray components data as a matrix, if components data are
available and ndarray dimension does not exceed 2.
Examples
========
>>> from sympy.tensor.tensor import TensorIndexType, tensorsymmetry, TensorType
>>> from sympy import ones
>>> Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
>>> sym2 = tensorsymmetry([1]*2)
>>> S2 = TensorType([Lorentz]*2, sym2)
>>> A = S2('A')
The tensor ``A`` is symmetric in its indices, as can be deduced by the
``[1, 1]`` Young tableau when constructing `sym2`. One has to be
careful to assign symmetric component data to ``A``, as the symmetry
properties of data are currently not checked to be compatible with the
defined tensor symmetry.
>>> from sympy.tensor.tensor import tensor_indices, tensorhead
>>> Lorentz.data = [1, -1, -1, -1]
>>> i0, i1 = tensor_indices('i0:2', Lorentz)
>>> A.data = [[j+i for j in range(4)] for i in range(4)]
>>> A(i0, i1).get_matrix()
Matrix([
[0, 1, 2, 3],
[1, 2, 3, 4],
[2, 3, 4, 5],
[3, 4, 5, 6]])
It is possible to perform usual operation on matrices, such as the
matrix multiplication:
>>> A(i0, i1).get_matrix()*ones(4, 1)
Matrix([
[ 6],
[10],
[14],
[18]])
"""
if 0 < self.rank <= 2:
rows = self.data.shape[0]
columns = self.data.shape[1] if self.rank == 2 else 1
if self.rank == 2:
mat_list = [] * rows
for i in range(rows):
mat_list.append([])
for j in range(columns):
mat_list[i].append(self[i, j])
else:
mat_list = [None] * rows
for i in range(rows):
mat_list[i] = self[i]
return Matrix(mat_list)
else:
raise NotImplementedError(
"missing multidimensional reduction to matrix.")
def _eval_simplify(self, ratio, measure):
# this is a way to simplify a tensor expression.
# This part walks for all `TensorHead`s appearing in the tensor expr
# and looks for `simplify_this_type`, to specifically act on a subexpr
# containing one type of `TensorHead` instance only:
expr = self
for i in list(set(self.components)):
if hasattr(i, 'simplify_this_type'):
expr = i.simplify_this_type(expr)
# TODO: missing feature, perform metric contraction.
return expr
@doctest_depends_on(modules=('numpy',))
[docs]class TensAdd(TensExpr):
"""
Sum of tensors
Parameters
==========
free_args : list of the free indices
Attributes
==========
``args`` : tuple of addends
``rank`` : rank of the tensor
``free_args`` : list of the free indices in sorted order
Notes
=====
Sum of more than one tensor are put automatically in canonical form.
Examples
========
>>> from sympy.tensor.tensor import TensorIndexType, tensorhead, tensor_indices
>>> Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
>>> a, b = tensor_indices('a,b', Lorentz)
>>> p, q = tensorhead('p,q', [Lorentz], [[1]])
>>> t = p(a) + q(a); t
p(a) + q(a)
>>> t(b)
p(b) + q(b)
Examples with components data added to the tensor expression:
>>> from sympy import eye
>>> Lorentz.data = [1, -1, -1, -1]
>>> a, b = tensor_indices('a, b', Lorentz)
>>> p.data = [2, 3, -2, 7]
>>> q.data = [2, 3, -2, 7]
>>> t = p(a) + q(a); t
p(a) + q(a)
>>> t(b)
p(b) + q(b)
The following are: 2**2 - 3**2 - 2**2 - 7**2 ==> -58
>>> (p(a)*p(-a)).data
-58
>>> p(a)**2
-58
"""
def __new__(cls, *args, **kw_args):
args = [sympify(x) for x in args if x]
args = TensAdd._tensAdd_flatten(args)
if not args:
return S.Zero
if len(args) == 1 and not isinstance(args[0], TensExpr):
return args[0]
# replace auto-matrix indices so that they are the same in all addends
args = TensAdd._tensAdd_check_automatrix(args)
# now check that all addends have the same indices:
TensAdd._tensAdd_check(args)
# if TensAdd has only 1 TensMul element in its `args`:
if len(args) == 1 and isinstance(args[0], TensMul):
obj = Basic.__new__(cls, *args, **kw_args)
return obj
# TODO: do not or do canonicalize by default?
# Technically, one may wish to have additions of non-canonicalized
# tensors. This feature should be removed in the future.
# Unfortunately this would require to rewrite a lot of tests.
# canonicalize all TensMul
args = [canon_bp(x) for x in args if x]
args = [x for x in args if x]
# if there are no more args (i.e. have cancelled out),
# just return zero:
if not args:
return S.Zero
if len(args) == 1:
return args[0]
# collect canonicalized terms
def sort_key(t):
x = get_tids(t)
return (x.components, x.free, x.dum)
args.sort(key=sort_key)
args = TensAdd._tensAdd_collect_terms(args)
if not args:
return S.Zero
# it there is only a component tensor return it
if len(args) == 1:
return args[0]
obj = Basic.__new__(cls, *args, **kw_args)
return obj
@staticmethod
def _tensAdd_flatten(args):
# flatten TensAdd, coerce terms which are not tensors to tensors
if not all(isinstance(x, TensExpr) for x in args):
args1 = []
for x in args:
if isinstance(x, TensExpr):
if isinstance(x, TensAdd):
args1.extend(list(x.args))
else:
args1.append(x)
args1 = [x for x in args1 if isinstance(x, TensExpr) and x.coeff]
args2 = [x for x in args if not isinstance(x, TensExpr)]
t1 = TensMul.from_data(Add(*args2), [], [], [])
args = [t1] + args1
a = []
for x in args:
if isinstance(x, TensAdd):
a.extend(list(x.args))
else:
a.append(x)
args = [x for x in a if x.coeff]
return args
@staticmethod
def _tensAdd_check_automatrix(args):
