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from __future__ import annotations
import collections.abc
import operator
from collections import defaultdict, Counter
from functools import reduce
import itertools
from itertools import accumulate
import typing
from sympy.core.numbers import Integer
from sympy.core.relational import Equality
from sympy.functions.special.tensor_functions import KroneckerDelta
from sympy.core.basic import Basic
from sympy.core.containers import Tuple
from sympy.core.expr import Expr
from sympy.core.function import (Function, Lambda)
from sympy.core.mul import Mul
from sympy.core.singleton import S
from sympy.core.sorting import default_sort_key
from sympy.core.symbol import (Dummy, Symbol)
from sympy.matrices.matrixbase import MatrixBase
from sympy.matrices.expressions.diagonal import diagonalize_vector
from sympy.matrices.expressions.matexpr import MatrixExpr
from sympy.matrices.expressions.special import ZeroMatrix
from sympy.tensor.array.arrayop import (permutedims, tensorcontraction, tensordiagonal, tensorproduct)
from sympy.tensor.array.dense_ndim_array import ImmutableDenseNDimArray
from sympy.tensor.array.ndim_array import NDimArray
from sympy.tensor.indexed import (Indexed, IndexedBase)
from sympy.matrices.expressions.matexpr import MatrixElement
from sympy.tensor.array.expressions.utils import _apply_recursively_over_nested_lists, _sort_contraction_indices, \
_get_mapping_from_subranks, _build_push_indices_up_func_transformation, _get_contraction_links, \
_build_push_indices_down_func_transformation
from sympy.combinatorics import Permutation
from sympy.combinatorics.permutations import _af_invert
from sympy.core.sympify import _sympify
class _ArrayExpr(Expr):
shape: tuple[Expr, ...]
def __getitem__(self, item):
if not isinstance(item, collections.abc.Iterable):
item = (item,)
ArrayElement._check_shape(self, item)
return self._get(item)
def _get(self, item):
return _get_array_element_or_slice(self, item)
class ArraySymbol(_ArrayExpr):
"""
Symbol representing an array expression
"""
_iterable = False
def __new__(cls, symbol, shape: typing.Iterable) -> "ArraySymbol":
if isinstance(symbol, str):
symbol = Symbol(symbol)
# symbol = _sympify(symbol)
shape = Tuple(*map(_sympify, shape))
obj = Expr.__new__(cls, symbol, shape)
return obj
@property
def name(self):
return self._args[0]
@property
def shape(self):
return self._args[1]
def as_explicit(self):
if not all(i.is_Integer for i in self.shape):
raise ValueError("cannot express explicit array with symbolic shape")
data = [self[i] for i in itertools.product(*[range(j) for j in self.shape])]
return ImmutableDenseNDimArray(data).reshape(*self.shape)
class ArrayElement(Expr):
"""
An element of an array.
"""
_diff_wrt = True
is_symbol = True
is_commutative = True
def __new__(cls, name, indices):
if isinstance(name, str):
name = Symbol(name)
name = _sympify(name)
if not isinstance(indices, collections.abc.Iterable):
indices = (indices,)
indices = _sympify(tuple(indices))
cls._check_shape(name, indices)
obj = Expr.__new__(cls, name, indices)
return obj
@classmethod
def _check_shape(cls, name, indices):
indices = tuple(indices)
if hasattr(name, "shape"):
index_error = IndexError("number of indices does not match shape of the array")
if len(indices) != len(name.shape):
raise index_error
if any((i >= s) == True for i, s in zip(indices, name.shape)):
raise ValueError("shape is out of bounds")
if any((i < 0) == True for i in indices):
raise ValueError("shape contains negative values")
@property
def name(self):
return self._args[0]
@property
def indices(self):
return self._args[1]
def _eval_derivative(self, s):
if not isinstance(s, ArrayElement):
return S.Zero
if s == self:
return S.One
if s.name != self.name:
return S.Zero
return Mul.fromiter(KroneckerDelta(i, j) for i, j in zip(self.indices, s.indices))
class ZeroArray(_ArrayExpr):
"""
Symbolic array of zeros. Equivalent to ``ZeroMatrix`` for matrices.
"""
def __new__(cls, *shape):
if len(shape) == 0:
return S.Zero
shape = map(_sympify, shape)
obj = Expr.__new__(cls, *shape)
return obj
@property
def shape(self):
return self._args
def as_explicit(self):
if not all(i.is_Integer for i in self.shape):
raise ValueError("Cannot return explicit form for symbolic shape.")
return ImmutableDenseNDimArray.zeros(*self.shape)
def _get(self, item):
return S.Zero
class OneArray(_ArrayExpr):
"""
Symbolic array of ones.
"""
def __new__(cls, *shape):
if len(shape) == 0:
return S.One
shape = map(_sympify, shape)
obj = Expr.__new__(cls, *shape)
return obj
@property
def shape(self):
return self._args
def as_explicit(self):
if not all(i.is_Integer for i in self.shape):
raise ValueError("Cannot return explicit form for symbolic shape.")
return ImmutableDenseNDimArray([S.One for i in range(reduce(operator.mul, self.shape))]).reshape(*self.shape)
def _get(self, item):
return S.One
class _CodegenArrayAbstract(Basic):
@property
def subranks(self):
"""
Returns the ranks of the objects in the uppermost tensor product inside
the current object. In case no tensor products are contained, return
the atomic ranks.
Examples
========
>>> from sympy.tensor.array import tensorproduct, tensorcontraction
>>> from sympy import MatrixSymbol
>>> M = MatrixSymbol("M", 3, 3)
>>> N = MatrixSymbol("N", 3, 3)
>>> P = MatrixSymbol("P", 3, 3)
Important: do not confuse the rank of the matrix with the rank of an array.
>>> tp = tensorproduct(M, N, P)
>>> tp.subranks
[2, 2, 2]
>>> co = tensorcontraction(tp, (1, 2), (3, 4))
>>> co.subranks
[2, 2, 2]
"""
return self._subranks[:]
def subrank(self):
"""
The sum of ``subranks``.
"""
return sum(self.subranks)
@property
def shape(self):
return self._shape
def doit(self, **hints):
deep = hints.get("deep", True)
if deep:
return self.func(*[arg.doit(**hints) for arg in self.args])._canonicalize()
else:
return self._canonicalize()
class ArrayTensorProduct(_CodegenArrayAbstract):
r"""
Class to represent the tensor product of array-like objects.