# check that all automatrix indices are the same.
# if there are no addends, just return.
if not args:
return args
# @type auto_left_types: set
auto_left_types = set([])
auto_right_types = set([])
args_auto_left_types = []
args_auto_right_types = []
for i, arg in enumerate(args):
arg_auto_left_types = set([])
arg_auto_right_types = set([])
for index in get_indices(arg):
# @type index: TensorIndex
if index in (index._tensortype.auto_left, -index._tensortype.auto_left):
auto_left_types.add(index._tensortype)
arg_auto_left_types.add(index._tensortype)
if index in (index._tensortype.auto_right, -index._tensortype.auto_right):
auto_right_types.add(index._tensortype)
arg_auto_right_types.add(index._tensortype)
args_auto_left_types.append(arg_auto_left_types)
args_auto_right_types.append(arg_auto_right_types)
for arg, aas_left, aas_right in zip(args, args_auto_left_types, args_auto_right_types):
missing_left = auto_left_types - aas_left
missing_right = auto_right_types - aas_right
missing_intersection = missing_left & missing_right
for j in missing_intersection:
args[i] *= j.delta(j.auto_left, -j.auto_right)
if missing_left != missing_right:
raise ValueError("cannot determine how to add auto-matrix indices on some args")
return args
@staticmethod
def _tensAdd_check(args):
# check that all addends have the same free indices
indices0 = set([x[0] for x in get_tids(args[0]).free])
list_indices = [set([y[0] for y in get_tids(x).free]) for x in args[1:]]
if not all(x == indices0 for x in list_indices):
raise ValueError('all tensors must have the same indices')
@staticmethod
def _tensAdd_collect_terms(args):
# collect TensMul terms differing at most by their coefficient
a = []
prev = args[0]
prev_coeff = get_coeff(prev)
changed = False
for x in args[1:]:
# if x and prev have the same tensor, update the coeff of prev
x_tids = get_tids(x)
prev_tids = get_tids(prev)
if x_tids.components == prev_tids.components \
and x_tids.free == prev_tids.free and x_tids.dum == prev_tids.dum:
prev_coeff = prev_coeff + get_coeff(x)
changed = True
op = 0
else:
# x and prev are different; if not changed, prev has not
# been updated; store it
if not changed:
a.append(prev)
else:
# get a tensor from prev with coeff=prev_coeff and store it
if prev_coeff:
t = TensMul.from_data(prev_coeff, prev_tids.components,
prev_tids.free, prev_tids.dum)
a.append(t)
# move x to prev
op = 1
pprev, prev = prev, x
pprev_coeff, prev_coeff = prev_coeff, get_coeff(x)
changed = False
# if the case op=0 prev was not stored; store it now
# in the case op=1 x was not stored; store it now (as prev)
if op == 0 and prev_coeff:
prev = TensMul.from_data(prev_coeff, prev_tids.components, prev_tids.free, prev_tids.dum)
a.append(prev)
elif op == 1:
a.append(prev)
return a
@property
def rank(self):
return self.args[0].rank
@property
def free_args(self):
return self.args[0].free_args
def __call__(self, *indices):
"""Returns tensor with ordered free indices replaced by ``indices``
Parameters
==========
indices
Examples
========
>>> from sympy import Symbol
>>> from sympy.tensor.tensor import TensorIndexType, tensor_indices, tensorhead
>>> D = Symbol('D')
>>> Lorentz = TensorIndexType('Lorentz', dim=D, dummy_fmt='L')
>>> i0,i1,i2,i3,i4 = tensor_indices('i0:5', Lorentz)
>>> p, q = tensorhead('p,q', [Lorentz], [[1]])
>>> g = Lorentz.metric
>>> t = p(i0)*p(i1) + g(i0,i1)*q(i2)*q(-i2)
>>> t(i0,i2)
metric(i0, i2)*q(L_0)*q(-L_0) + p(i0)*p(i2)
>>> t(i0,i1) - t(i1,i0)
0
"""
free_args = self.free_args
indices = list(indices)
if [x._tensortype for x in indices] != [x._tensortype for x in free_args]:
raise ValueError('incompatible types')
if indices == free_args:
return self
index_tuples = list(zip(free_args, indices))
a = [x.func(*x.fun_eval(*index_tuples).args) for x in self.args]
res = TensAdd(*a)
return res
[docs] def canon_bp(self):
"""
canonicalize using the Butler-Portugal algorithm for canonicalization
under monoterm symmetries.