"""
def __new__(cls, *args, **kwargs):
args = [_sympify(arg) for arg in args]
canonicalize = kwargs.pop("canonicalize", False)
ranks = [get_rank(arg) for arg in args]
obj = Basic.__new__(cls, *args)
obj._subranks = ranks
shapes = [get_shape(i) for i in args]
if any(i is None for i in shapes):
obj._shape = None
else:
obj._shape = tuple(j for i in shapes for j in i)
if canonicalize:
return obj._canonicalize()
return obj
def _canonicalize(self):
args = self.args
args = self._flatten(args)
ranks = [get_rank(arg) for arg in args]
# Check if there are nested permutation and lift them up:
permutation_cycles = []
for i, arg in enumerate(args):
if not isinstance(arg, PermuteDims):
continue
permutation_cycles.extend([[k + sum(ranks[:i]) for k in j] for j in arg.permutation.cyclic_form])
args[i] = arg.expr
if permutation_cycles:
return _permute_dims(_array_tensor_product(*args), Permutation(sum(ranks)-1)*Permutation(permutation_cycles))
if len(args) == 1:
return args[0]
# If any object is a ZeroArray, return a ZeroArray:
if any(isinstance(arg, (ZeroArray, ZeroMatrix)) for arg in args):
shapes = reduce(operator.add, [get_shape(i) for i in args], ())
return ZeroArray(*shapes)
# If there are contraction objects inside, transform the whole
# expression into `ArrayContraction`:
contractions = {i: arg for i, arg in enumerate(args) if isinstance(arg, ArrayContraction)}
if contractions:
ranks = [_get_subrank(arg) if isinstance(arg, ArrayContraction) else get_rank(arg) for arg in args]
cumulative_ranks = list(accumulate([0] + ranks))[:-1]
tp = _array_tensor_product(*[arg.expr if isinstance(arg, ArrayContraction) else arg for arg in args])
contraction_indices = [tuple(cumulative_ranks[i] + k for k in j) for i, arg in contractions.items() for j in arg.contraction_indices]
return _array_contraction(tp, *contraction_indices)
diagonals = {i: arg for i, arg in enumerate(args) if isinstance(arg, ArrayDiagonal)}
if diagonals:
inverse_permutation = []
last_perm = []
ranks = [get_rank(arg) for arg in args]
cumulative_ranks = list(accumulate([0] + ranks))[:-1]
for i, arg in enumerate(args):
if isinstance(arg, ArrayDiagonal):
i1 = get_rank(arg) - len(arg.diagonal_indices)
i2 = len(arg.diagonal_indices)
inverse_permutation.extend([cumulative_ranks[i] + j for j in range(i1)])
last_perm.extend([cumulative_ranks[i] + j for j in range(i1, i1 + i2)])
else:
inverse_permutation.extend([cumulative_ranks[i] + j for j in range(get_rank(arg))])
inverse_permutation.extend(last_perm)
tp = _array_tensor_product(*[arg.expr if isinstance(arg, ArrayDiagonal) else arg for arg in args])
ranks2 = [_get_subrank(arg) if isinstance(arg, ArrayDiagonal) else get_rank(arg) for arg in args]
cumulative_ranks2 = list(accumulate([0] + ranks2))[:-1]
diagonal_indices = [tuple(cumulative_ranks2[i] + k for k in j) for i, arg in diagonals.items() for j in arg.diagonal_indices]
return _permute_dims(_array_diagonal(tp, *diagonal_indices), _af_invert(inverse_permutation))
return self.func(*args, canonicalize=False)
@classmethod
def _flatten(cls, args):
args = [i for arg in args for i in (arg.args if isinstance(arg, cls) else [arg])]
return args
def as_explicit(self):
return tensorproduct(*[arg.as_explicit() if hasattr(arg, "as_explicit") else arg for arg in self.args])
class ArrayAdd(_CodegenArrayAbstract):
r"""
Class for elementwise array additions.
"""
def __new__(cls, *args, **kwargs):
args = [_sympify(arg) for arg in args]
ranks = [get_rank(arg) for arg in args]
ranks = list(set(ranks))
if len(ranks) != 1:
raise ValueError("summing arrays of different ranks")
shapes = [arg.shape for arg in args]
if len({i for i in shapes if i is not None}) > 1:
raise ValueError("mismatching shapes in addition")
canonicalize = kwargs.pop("canonicalize", False)
obj = Basic.__new__(cls, *args)
obj._subranks = ranks
if any(i is None for i in shapes):
obj._shape = None
else:
obj._shape = shapes[0]
if canonicalize:
return obj._canonicalize()
return obj
def _canonicalize(self):
args = self.args
# Flatten:
args = self._flatten_args(args)
shapes = [get_shape(arg) for arg in args]
args = [arg for arg in args if not isinstance(arg, (ZeroArray, ZeroMatrix))]
if len(args) == 0:
if any(i for i in shapes if i is None):
raise NotImplementedError("cannot handle addition of ZeroMatrix/ZeroArray and undefined shape object")
return ZeroArray(*shapes[0])
elif len(args) == 1:
return args[0]
return self.func(*args, canonicalize=False)
@classmethod
def _flatten_args(cls, args):
new_args = []
for arg in args:
if isinstance(arg, ArrayAdd):
new_args.extend(arg.args)
else:
new_args.append(arg)
return new_args
def as_explicit(self):
return reduce(
operator.add,
[arg.as_explicit() if hasattr(arg, "as_explicit") else arg for arg in self.args])
class PermuteDims(_CodegenArrayAbstract):
r"""
Class to represent permutation of axes of arrays.