"""
args = [x.canon_bp() for x in self.args]
res = TensAdd(*args)
return res
def equals(self, other):
other = sympify(other)
if isinstance(other, TensMul) and other._coeff == 0:
return all(x._coeff == 0 for x in self.args)
if isinstance(other, TensExpr):
if self.rank != other.rank:
return False
if isinstance(other, TensAdd):
if set(self.args) != set(other.args):
return False
else:
return True
t = self - other
if not isinstance(t, TensExpr):
return t == 0
else:
if isinstance(t, TensMul):
return t._coeff == 0
else:
return all(x._coeff == 0 for x in t.args)
def __add__(self, other):
return TensAdd(self, other)
def __radd__(self, other):
return TensAdd(other, self)
def __sub__(self, other):
return TensAdd(self, -other)
def __rsub__(self, other):
return TensAdd(other, -self)
def __mul__(self, other):
return TensAdd(*(x*other for x in self.args))
def __rmul__(self, other):
return self*other
def __div__(self, other):
other = sympify(other)
if isinstance(other, TensExpr):
raise ValueError('cannot divide by a tensor')
return TensAdd(*(x/other for x in self.args))
def __rdiv__(self, other):
raise ValueError('cannot divide by a tensor')
def __getitem__(self, item):
return self.data[item]
__truediv__ = __div__
__truerdiv__ = __rdiv__
def contract_delta(self, delta):
args = [x.contract_delta(delta) for x in self.args]
t = TensAdd(*args)
return canon_bp(t)
[docs] def contract_metric(self, g):
"""
Raise or lower indices with the metric ``g``
Parameters
==========
g : metric
contract_all : if True, eliminate all ``g`` which are contracted
Notes
=====
see the ``TensorIndexType`` docstring for the contraction conventions
"""
args = [contract_metric(x, g) for x in self.args]
t = TensAdd(*args)
return canon_bp(t)
[docs] def fun_eval(self, *index_tuples):
"""
Return a tensor with free indices substituted according to ``index_tuples``
Parameters
==========
index_types : list of tuples ``(old_index, new_index)``
Examples
========
>>> from sympy.tensor.tensor import TensorIndexType, tensor_indices, tensorhead
>>> Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
>>> i, j, k, l = tensor_indices('i,j,k,l', Lorentz)
>>> A, B = tensorhead('A,B', [Lorentz]*2, [[1]*2])
>>> t = A(i, k)*B(-k, -j) + A(i, -j)
>>> t.fun_eval((i, k),(-j, l))
A(k, L_0)*B(l, -L_0) + A(k, l)
"""
args = self.args
args1 = []
for x in args:
y = x.fun_eval(*index_tuples)
args1.append(y)
return TensAdd(*args1)
[docs] def substitute_indices(self, *index_tuples):
"""
Return a tensor with free indices substituted according to ``index_tuples``
Parameters
==========
index_types : list of tuples ``(old_index, new_index)``
Examples
========
>>> from sympy.tensor.tensor import TensorIndexType, tensor_indices, tensorhead
>>> Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
>>> i, j, k, l = tensor_indices('i,j,k,l', Lorentz)
>>> A, B = tensorhead('A,B', [Lorentz]*2, [[1]*2])
>>> t = A(i, k)*B(-k, -j); t
A(i, L_0)*B(-L_0, -j)
>>> t.substitute_indices((i,j), (j, k))
A(j, L_0)*B(-L_0, -k)
"""
args = self.args
args1 = []
for x in args:
y = x.substitute_indices(*index_tuples)
args1.append(y)
return TensAdd(*args1)
def _print(self):
a = []
args = self.args
for x in args:
a.append(str(x))
a.sort()
s = ' + '.join(a)
s = s.replace('+ -', '- ')
return s
@staticmethod
[docs] def from_TIDS_list(coeff, tids_list):
"""
Given a list of coefficients and a list of ``TIDS`` objects, construct
a ``TensAdd`` instance, equivalent to the one that would result from
creating single instances of ``TensMul`` and then adding them.
Examples
========
>>> from sympy.tensor.tensor import TensorIndexType, tensor_indices, tensorhead, TensAdd
>>> Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
>>> i, j = tensor_indices('i,j', Lorentz)
>>> A, B = tensorhead('A,B', [Lorentz]*2, [[1]*2])
>>> eA = 3*A(i, j)
>>> eB = 2*B(j, i)
>>> t1 = eA._tids
>>> t2 = eB._tids
>>> c1 = eA.coeff
>>> c2 = eB.coeff
>>> TensAdd.from_TIDS_list([c1, c2], [t1, t2])
2*B(i, j) + 3*A(i, j)
If the coefficient parameter is a scalar, then it will be applied
as a coefficient on all ``TIDS`` objects.
>>> TensAdd.from_TIDS_list(4, [t1, t2])
4*A(i, j) + 4*B(i, j)
"""
if not isinstance(coeff, (list, tuple, Tuple)):
coeff = [coeff] * len(tids_list)
tensmul_list = [TensMul.from_TIDS(c, t) for c, t in zip(coeff, tids_list)]
return TensAdd(*tensmul_list)
@property
def data(self):
return _tensor_data_substitution_dict[self]
@data.setter
def data(self, data):
# TODO: check data compatibility with properties of tensor.
_tensor_data_substitution_dict[self] = data
@data.deleter
def data(self):
if self in _tensor_data_substitution_dict:
del _tensor_data_substitution_dict[self]
def __iter__(self):
if not self.data:
raise ValueError("No iteration on abstract tensors")
return self.data.flatten().__iter__()
@doctest_depends_on(modules=('numpy',))
class Tensor(TensExpr):
"""
Base tensor class, i.e. this represents a tensor, the single unit to be
put into an expression.
This object is usually created from a ``TensorHead``, by attaching indices
to it. Indices preceded by a minus sign are considered contravariant,
otherwise covariant.
Examples
========
>>> from sympy.tensor.tensor import TensorIndexType, tensor_indices, tensorhead
>>> Lorentz = TensorIndexType("Lorentz", dummy_fmt="L")
>>> mu, nu = tensor_indices('mu nu', Lorentz)
>>> A = tensorhead("A", [Lorentz, Lorentz], [[1], [1]])
>>> A(mu, -nu)
A(mu, -nu)
>>> A(mu, -mu)
A(L_0, -L_0)
"""
is_commutative = False
def __new__(cls, tensor_head, indices, **kw_args):
tids = TIDS.from_components_and_indices((tensor_head,), indices)
obj = Basic.__new__(cls, tensor_head, Tuple(*indices), **kw_args)
obj._tids = tids
obj._indices = indices
obj._is_canon_bp = kw_args.get('is_canon_bp', False)
return obj
@staticmethod
def _new_with_dummy_replacement(tensor_head, indices, **kw_args):
tids = TIDS.from_components_and_indices((tensor_head,), indices)
indices = tids.get_indices()
return Tensor(tensor_head, indices, **kw_args)
@property
def is_canon_bp(self):
return self._is_canon_bp
@property
def indices(self):
return self._indices
@property
def free(self):
return self._tids.free
@property
def dum(self):
return self._tids.dum
@property
def rank(self):
return len(self.free)
@property
def free_args(self):
return sorted([x[0] for x in self.free])
def perm2tensor(self, g, canon_bp=False):
"""
Returns the tensor corresponding to the permutation ``g``
For further details, see the method in ``TIDS`` with the same name.
"""
return perm2tensor(self, g, canon_bp)
def canon_bp(self):
if self._is_canon_bp:
return self
g, dummies, msym, v = self._tids.canon_args()
can = canonicalize(g, dummies, msym, *v)
if can == 0:
return S.Zero
tensor = self.perm2tensor(can, True)
return tensor
@property
def types(self):
return get_tids(self).components[0]._types
@property
def coeff(self):
return S.One
@property
def component(self):
return self.args[0]
@property
def components(self):
return [self.args[0]]
def split(self):
return [self]
def expand(self):
return self
def sorted_components(self):
return self
def get_indices(self):
"""
Get a list of indices, corresponding to those of the tensor.
"""
return self._tids.get_indices()
def as_base_exp(self):
return self, S.One
def substitute_indices(self, *index_tuples):
return substitute_indices(self, *index_tuples)
def __call__(self, *indices):
"""Returns tensor with ordered free indices replaced by ``indices``
Examples
========
>>> from sympy.tensor.tensor import TensorIndexType, tensor_indices, tensorhead
>>> Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
>>> i0,i1,i2,i3,i4 = tensor_indices('i0:5', Lorentz)
>>> A = tensorhead('A', [Lorentz]*5, [[1]*5])
>>> t = A(i2, i1, -i2, -i3, i4)
>>> t
A(L_0, i1, -L_0, -i3, i4)
>>> t(i1, i2, i3)
A(L_0, i1, -L_0, i2, i3)
"""
free_args = self.free_args
indices = list(indices)
if [x._tensortype for x in indices] != [x._tensortype for x in free_args]:
raise ValueError('incompatible types')
if indices == free_args:
return self
t = self.fun_eval(*list(zip(free_args, indices)))
# object is rebuilt in order to make sure that all contracted indices
# get recognized as dummies, but only if there are contracted indices.
if len(set(i if i.is_up else -i for i in indices)) != len(indices):
return t.func(*t.args)
return t
def fun_eval(self, *index_tuples):
free = self.free
free1 = []
for j, ipos, cpos in free:
# search j in index_tuples
for i, v in index_tuples:
if i == j:
free1.append((v, ipos, cpos))
break
else:
free1.append((j, ipos, cpos))
return TensMul.from_data(self.coeff, self.components, free1, self.dum)
# TODO: put this into TensExpr?
def __iter__(self):
return self.data.flatten().__iter__()
# TODO: put this into TensExpr?
def __getitem__(self, item):
return self.data[item]
@property
def data(self):
return _tensor_data_substitution_dict[self]
@data.setter
def data(self, data):