Examples
========
>>> from sympy.tensor.array import permutedims
>>> from sympy import MatrixSymbol
>>> M = MatrixSymbol("M", 3, 3)
>>> cg = permutedims(M, [1, 0])
The object ``cg`` represents the transposition of ``M``, as the permutation
``[1, 0]`` will act on its indices by switching them:
`M_{ij} \Rightarrow M_{ji}`
This is evident when transforming back to matrix form:
>>> from sympy.tensor.array.expressions.from_array_to_matrix import convert_array_to_matrix
>>> convert_array_to_matrix(cg)
M.T
>>> N = MatrixSymbol("N", 3, 2)
>>> cg = permutedims(N, [1, 0])
>>> cg.shape
(2, 3)
There are optional parameters that can be used as alternative to the permutation:
>>> from sympy.tensor.array.expressions import ArraySymbol, PermuteDims
>>> M = ArraySymbol("M", (1, 2, 3, 4, 5))
>>> expr = PermuteDims(M, index_order_old="ijklm", index_order_new="kijml")
>>> expr
PermuteDims(M, (0 2 1)(3 4))
>>> expr.shape
(3, 1, 2, 5, 4)
Permutations of tensor products are simplified in order to achieve a
standard form:
>>> from sympy.tensor.array import tensorproduct
>>> M = MatrixSymbol("M", 4, 5)
>>> tp = tensorproduct(M, N)
>>> tp.shape
(4, 5, 3, 2)
>>> perm1 = permutedims(tp, [2, 3, 1, 0])
The args ``(M, N)`` have been sorted and the permutation has been
simplified, the expression is equivalent:
>>> perm1.expr.args
(N, M)
>>> perm1.shape
(3, 2, 5, 4)
>>> perm1.permutation
(2 3)
The permutation in its array form has been simplified from
``[2, 3, 1, 0]`` to ``[0, 1, 3, 2]``, as the arguments of the tensor
product `M` and `N` have been switched:
>>> perm1.permutation.array_form
[0, 1, 3, 2]
We can nest a second permutation:
>>> perm2 = permutedims(perm1, [1, 0, 2, 3])
>>> perm2.shape
(2, 3, 5, 4)
>>> perm2.permutation.array_form
[1, 0, 3, 2]
"""
def __new__(cls, expr, permutation=None, index_order_old=None, index_order_new=None, **kwargs):
from sympy.combinatorics import Permutation
expr = _sympify(expr)
expr_rank = get_rank(expr)
permutation = cls._get_permutation_from_arguments(permutation, index_order_old, index_order_new, expr_rank)
permutation = Permutation(permutation)
permutation_size = permutation.size
if permutation_size != expr_rank:
raise ValueError("Permutation size must be the length of the shape of expr")
canonicalize = kwargs.pop("canonicalize", False)
obj = Basic.__new__(cls, expr, permutation)
obj._subranks = [get_rank(expr)]
shape = get_shape(expr)
if shape is None:
obj._shape = None
else:
obj._shape = tuple(shape[permutation(i)] for i in range(len(shape)))
if canonicalize:
return obj._canonicalize()
return obj
def _canonicalize(self):
expr = self.expr
permutation = self.permutation
if isinstance(expr, PermuteDims):
subexpr = expr.expr
subperm = expr.permutation
permutation = permutation * subperm
expr = subexpr
if isinstance(expr, ArrayContraction):
expr, permutation = self._PermuteDims_denestarg_ArrayContraction(expr, permutation)
if isinstance(expr, ArrayTensorProduct):
expr, permutation = self._PermuteDims_denestarg_ArrayTensorProduct(expr, permutation)
if isinstance(expr, (ZeroArray, ZeroMatrix)):
return ZeroArray(*[expr.shape[i] for i in permutation.array_form])
plist = permutation.array_form
if plist == sorted(plist):
return expr
return self.func(expr, permutation, canonicalize=False)
@property
def expr(self):
return self.args[0]
@property
def permutation(self):
return self.args[1]
@classmethod
def _PermuteDims_denestarg_ArrayTensorProduct(cls, expr, permutation):
# Get the permutation in its image-form:
perm_image_form = _af_invert(permutation.array_form)
args = list(expr.args)
# Starting index global position for every arg:
cumul = list(accumulate([0] + expr.subranks))
# Split `perm_image_form` into a list of list corresponding to the indices
# of every argument:
perm_image_form_in_components = [perm_image_form[cumul[i]:cumul[i+1]] for i in range(len(args))]
# Create an index, target-position-key array:
ps = [(i, sorted(comp)) for i, comp in enumerate(perm_image_form_in_components)]
# Sort the array according to the target-position-key:
# In this way, we define a canonical way to sort the arguments according
# to the permutation.
ps.sort(key=lambda x: x[1])
# Read the inverse-permutation (i.e. image-form) of the args:
perm_args_image_form = [i[0] for i in ps]
# Apply the args-permutation to the `args`:
args_sorted = [args[i] for i in perm_args_image_form]
# Apply the args-permutation to the array-form of the permutation of the axes (of `expr`):
perm_image_form_sorted_args = [perm_image_form_in_components[i] for i in perm_args_image_form]
new_permutation = Permutation(_af_invert([j for i in perm_image_form_sorted_args for j in i]))
return _array_tensor_product(*args_sorted), new_permutation
@classmethod
def _PermuteDims_denestarg_ArrayContraction(cls, expr, permutation):
if not isinstance(expr, ArrayContraction):
return expr, permutation
if not isinstance(expr.expr, ArrayTensorProduct):
return expr, permutation
args = expr.expr.args
subranks = [get_rank(arg) for arg in expr.expr.args]
contraction_indices = expr.contraction_indices
contraction_indices_flat = [j for i in contraction_indices for j in i]
cumul = list(accumulate([0] + subranks))
# Spread the permutation in its array form across the args in the corresponding
# tensor-product arguments with free indices:
permutation_array_blocks_up = []
image_form = _af_invert(permutation.array_form)
counter = 0
for i in range(len(subranks)):
current = []
for j in range(cumul[i], cumul[i+1]):
if j in contraction_indices_flat:
continue
current.append(image_form[counter])
counter += 1
permutation_array_blocks_up.append(current)
# Get the map of axis repositioning for every argument of tensor-product:
index_blocks = [list(range(cumul[i], cumul[i+1])) for i, e in enumerate(expr.subranks)]
index_blocks_up = expr._push_indices_up(expr.contraction_indices, index_blocks)
inverse_permutation = permutation**(-1)
index_blocks_up_permuted = [[inverse_permutation(j) for j in i if j is not None] for i in index_blocks_up]
# Sorting key is a list of tuple, first element is the index of `args`, second element of
# the tuple is the sorting key to sort `args` of the tensor product:
sorting_keys = list(enumerate(index_blocks_up_permuted))