# TODO: check data compatibility with properties of tensor.
_tensor_data_substitution_dict[self] = data
@data.deleter
def data(self):
if self in _tensor_data_substitution_dict:
del _tensor_data_substitution_dict[self]
if self.metric in _tensor_data_substitution_dict:
del _tensor_data_substitution_dict[self.metric]
def __mul__(self, other):
if isinstance(other, TensAdd):
return TensAdd(*[self*arg for arg in other.args])
tmul = TensMul(self, other)
return tmul
def __rmul__(self, other):
return TensMul(other, self)
def __div__(self, other):
if isinstance(other, TensExpr):
raise ValueError('cannot divide by a tensor')
return TensMul(self, S.One/other, is_canon_bp=self.is_canon_bp)
def __rdiv__(self, other):
raise ValueError('cannot divide by a tensor')
def __add__(self, other):
return TensAdd(self, other)
def __radd__(self, other):
return TensAdd(other, self)
def __sub__(self, other):
return TensAdd(self, -other)
def __rsub__(self, other):
return TensAdd(other, self)
__truediv__ = __div__
__rtruediv__ = __rdiv__
def __neg__(self):
return TensMul(S.NegativeOne, self)
def _print(self):
indices = [str(ind) for ind in self.indices]
component = self.component
if component.rank > 0:
return ('%s(%s)' % (component.name, ', '.join(indices)))
else:
return ('%s' % component.name)
def equals(self, other):
if other == 0:
return self.coeff == 0
other = sympify(other)
if not isinstance(other, TensExpr):
assert not self.components
return S.One == other
def _get_compar_comp(self):
t = self.canon_bp()
r = (t.coeff, tuple(t.components), \
tuple(sorted(t.free)), tuple(sorted(t.dum)))
return r
return _get_compar_comp(self) == _get_compar_comp(other)
def contract_metric(self, metric):
tids, sign = get_tids(self).contract_metric(metric)
return TensMul.from_TIDS(sign, tids)
def contract_delta(self, metric):
return self.contract_metric(metric)
@doctest_depends_on(modules=('numpy',))
[docs]class TensMul(TensExpr):
"""
Product of tensors
Parameters
==========
coeff : SymPy coefficient of the tensor
args
Attributes
==========
``components`` : list of ``TensorHead`` of the component tensors
``types`` : list of nonrepeated ``TensorIndexType``
``free`` : list of ``(ind, ipos, icomp)``, see Notes
``dum`` : list of ``(ipos1, ipos2, icomp1, icomp2)``, see Notes
``ext_rank`` : rank of the tensor counting the dummy indices
``rank`` : rank of the tensor
``coeff`` : SymPy coefficient of the tensor
``free_args`` : list of the free indices in sorted order
``is_canon_bp`` : ``True`` if the tensor in in canonical form
Notes
=====
``args[0]`` list of ``TensorHead`` of the component tensors.
``args[1]`` list of ``(ind, ipos, icomp)``
where ``ind`` is a free index, ``ipos`` is the slot position
of ``ind`` in the ``icomp``-th component tensor.
``args[2]`` list of tuples representing dummy indices.
``(ipos1, ipos2, icomp1, icomp2)`` indicates that the contravariant
dummy index is the ``ipos1``-th slot position in the ``icomp1``-th
component tensor; the corresponding covariant index is
in the ``ipos2`` slot position in the ``icomp2``-th component tensor.
"""
def __new__(cls, *args, **kw_args):
# make sure everything is sympified:
args = [sympify(arg) for arg in args]
# flatten:
args = TensMul._flatten(args)
is_canon_bp = kw_args.get('is_canon_bp', False)
if not any([isinstance(arg, TensExpr) for arg in args]):
tids = TIDS([], [], [])
else:
tids_list = [arg._tids for arg in args if isinstance(arg, (Tensor, TensMul))]
if len(tids_list) == 1:
for arg in args:
if not isinstance(arg, Tensor):
continue
is_canon_bp = kw_args.get('is_canon_bp', arg._is_canon_bp)
tids = reduce(lambda a, b: a*b, tids_list)
if any([isinstance(arg, TensAdd) for arg in args]):
add_args = TensAdd._tensAdd_flatten(args)
return TensAdd(*add_args)
coeff = reduce(lambda a, b: a*b, [S.One] + [arg for arg in args if not isinstance(arg, TensExpr)])
args = tids.get_tensors()
if coeff != 1:
args = [coeff] + args
if len(args) == 1:
return args[0]
obj = Basic.__new__(cls, *args)
obj._types = []
for t in tids.components:
obj._types.extend(t._types)
obj._tids = tids
obj._ext_rank = len(obj._tids.free) + 2*len(obj._tids.dum)
obj._coeff = coeff
obj._is_canon_bp = is_canon_bp
return obj
@staticmethod
def _flatten(args):
a = []
for arg in args:
if isinstance(arg, TensMul):
a.extend(arg.args)
else:
a.append(arg)
return a
@staticmethod
def from_data(coeff, components, free, dum, **kw_args):
tids = TIDS(components, free, dum)
return TensMul.from_TIDS(coeff, tids, **kw_args)
@staticmethod
def from_TIDS(coeff, tids, **kw_args):
return TensMul(coeff, *tids.get_tensors(), **kw_args)
@property
def free_args(self):
return sorted([x[0] for x in self.free])
@property
def components(self):
return self._tids.components[:]
@property
def free(self):
return self._tids.free[:]
@property
def coeff(self):
return self._coeff
@property
def dum(self):
return self._tids.dum[:]
@property
def rank(self):
return len(self.free)
@property
def types(self):
return self._types[:]
def equals(self, other):
if other == 0:
return self.coeff == 0
other = sympify(other)
if not isinstance(other, TensExpr):
assert not self.components
return self._coeff == other
def _get_compar_comp(self):
t = self.canon_bp()
r = (get_coeff(t), tuple(t.components), \
tuple(sorted(t.free)), tuple(sorted(t.dum)))
return r
return _get_compar_comp(self) == _get_compar_comp(other)
[docs] def get_indices(self):
"""
Returns the list of indices of the tensor
The indices are listed in the order in which they appear in the
component tensors.
The dummy indices are given a name which does not collide with
the names of the free indices.
Examples
========
>>> from sympy.tensor.tensor import TensorIndexType, tensor_indices, tensorhead
>>> Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
>>> m0, m1, m2 = tensor_indices('m0,m1,m2', Lorentz)
>>> g = Lorentz.metric
>>> p, q = tensorhead('p,q', [Lorentz], [[1]])
>>> t = p(m1)*g(m0,m2)
>>> t.get_indices()
[m1, m0, m2]
"""
return self._tids.get_indices()
[docs] def split(self):
"""
Returns a list of tensors, whose product is ``self``
Dummy indices contracted among different tensor components
become free indices with the same name as the one used to
represent the dummy indices.
Examples
========
>>> from sympy.tensor.tensor import TensorIndexType, tensor_indices, tensorhead
>>> Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
>>> a, b, c, d = tensor_indices('a,b,c,d', Lorentz)
>>> A, B = tensorhead('A,B', [Lorentz]*2, [[1]*2])
>>> t = A(a,b)*B(-b,c)
>>> t
A(a, L_0)*B(-L_0, c)
>>> t.split()
[A(a, L_0), B(-L_0, c)]
"""
if self.args == ():
return [self]
splitp = []
res = 1
for arg in self.args:
if isinstance(arg, Tensor):
splitp.append(res*arg)
res = 1
else:
res *= arg
return splitp
def __add__(self, other):
return TensAdd(self, other)
def __radd__(self, other):
return TensAdd(other, self)
def __sub__(self, other):
return TensAdd(self, -other)
def __rsub__(self, other):
return TensAdd(other, -self)
def __mul__(self, other):
"""
Multiply two tensors using Einstein summation convention.
If the two tensors have an index in common, one contravariant
and the other covariant, in their product the indices are summed
Examples
========
>>> from sympy.tensor.tensor import TensorIndexType, tensor_indices, tensorhead
>>> Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
>>> m0, m1, m2 = tensor_indices('m0,m1,m2', Lorentz)
>>> g = Lorentz.metric
>>> p, q = tensorhead('p,q', [Lorentz], [[1]])
>>> t1 = p(m0)
>>> t2 = q(-m0)
>>> t1*t2
p(L_0)*q(-L_0)
"""
other = sympify(other)
if not isinstance(other, TensExpr):
coeff = self.coeff*other
tmul = TensMul.from_TIDS(coeff, self._tids, is_canon_bp=self._is_canon_bp)
return tmul
if isinstance(other, TensAdd):
return TensAdd(*[self*x for x in other.args])
new_tids = self._tids*other._tids
coeff = self.coeff*other.coeff
tmul = TensMul.from_TIDS(coeff, new_tids)
return tmul
def __rmul__(self, other):
other = sympify(other)
coeff = other*self._coeff
tmul = TensMul.from_TIDS(coeff, self._tids)
return tmul
def __div__(self, other):
other = sympify(other)
if isinstance(other, TensExpr):
raise ValueError('cannot divide by a tensor')
coeff = self._coeff/other
tmul = TensMul.from_TIDS(coeff, self._tids, is_canon_bp=self._is_canon_bp)
return tmul
def __rdiv__(self, other):
raise ValueError('cannot divide by a tensor')
def __getitem__(self, item):
return self.data[item]
__truediv__ = __div__
__truerdiv__ = __rdiv__
[docs] def sorted_components(self):
"""
Returns a tensor with sorted components
calling the corresponding method in a ``TIDS`` object.
"""
new_tids, sign = self._tids.sorted_components()
coeff = -self.coeff if sign == -1 else self.coeff
t = TensMul.from_TIDS(coeff, new_tids)
return t
[docs] def perm2tensor(self, g, canon_bp=False):
"""
Returns the tensor corresponding to the permutation ``g``
For further details, see the method in ``TIDS`` with the same name.
"""
return perm2tensor(self, g, canon_bp)
[docs] def canon_bp(self):
"""
Canonicalize using the Butler-Portugal algorithm for canonicalization
under monoterm symmetries.
Examples
========
>>> from sympy.tensor.tensor import TensorIndexType, tensor_indices, tensorhead
>>> Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
>>> m0, m1, m2 = tensor_indices('m0,m1,m2', Lorentz)
>>> A = tensorhead('A', [Lorentz]*2, [[2]])
>>> t = A(m0,-m1)*A(m1,-m0)
>>> t.canon_bp()
-A(L_0, L_1)*A(-L_0, -L_1)
>>> t = A(m0,-m1)*A(m1,-m2)*A(m2,-m0)
>>> t.canon_bp()
0
"""
if self._is_canon_bp:
return self
if not self.components:
return self
t = self.sorted_components()
g, dummies, msym, v = t._tids.canon_args()
can = canonicalize(g, dummies, msym, *v)
if can == 0:
return S.Zero
tmul = t.perm2tensor(can, True)
return tmul
def contract_delta(self, delta):
t = self.contract_metric(delta)
return t
[docs] def contract_metric(self, g):
"""
Raise or lower indices with the metric ``g``
Parameters
==========
g : metric
Notes
=====
see the ``TensorIndexType`` docstring for the contraction conventions
Examples
========
>>> from sympy.tensor.tensor import TensorIndexType, tensor_indices, tensorhead
>>> Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
>>> m0, m1, m2 = tensor_indices('m0,m1,m2', Lorentz)
>>> g = Lorentz.metric
>>> p, q = tensorhead('p,q', [Lorentz], [[1]])
>>> t = p(m0)*q(m1)*g(-m0, -m1)
>>> t.canon_bp()
metric(L_0, L_1)*p(-L_0)*q(-L_1)
>>> t.contract_metric(g).canon_bp()