sorting_keys.sort(key=lambda x: x[1])
# Now we can get the permutation acting on the args in its image-form:
new_perm_image_form = [i[0] for i in sorting_keys]
# Apply the args-level permutation to various elements:
new_index_blocks = [index_blocks[i] for i in new_perm_image_form]
new_index_perm_array_form = _af_invert([j for i in new_index_blocks for j in i])
new_args = [args[i] for i in new_perm_image_form]
new_contraction_indices = [tuple(new_index_perm_array_form[j] for j in i) for i in contraction_indices]
new_expr = _array_contraction(_array_tensor_product(*new_args), *new_contraction_indices)
new_permutation = Permutation(_af_invert([j for i in [permutation_array_blocks_up[k] for k in new_perm_image_form] for j in i]))
return new_expr, new_permutation
@classmethod
def _check_permutation_mapping(cls, expr, permutation):
subranks = expr.subranks
index2arg = [i for i, arg in enumerate(expr.args) for j in range(expr.subranks[i])]
permuted_indices = [permutation(i) for i in range(expr.subrank())]
new_args = list(expr.args)
arg_candidate_index = index2arg[permuted_indices[0]]
current_indices = []
new_permutation = []
inserted_arg_cand_indices = set()
for i, idx in enumerate(permuted_indices):
if index2arg[idx] != arg_candidate_index:
new_permutation.extend(current_indices)
current_indices = []
arg_candidate_index = index2arg[idx]
current_indices.append(idx)
arg_candidate_rank = subranks[arg_candidate_index]
if len(current_indices) == arg_candidate_rank:
new_permutation.extend(sorted(current_indices))
local_current_indices = [j - min(current_indices) for j in current_indices]
i1 = index2arg[i]
new_args[i1] = _permute_dims(new_args[i1], Permutation(local_current_indices))
inserted_arg_cand_indices.add(arg_candidate_index)
current_indices = []
new_permutation.extend(current_indices)
# TODO: swap args positions in order to simplify the expression:
# TODO: this should be in a function
args_positions = list(range(len(new_args)))
# Get possible shifts:
maps = {}
cumulative_subranks = [0] + list(accumulate(subranks))
for i in range(len(subranks)):
s = {index2arg[new_permutation[j]] for j in range(cumulative_subranks[i], cumulative_subranks[i+1])}
if len(s) != 1:
continue
elem = next(iter(s))
if i != elem:
maps[i] = elem
# Find cycles in the map:
lines = []
current_line = []
while maps:
if len(current_line) == 0:
k, v = maps.popitem()
current_line.append(k)
else:
k = current_line[-1]
if k not in maps:
current_line = []
continue
v = maps.pop(k)
if v in current_line:
lines.append(current_line)
current_line = []
continue
current_line.append(v)
for line in lines:
for i, e in enumerate(line):
args_positions[line[(i + 1) % len(line)]] = e
# TODO: function in order to permute the args:
permutation_blocks = [[new_permutation[cumulative_subranks[i] + j] for j in range(e)] for i, e in enumerate(subranks)]
new_args = [new_args[i] for i in args_positions]
new_permutation_blocks = [permutation_blocks[i] for i in args_positions]
new_permutation2 = [j for i in new_permutation_blocks for j in i]
return _array_tensor_product(*new_args), Permutation(new_permutation2) # **(-1)
@classmethod
def _check_if_there_are_closed_cycles(cls, expr, permutation):
args = list(expr.args)
subranks = expr.subranks
cyclic_form = permutation.cyclic_form
cumulative_subranks = [0] + list(accumulate(subranks))
cyclic_min = [min(i) for i in cyclic_form]
cyclic_max = [max(i) for i in cyclic_form]
cyclic_keep = []
for i, cycle in enumerate(cyclic_form):
flag = True
for j in range(len(cumulative_subranks) - 1):
if cyclic_min[i] >= cumulative_subranks[j] and cyclic_max[i] < cumulative_subranks[j+1]:
# Found a sinkable cycle.
args[j] = _permute_dims(args[j], Permutation([[k - cumulative_subranks[j] for k in cycle]]))
flag = False
break
if flag:
cyclic_keep.append(cycle)
return _array_tensor_product(*args), Permutation(cyclic_keep, size=permutation.size)
def nest_permutation(self):
r"""
DEPRECATED.
"""
ret = self._nest_permutation(self.expr, self.permutation)
if ret is None:
return self
return ret
@classmethod
def _nest_permutation(cls, expr, permutation):
if isinstance(expr, ArrayTensorProduct):
return _permute_dims(*cls._check_if_there_are_closed_cycles(expr, permutation))
elif isinstance(expr, ArrayContraction):
# Invert tree hierarchy: put the contraction above.
cycles = permutation.cyclic_form
newcycles = ArrayContraction._convert_outer_indices_to_inner_indices(expr, *cycles)
newpermutation = Permutation(newcycles)
new_contr_indices = [tuple(newpermutation(j) for j in i) for i in expr.contraction_indices]
return _array_contraction(PermuteDims(expr.expr, newpermutation), *new_contr_indices)
elif isinstance(expr, ArrayAdd):
return _array_add(*[PermuteDims(arg, permutation) for arg in expr.args])
return None
def as_explicit(self):
expr = self.expr
if hasattr(expr, "as_explicit"):
expr = expr.as_explicit()
return permutedims(expr, self.permutation)
@classmethod
def _get_permutation_from_arguments(cls, permutation, index_order_old, index_order_new, dim):
if permutation is None:
if index_order_new is None or index_order_old is None:
raise ValueError("Permutation not defined")
return PermuteDims._get_permutation_from_index_orders(index_order_old, index_order_new, dim)
else:
if index_order_new is not None:
raise ValueError("index_order_new cannot be defined with permutation")
if index_order_old is not None:
raise ValueError("index_order_old cannot be defined with permutation")
return permutation
@classmethod
def _get_permutation_from_index_orders(cls, index_order_old, index_order_new, dim):
if len(set(index_order_new)) != dim:
raise ValueError("wrong number of indices in index_order_new")
if len(set(index_order_old)) != dim:
raise ValueError("wrong number of indices in index_order_old")
if len(set.symmetric_difference(set(index_order_new), set(index_order_old))) > 0:
raise ValueError("index_order_new and index_order_old must have the same indices")
permutation = [index_order_old.index(i) for i in index_order_new]
return permutation
class ArrayDiagonal(_CodegenArrayAbstract):
r"""
Class to represent the diagonal operator.