p(L_0)*q(-L_0)
"""
tids, sign = get_tids(self).contract_metric(g)
res = TensMul.from_TIDS(sign*self.coeff, tids)
return res
def substitute_indices(self, *index_tuples):
return substitute_indices(self, *index_tuples)
[docs] def fun_eval(self, *index_tuples):
"""
Return a tensor with free indices substituted according to ``index_tuples``
``index_types`` list of tuples ``(old_index, new_index)``
Examples
========
>>> from sympy.tensor.tensor import TensorIndexType, tensor_indices, tensorhead
>>> Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
>>> i, j, k, l = tensor_indices('i,j,k,l', Lorentz)
>>> A, B = tensorhead('A,B', [Lorentz]*2, [[1]*2])
>>> t = A(i, k)*B(-k, -j); t
A(i, L_0)*B(-L_0, -j)
>>> t.fun_eval((i, k),(-j, l))
A(k, L_0)*B(-L_0, l)
"""
free = self.free
free1 = []
for j, ipos, cpos in free:
# search j in index_tuples
for i, v in index_tuples:
if i == j:
free1.append((v, ipos, cpos))
break
else:
free1.append((j, ipos, cpos))
return TensMul.from_data(self.coeff, self.components, free1, self.dum)
def __call__(self, *indices):
"""Returns tensor product with ordered free indices replaced by ``indices``
Examples
========
>>> from sympy import Symbol
>>> from sympy.tensor.tensor import TensorIndexType, tensor_indices, tensorhead
>>> D = Symbol('D')
>>> Lorentz = TensorIndexType('Lorentz', dim=D, dummy_fmt='L')
>>> i0,i1,i2,i3,i4 = tensor_indices('i0:5', Lorentz)
>>> g = Lorentz.metric
>>> p, q = tensorhead('p,q', [Lorentz], [[1]])
>>> t = p(i0)*q(i1)*q(-i1)
>>> t(i1)
p(i1)*q(L_0)*q(-L_0)
"""
free_args = self.free_args
indices = list(indices)
if [x._tensortype for x in indices] != [x._tensortype for x in free_args]:
raise ValueError('incompatible types')
if indices == free_args:
return self
t = self.fun_eval(*list(zip(free_args, indices)))
# object is rebuilt in order to make sure that all contracted indices
# get recognized as dummies, but only if there are contracted indices.
if len(set(i if i.is_up else -i for i in indices)) != len(indices):
return t.func(*t.args)
return t
def _print(self):
args = self.args
get_str = lambda arg: str(arg) if arg.is_Atom or isinstance(arg, TensExpr) else ("(%s)" % str(arg))
if not args:
# no arguments is equivalent to "1", i.e. TensMul().
# If tensors are constructed correctly, this should never occur.
return "1"
if self.coeff == S.NegativeOne:
# expressions like "-A(a)"
return "-"+"*".join([get_str(arg) for arg in args[1:]])
# prints expressions like "A(a)", "3*A(a)", "(1+x)*A(a)"
return "*".join([get_str(arg) for arg in self.args])
@property
def data(self):
dat = _tensor_data_substitution_dict[self]
if dat is None:
return None
return self.coeff * dat
@data.setter
def data(self, data):
raise ValueError("Not possible to set component data to a tensor expression")
@data.deleter
def data(self):
raise ValueError("Not possible to delete component data to a tensor expression")
def __iter__(self):
if self.data is None:
raise ValueError("No iteration on abstract tensors")
return (self.data.flatten()).__iter__()
[docs]def canon_bp(p):
"""
Butler-Portugal canonicalization
"""
if isinstance(p, TensExpr):
return p.canon_bp()
return p
[docs]def tensor_mul(*a):
"""
product of tensors
"""
if not a:
return TensMul.from_data(S.One, [], [], [])
t = a[0]
for tx in a[1:]:
t = t*tx
return t
[docs]def riemann_cyclic_replace(t_r):
"""
replace Riemann tensor with an equivalent expression
``R(m,n,p,q) -> 2/3*R(m,n,p,q) - 1/3*R(m,q,n,p) + 1/3*R(m,p,n,q)``
"""
free = sorted(t_r.free, key=lambda x: x[1])
m, n, p, q = [x[0] for x in free]
t0 = S(2)/3*t_r
t1 = - S(1)/3*t_r.substitute_indices((m,m),(n,q),(p,n),(q,p))
t2 = S(1)/3*t_r.substitute_indices((m,m),(n,p),(p,n),(q,q))
t3 = t0 + t1 + t2
return t3
[docs]def riemann_cyclic(t2):
"""
replace each Riemann tensor with an equivalent expression
satisfying the cyclic identity.
This trick is discussed in the reference guide to Cadabra.
Examples
========
>>> from sympy.tensor.tensor import TensorIndexType, tensor_indices, tensorhead, riemann_cyclic
>>> Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
>>> i, j, k, l = tensor_indices('i,j,k,l', Lorentz)
>>> R = tensorhead('R', [Lorentz]*4, [[2, 2]])
>>> t = R(i,j,k,l)*(R(-i,-j,-k,-l) - 2*R(-i,-k,-j,-l))
>>> riemann_cyclic(t)
0
"""
if isinstance(t2, (TensMul, Tensor)):
args = [t2]
else:
args = t2.args
a1 = [x.split() for x in args]
a2 = [[riemann_cyclic_replace(tx) for tx in y] for y in a1]
a3 = [tensor_mul(*v) for v in a2]
t3 = TensAdd(*a3)
if not t3:
return t3
else:
return canon_bp(t3)
def get_lines(ex, index_type):
"""
returns ``(lines, traces, rest)`` for an index type,
where ``lines`` is the list of list of positions of a matrix line,
``traces`` is the list of list of traced matrix lines,
``rest`` is the rest of the elements ot the tensor.