Explanation
===========
In a 2-dimensional array it returns the diagonal, this looks like the
operation:
`A_{ij} \rightarrow A_{ii}`
The diagonal over axes 1 and 2 (the second and third) of the tensor product
of two 2-dimensional arrays `A \otimes B` is
`\Big[ A_{ab} B_{cd} \Big]_{abcd} \rightarrow \Big[ A_{ai} B_{id} \Big]_{adi}`
In this last example the array expression has been reduced from
4-dimensional to 3-dimensional. Notice that no contraction has occurred,
rather there is a new index `i` for the diagonal, contraction would have
reduced the array to 2 dimensions.
Notice that the diagonalized out dimensions are added as new dimensions at
the end of the indices.
"""
def __new__(cls, expr, *diagonal_indices, **kwargs):
expr = _sympify(expr)
diagonal_indices = [Tuple(*sorted(i)) for i in diagonal_indices]
canonicalize = kwargs.get("canonicalize", False)
shape = get_shape(expr)
if shape is not None:
cls._validate(expr, *diagonal_indices, **kwargs)
# Get new shape:
positions, shape = cls._get_positions_shape(shape, diagonal_indices)
else:
positions = None
if len(diagonal_indices) == 0:
return expr
obj = Basic.__new__(cls, expr, *diagonal_indices)
obj._positions = positions
obj._subranks = _get_subranks(expr)
obj._shape = shape
if canonicalize:
return obj._canonicalize()
return obj
def _canonicalize(self):
expr = self.expr
diagonal_indices = self.diagonal_indices
trivial_diags = [i for i in diagonal_indices if len(i) == 1]
if len(trivial_diags) > 0:
trivial_pos = {e[0]: i for i, e in enumerate(diagonal_indices) if len(e) == 1}
diag_pos = {e: i for i, e in enumerate(diagonal_indices) if len(e) > 1}
diagonal_indices_short = [i for i in diagonal_indices if len(i) > 1]
rank1 = get_rank(self)
rank2 = len(diagonal_indices)
rank3 = rank1 - rank2
inv_permutation = []
counter1 = 0
indices_down = ArrayDiagonal._push_indices_down(diagonal_indices_short, list(range(rank1)), get_rank(expr))
for i in indices_down:
if i in trivial_pos:
inv_permutation.append(rank3 + trivial_pos[i])
elif isinstance(i, (Integer, int)):
inv_permutation.append(counter1)
counter1 += 1
else:
inv_permutation.append(rank3 + diag_pos[i])
permutation = _af_invert(inv_permutation)
if len(diagonal_indices_short) > 0:
return _permute_dims(_array_diagonal(expr, *diagonal_indices_short), permutation)
else:
return _permute_dims(expr, permutation)
if isinstance(expr, ArrayAdd):
return self._ArrayDiagonal_denest_ArrayAdd(expr, *diagonal_indices)
if isinstance(expr, ArrayDiagonal):
return self._ArrayDiagonal_denest_ArrayDiagonal(expr, *diagonal_indices)
if isinstance(expr, PermuteDims):
return self._ArrayDiagonal_denest_PermuteDims(expr, *diagonal_indices)
if isinstance(expr, (ZeroArray, ZeroMatrix)):
positions, shape = self._get_positions_shape(expr.shape, diagonal_indices)
return ZeroArray(*shape)
return self.func(expr, *diagonal_indices, canonicalize=False)
@staticmethod
def _validate(expr, *diagonal_indices, **kwargs):
# Check that no diagonalization happens on indices with mismatched
# dimensions:
shape = get_shape(expr)
for i in diagonal_indices:
if any(j >= len(shape) for j in i):
raise ValueError("index is larger than expression shape")
if len({shape[j] for j in i}) != 1:
raise ValueError("diagonalizing indices of different dimensions")
if not kwargs.get("allow_trivial_diags", False) and len(i) <= 1:
raise ValueError("need at least two axes to diagonalize")
if len(set(i)) != len(i):
raise ValueError("axis index cannot be repeated")
@staticmethod
def _remove_trivial_dimensions(shape, *diagonal_indices):
return [tuple(j for j in i) for i in diagonal_indices if shape[i[0]] != 1]
@property
def expr(self):
return self.args[0]
@property
def diagonal_indices(self):
return self.args[1:]
@staticmethod
def _flatten(expr, *outer_diagonal_indices):
inner_diagonal_indices = expr.diagonal_indices
all_inner = [j for i in inner_diagonal_indices for j in i]
all_inner.sort()
# TODO: add API for total rank and cumulative rank:
total_rank = _get_subrank(expr)
inner_rank = len(all_inner)
outer_rank = total_rank - inner_rank
shifts = [0 for i in range(outer_rank)]
counter = 0
pointer = 0
for i in range(outer_rank):
while pointer < inner_rank and counter >= all_inner[pointer]:
counter += 1
pointer += 1
shifts[i] += pointer
counter += 1
outer_diagonal_indices = tuple(tuple(shifts[j] + j for j in i) for i in outer_diagonal_indices)
diagonal_indices = inner_diagonal_indices + outer_diagonal_indices
return _array_diagonal(expr.expr, *diagonal_indices)
@classmethod
def _ArrayDiagonal_denest_ArrayAdd(cls, expr, *diagonal_indices):
return _array_add(*[_array_diagonal(arg, *diagonal_indices) for arg in expr.args])
@classmethod
def _ArrayDiagonal_denest_ArrayDiagonal(cls, expr, *diagonal_indices):
return cls._flatten(expr, *diagonal_indices)
@classmethod
def _ArrayDiagonal_denest_PermuteDims(cls, expr: PermuteDims, *diagonal_indices):
back_diagonal_indices = [[expr.permutation(j) for j in i] for i in diagonal_indices]
nondiag = [i for i in range(get_rank(expr)) if not any(i in j for j in diagonal_indices)]
back_nondiag = [expr.permutation(i) for i in nondiag]
remap = {e: i for i, e in enumerate(sorted(back_nondiag))}
new_permutation1 = [remap[i] for i in back_nondiag]
shift = len(new_permutation1)
diag_block_perm = [i + shift for i in range(len(back_diagonal_indices))]
new_permutation = new_permutation1 + diag_block_perm
return _permute_dims(
_array_diagonal(
expr.expr,
*back_diagonal_indices
),
new_permutation
)
def _push_indices_down_nonstatic(self, indices):
transform = lambda x: self._positions[x] if x < len(self._positions) else None
return _apply_recursively_over_nested_lists(transform, indices)
def _push_indices_up_nonstatic(self, indices):
def transform(x):
for i, e in enumerate(self._positions):
if (isinstance(e, int) and x == e) or (isinstance(e, tuple) and x in e):
return i
return _apply_recursively_over_nested_lists(transform, indices)
@classmethod
def _push_indices_down(cls, diagonal_indices, indices, rank):
positions, shape = cls._get_positions_shape(range(rank), diagonal_indices)
transform = lambda x: positions[x] if x < len(positions) else None
return _apply_recursively_over_nested_lists(transform, indices)
@classmethod
def _push_indices_up(cls, diagonal_indices, indices, rank):
positions, shape = cls._get_positions_shape(range(rank), diagonal_indices)
def transform(x):
for i, e in enumerate(positions):
if (isinstance(e, int) and x == e) or (isinstance(e, (tuple, Tuple)) and (x in e)):
return i
return _apply_recursively_over_nested_lists(transform, indices)
@classmethod
def _get_positions_shape(cls, shape, diagonal_indices):
data1 = tuple((i, shp) for i, shp in enumerate(shape) if not any(i in j for j in diagonal_indices))
pos1, shp1 = zip(*data1) if data1 else ((), ())
data2 = tuple((i, shape[i[0]]) for i in diagonal_indices)
pos2, shp2 = zip(*data2) if data2 else ((), ())
positions = pos1 + pos2
shape = shp1 + shp2
return positions, shape
def as_explicit(self):
expr = self.expr
if hasattr(expr, "as_explicit"):
expr = expr.as_explicit()
return tensordiagonal(expr, *self.diagonal_indices)
class ArrayElementwiseApplyFunc(_CodegenArrayAbstract):
def __new__(cls, function, element):
if not isinstance(function, Lambda):
d = Dummy('d')
function = Lambda(d, function(d))
obj = _CodegenArrayAbstract.__new__(cls, function, element)
obj._subranks = _get_subranks(element)
return obj
@property
def function(self):
return self.args[0]
@property
def expr(self):
return self.args[1]
@property
def shape(self):
return self.expr.shape
def _get_function_fdiff(self):
d = Dummy("d")
function = self.function(d)
fdiff = function.diff(d)
if isinstance(fdiff, Function):
fdiff = type(fdiff)
else:
fdiff = Lambda(d, fdiff)
return fdiff
def as_explicit(self):
expr = self.expr
if hasattr(expr, "as_explicit"):
expr = expr.as_explicit()
return expr.applyfunc(self.function)
class ArrayContraction(_CodegenArrayAbstract):
r"""
This class is meant to represent contractions of arrays in a form easily
processable by the code printers.
"""
def __new__(cls, expr, *contraction_indices, **kwargs):
contraction_indices = _sort_contraction_indices(contraction_indices)
expr = _sympify(expr)
canonicalize = kwargs.get("canonicalize", False)
obj = Basic.__new__(cls, expr, *contraction_indices)
obj._subranks = _get_subranks(expr)
obj._mapping = _get_mapping_from_subranks(obj._subranks)
free_indices_to_position = {i: i for i in range(sum(obj._subranks)) if all(i not in cind for cind in contraction_indices)}
obj._free_indices_to_position = free_indices_to_position
shape = get_shape(expr)
cls._validate(expr, *contraction_indices)
if shape:
shape = tuple(shp for i, shp in enumerate(shape) if not any(i in j for j in contraction_indices))
obj._shape = shape
if canonicalize:
return obj._canonicalize()
return obj
def _canonicalize(self):
expr = self.expr
contraction_indices = self.contraction_indices
if len(contraction_indices) == 0:
return expr
if isinstance(expr, ArrayContraction):
return self._ArrayContraction_denest_ArrayContraction(expr, *contraction_indices)
if isinstance(expr, (ZeroArray, ZeroMatrix)):
return self._ArrayContraction_denest_ZeroArray(expr, *contraction_indices)
if isinstance(expr, PermuteDims):
return self._ArrayContraction_denest_PermuteDims(expr, *contraction_indices)
if isinstance(expr, ArrayTensorProduct):
expr, contraction_indices = self._sort_fully_contracted_args(expr, contraction_indices)
expr, contraction_indices = self._lower_contraction_to_addends(expr, contraction_indices)
if len(contraction_indices) == 0:
return expr
if isinstance(expr, ArrayDiagonal):
return self._ArrayContraction_denest_ArrayDiagonal(expr, *contraction_indices)
if isinstance(expr, ArrayAdd):
return self._ArrayContraction_denest_ArrayAdd(expr, *contraction_indices)
# Check single index contractions on 1-dimensional axes:
contraction_indices = [i for i in contraction_indices if len(i) > 1 or get_shape(expr)[i[0]] != 1]
if len(contraction_indices) == 0:
return expr
return self.func(expr, *contraction_indices, canonicalize=False)
def __mul__(self, other):
if other == 1:
return self
else:
raise NotImplementedError("Product of N-dim arrays is not uniquely defined. Use another method.")
def __rmul__(self, other):
if other == 1:
return self
else:
raise NotImplementedError("Product of N-dim arrays is not uniquely defined. Use another method.")
@staticmethod
def _validate(expr, *contraction_indices):
shape = get_shape(expr)
if shape is None:
return
# Check that no contraction happens when the shape is mismatched:
for i in contraction_indices:
if len({shape[j] for j in i if shape[j] != -1}) != 1:
raise ValueError("contracting indices of different dimensions")
@classmethod
def _push_indices_down(cls, contraction_indices, indices):
flattened_contraction_indices = [j for i in contraction_indices for j in i]
flattened_contraction_indices.sort()
transform = _build_push_indices_down_func_transformation(flattened_contraction_indices)
return _apply_recursively_over_nested_lists(transform, indices)
@classmethod
def _push_indices_up(cls, contraction_indices, indices):
flattened_contraction_indices = [j for i in contraction_indices for j in i]
flattened_contraction_indices.sort()
transform = _build_push_indices_up_func_transformation(flattened_contraction_indices)
return _apply_recursively_over_nested_lists(transform, indices)
@classmethod
def _lower_contraction_to_addends(cls, expr, contraction_indices):
if isinstance(expr, ArrayAdd):
raise NotImplementedError()
if not isinstance(expr, ArrayTensorProduct):
return expr, contraction_indices
subranks = expr.subranks
cumranks = list(accumulate([0] + subranks))
contraction_indices_remaining = []
contraction_indices_args = [[] for i in expr.args]
backshift = set()
for contraction_group in contraction_indices:
for j in range(len(expr.args)):
if not isinstance(expr.args[j], ArrayAdd):
continue
if all(cumranks[j] <= k < cumranks[j+1] for k in contraction_group):
contraction_indices_args[j].append([k - cumranks[j] for k in contraction_group])
backshift.update(contraction_group)
break
else:
contraction_indices_remaining.append(contraction_group)
if len(contraction_indices_remaining) == len(contraction_indices):
return expr, contraction_indices
total_rank = get_rank(expr)
shifts = list(accumulate([1 if i in backshift else 0 for i in range(total_rank)]))
contraction_indices_remaining = [Tuple.fromiter(j - shifts[j] for j in i) for i in contraction_indices_remaining]
ret = _array_tensor_product(*[
_array_contraction(arg, *contr) for arg, contr in zip(expr.args, contraction_indices_args)
])
return ret, contraction_indices_remaining
def split_multiple_contractions(self):
"""
Recognize multiple contractions and attempt at rewriting them as paired-contractions.