"""
def _join_lines(a):
i = 0
while i < len(a):
x = a[i]
xend = x[-1]
xstart = x[0]
hit = True
while hit:
hit = False
for j in range(i + 1, len(a)):
if j >= len(a):
break
if a[j][0] == xend:
hit = True
x.extend(a[j][1:])
xend = x[-1]
a.pop(j)
continue
if a[j][0] == xstart:
hit = True
a[i] = reversed(a[j][1:]) + x
x = a[i]
xstart = a[i][0]
a.pop(j)
continue
if a[j][-1] == xend:
hit = True
x.extend(reversed(a[j][:-1]))
xend = x[-1]
a.pop(j)
continue
if a[j][-1] == xstart:
hit = True
a[i] = a[j][:-1] + x
x = a[i]
xstart = x[0]
a.pop(j)
continue
i += 1
return a
tids = ex._tids
components = tids.components
dt = {}
for c in components:
if c in dt:
continue
index_types = c.index_types
a = []
for i in range(len(index_types)):
if index_types[i] is index_type:
a.append(i)
if len(a) > 2:
raise ValueError('at most two indices of type %s allowed' % index_type)
if len(a) == 2:
dt[c] = a
dum = tids.dum
lines = []
traces = []
traces1 = []
for p0, p1, c0, c1 in dum:
if components[c0] not in dt:
continue
if c0 == c1:
traces.append([c0])
continue
ta0 = dt[components[c0]]
ta1 = dt[components[c1]]
if p0 not in ta0:
continue
if ta0.index(p0) == ta1.index(p1):
# case gamma(i,s0,-s1)in c0, gamma(j,-s0,s2) in c1;
# to deal with this case one could add to the position
# a flag for transposition;
# one could write [(c0, False), (c1, True)]
raise NotImplementedError
# if p0 == ta0[1] then G in pos c0 is mult on the right by G in c1
# if p0 == ta0[0] then G in pos c1 is mult on the right by G in c0
ta0 = dt[components[c0]]
b0, b1 = (c0, c1) if p0 == ta0[1] else (c1, c0)
lines1 = lines[:]
for line in lines:
if line[-1] == b0:
if line[0] == b1:
n = line.index(min(line))
traces1.append(line)
traces.append(line[n:] + line[:n])
else:
line.append(b1)
break
elif line[0] == b1:
line.insert(0, b0)
break
else:
lines1.append([b0, b1])
lines = [x for x in lines1 if x not in traces1]
lines = _join_lines(lines)
rest = []
for line in lines:
for y in line:
rest.append(y)
for line in traces:
for y in line:
rest.append(y)
rest = [x for x in range(len(components)) if x not in rest]
return lines, traces, rest
def get_indices(t):
if not isinstance(t, TensExpr):
return ()
return t.get_indices()
def get_tids(t):
if isinstance(t, TensExpr):
return t._tids
return TIDS([], [], [])
def get_coeff(t):
if isinstance(t, Tensor):
return S.One
if isinstance(t, TensMul):
return t.coeff
if isinstance(t, TensExpr):
raise ValueError("no coefficient associated to this tensor expression")
return t
def contract_metric(t, g):
if isinstance(t, TensExpr):
return t.contract_metric(g)
return t
def perm2tensor(t, g, canon_bp=False):
"""
Returns the tensor corresponding to the permutation ``g``
For further details, see the method in ``TIDS`` with the same name.
"""
if not isinstance(t, TensExpr):
return t
new_tids = get_tids(t).perm2tensor(g, canon_bp)
coeff = get_coeff(t)
if g[-1] != len(g) - 1:
coeff = -coeff
res = TensMul.from_TIDS(coeff, new_tids, is_canon_bp=canon_bp)
return res
def substitute_indices(t, *index_tuples):
"""
Return a tensor with free indices substituted according to ``index_tuples``
``index_types`` list of tuples ``(old_index, new_index)``
Note: this method will neither raise or lower the indices, it will just replace their symbol.
Examples
========
>>> from sympy.tensor.tensor import TensorIndexType, tensor_indices, tensorhead
>>> Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
>>> i, j, k, l = tensor_indices('i,j,k,l', Lorentz)
>>> A, B = tensorhead('A,B', [Lorentz]*2, [[1]*2])
>>> t = A(i, k)*B(-k, -j); t
A(i, L_0)*B(-L_0, -j)
>>> t.substitute_indices((i,j), (j, k))
A(j, L_0)*B(-L_0, -k)
"""
if not isinstance(t, TensExpr):
return t
free = t.free
free1 = []
for j, ipos, cpos in free:
for i, v in index_tuples:
if i._name == j._name and i._tensortype == j._tensortype:
if i._is_up == j._is_up:
free1.append((v, ipos, cpos))
else:
free1.append((-v, ipos, cpos))
break
else:
free1.append((j, ipos, cpos))
t = TensMul.from_data(t.coeff, t.components, free1, t.dum)
return t