This allows some contractions involving more than two indices to be
rewritten as multiple contractions involving two indices, thus allowing
the expression to be rewritten as a matrix multiplication line.
Examples:
* `A_ij b_j0 C_jk` ===> `A*DiagMatrix(b)*C`
Care for:
- matrix being diagonalized (i.e. `A_ii`)
- vectors being diagonalized (i.e. `a_i0`)
Multiple contractions can be split into matrix multiplications if
not more than two arguments are non-diagonals or non-vectors.
Vectors get diagonalized while diagonal matrices remain diagonal.
The non-diagonal matrices can be at the beginning or at the end
of the final matrix multiplication line.
"""
editor = _EditArrayContraction(self)
contraction_indices = self.contraction_indices
onearray_insert = []
for indl, links in enumerate(contraction_indices):
if len(links) <= 2:
continue
# Check multiple contractions:
#
# Examples:
#
# * `A_ij b_j0 C_jk` ===> `A*DiagMatrix(b)*C \otimes OneArray(1)` with permutation (1 2)
#
# Care for:
# - matrix being diagonalized (i.e. `A_ii`)
# - vectors being diagonalized (i.e. `a_i0`)
# Multiple contractions can be split into matrix multiplications if
# not more than three arguments are non-diagonals or non-vectors.
#
# Vectors get diagonalized while diagonal matrices remain diagonal.
# The non-diagonal matrices can be at the beginning or at the end
# of the final matrix multiplication line.
positions = editor.get_mapping_for_index(indl)
# Also consider the case of diagonal matrices being contracted:
current_dimension = self.expr.shape[links[0]]
not_vectors = []
vectors = []
for arg_ind, rel_ind in positions:
arg = editor.args_with_ind[arg_ind]
mat = arg.element
abs_arg_start, abs_arg_end = editor.get_absolute_range(arg)
other_arg_pos = 1-rel_ind
other_arg_abs = abs_arg_start + other_arg_pos
if ((1 not in mat.shape) or
((current_dimension == 1) is True and mat.shape != (1, 1)) or
any(other_arg_abs in l for li, l in enumerate(contraction_indices) if li != indl)
):
not_vectors.append((arg, rel_ind))
else:
vectors.append((arg, rel_ind))
if len(not_vectors) > 2:
# If more than two arguments in the multiple contraction are
# non-vectors and non-diagonal matrices, we cannot find a way
# to split this contraction into a matrix multiplication line:
continue
# Three cases to handle:
# - zero non-vectors
# - one non-vector
# - two non-vectors
for v, rel_ind in vectors:
v.element = diagonalize_vector(v.element)
vectors_to_loop = not_vectors[:1] + vectors + not_vectors[1:]
first_not_vector, rel_ind = vectors_to_loop[0]
new_index = first_not_vector.indices[rel_ind]
for v, rel_ind in vectors_to_loop[1:-1]:
v.indices[rel_ind] = new_index
new_index = editor.get_new_contraction_index()
assert v.indices.index(None) == 1 - rel_ind
v.indices[v.indices.index(None)] = new_index
onearray_insert.append(v)
last_vec, rel_ind = vectors_to_loop[-1]
last_vec.indices[rel_ind] = new_index
for v in onearray_insert:
editor.insert_after(v, _ArgE(OneArray(1), [None]))
return editor.to_array_contraction()
def flatten_contraction_of_diagonal(self):
if not isinstance(self.expr, ArrayDiagonal):
return self
contraction_down = self.expr._push_indices_down(self.expr.diagonal_indices, self.contraction_indices)
new_contraction_indices = []
diagonal_indices = self.expr.diagonal_indices[:]
for i in contraction_down:
contraction_group = list(i)
for j in i:
diagonal_with = [k for k in diagonal_indices if j in k]
contraction_group.extend([l for k in diagonal_with for l in k])
diagonal_indices = [k for k in diagonal_indices if k not in diagonal_with]
new_contraction_indices.append(sorted(set(contraction_group)))
new_contraction_indices = ArrayDiagonal._push_indices_up(diagonal_indices, new_contraction_indices)
return _array_contraction(
_array_diagonal(
self.expr.expr,
*diagonal_indices
),
*new_contraction_indices
)
@staticmethod
def _get_free_indices_to_position_map(free_indices, contraction_indices):
free_indices_to_position = {}
flattened_contraction_indices = [j for i in contraction_indices for j in i]
counter = 0
for ind in free_indices:
while counter in flattened_contraction_indices:
counter += 1
free_indices_to_position[ind] = counter
counter += 1
return free_indices_to_position
@staticmethod
def _get_index_shifts(expr):
"""
Get the mapping of indices at the positions before the contraction
occurs.
Examples
========
>>> from sympy.tensor.array import tensorproduct, tensorcontraction
>>> from sympy import MatrixSymbol
>>> M = MatrixSymbol("M", 3, 3)
>>> N = MatrixSymbol("N", 3, 3)
>>> cg = tensorcontraction(tensorproduct(M, N), [1, 2])
>>> cg._get_index_shifts(cg)
[0, 2]
Indeed, ``cg`` after the contraction has two dimensions, 0 and 1. They
need to be shifted by 0 and 2 to get the corresponding positions before
the contraction (that is, 0 and 3).
"""
inner_contraction_indices = expr.contraction_indices
all_inner = [j for i in inner_contraction_indices for j in i]
all_inner.sort()
# TODO: add API for total rank and cumulative rank:
total_rank = _get_subrank(expr)
inner_rank = len(all_inner)
outer_rank = total_rank - inner_rank
shifts = [0 for i in range(outer_rank)]
counter = 0
pointer = 0
for i in range(outer_rank):
while pointer < inner_rank and counter >= all_inner[pointer]:
counter += 1
pointer += 1
shifts[i] += pointer
counter += 1
return shifts
@staticmethod
def _convert_outer_indices_to_inner_indices(expr, *outer_contraction_indices):
shifts = ArrayContraction._get_index_shifts(expr)
outer_contraction_indices = tuple(tuple(shifts[j] + j for j in i) for i in outer_contraction_indices)
return outer_contraction_indices
@staticmethod
def _flatten(expr, *outer_contraction_indices):
inner_contraction_indices = expr.contraction_indices
outer_contraction_indices = ArrayContraction._convert_outer_indices_to_inner_indices(expr, *outer_contraction_indices)
contraction_indices = inner_contraction_indices + outer_contraction_indices
return _array_contraction(expr.expr, *contraction_indices)
@classmethod
def _ArrayContraction_denest_ArrayContraction(cls, expr, *contraction_indices):
return cls._flatten(expr, *contraction_indices)
@classmethod
def _ArrayContraction_denest_ZeroArray(cls, expr, *contraction_indices):
contraction_indices_flat = [j for i in contraction_indices for j in i]
shape = [e for i, e in enumerate(expr.shape) if i not in contraction_indices_flat]
return ZeroArray(*shape)
@classmethod
def _ArrayContraction_denest_ArrayAdd(cls, expr, *contraction_indices):
return _array_add(*[_array_contraction(i, *contraction_indices) for i in expr.args])
@classmethod
def _ArrayContraction_denest_PermuteDims(cls, expr, *contraction_indices):
permutation = expr.permutation
plist = permutation.array_form
new_contraction_indices = [tuple(permutation(j) for j in i) for i in contraction_indices]
new_plist = [i for i in plist if not any(i in j for j in new_contraction_indices)]
new_plist = cls._push_indices_up(new_contraction_indices, new_plist)
return _permute_dims(
_array_contraction(expr.expr, *new_contraction_indices),
Permutation(new_plist)
)
@classmethod
def _ArrayContraction_denest_ArrayDiagonal(cls, expr: 'ArrayDiagonal', *contraction_indices):
diagonal_indices = list(expr.diagonal_indices)
down_contraction_indices = expr._push_indices_down(expr.diagonal_indices, contraction_indices, get_rank(expr.expr))
# Flatten diagonally contracted indices:
down_contraction_indices = [[k for j in i for k in (j if isinstance(j, (tuple, Tuple)) else [j])] for i in down_contraction_indices]
new_contraction_indices = []
for contr_indgrp in down_contraction_indices:
ind = contr_indgrp[:]
for j, diag_indgrp in enumerate(diagonal_indices):
if diag_indgrp is None:
continue
if any(i in diag_indgrp for i in contr_indgrp):
ind.extend(diag_indgrp)
diagonal_indices[j] = None
new_contraction_indices.append(sorted(set(ind)))
new_diagonal_indices_down = [i for i in diagonal_indices if i is not None]
new_diagonal_indices = ArrayContraction._push_indices_up(new_contraction_indices, new_diagonal_indices_down)
return _array_diagonal(
_array_contraction(expr.expr, *new_contraction_indices),
*new_diagonal_indices
)
@classmethod
def _sort_fully_contracted_args(cls, expr, contraction_indices):
if expr.shape is None:
return expr, contraction_indices
cumul = list(accumulate([0] + expr.subranks))
index_blocks = [list(range(cumul[i], cumul[i+1])) for i in range(len(expr.args))]
contraction_indices_flat = {j for i in contraction_indices for j in i}
fully_contracted = [all(j in contraction_indices_flat for j in range(cumul[i], cumul[i+1])) for i, arg in enumerate(expr.args)]
new_pos = sorted(range(len(expr.args)), key=lambda x: (0, default_sort_key(expr.args[x])) if fully_contracted[x] else (1,))
new_args = [expr.args[i] for i in new_pos]
new_index_blocks_flat = [j for i in new_pos for j in index_blocks[i]]
index_permutation_array_form = _af_invert(new_index_blocks_flat)
new_contraction_indices = [tuple(index_permutation_array_form[j] for j in i) for i in contraction_indices]
new_contraction_indices = _sort_contraction_indices(new_contraction_indices)
return _array_tensor_product(*new_args), new_contraction_indices
def _get_contraction_tuples(self):
r"""
Return tuples containing the argument index and position within the
argument of the index position.
Examples
========
>>> from sympy import MatrixSymbol
>>> from sympy.abc import N
>>> from sympy.tensor.array import tensorproduct, tensorcontraction
>>> A = MatrixSymbol("A", N, N)
>>> B = MatrixSymbol("B", N, N)
>>> cg = tensorcontraction(tensorproduct(A, B), (1, 2))
>>> cg._get_contraction_tuples()
[[(0, 1), (1, 0)]]
Notes
=====
Here the contraction pair `(1, 2)` meaning that the 2nd and 3rd indices
of the tensor product `A\otimes B` are contracted, has been transformed
into `(0, 1)` and `(1, 0)`, identifying the same indices in a different
notation. `(0, 1)` is the second index (1) of the first argument (i.e.
0 or `A`). `(1, 0)` is the first index (i.e. 0) of the second
argument (i.e. 1 or `B`).
"""
mapping = self._mapping
return [[mapping[j] for j in i] for i in self.contraction_indices]
@staticmethod
def _contraction_tuples_to_contraction_indices(expr, contraction_tuples):
# TODO: check that `expr` has `.subranks`:
ranks = expr.subranks
cumulative_ranks = [0] + list(accumulate(ranks))
return [tuple(cumulative_ranks[j]+k for j, k in i) for i in contraction_tuples]
@property
def free_indices(self):
return self._free_indices[:]
@property
def free_indices_to_position(self):
return dict(self._free_indices_to_position)
@property
def expr(self):
return self.args[0]
@property
def contraction_indices(self):
return self.args[1:]
def _contraction_indices_to_components(self):
expr = self.expr
if not isinstance(expr, ArrayTensorProduct):
raise NotImplementedError("only for contractions of tensor products")
ranks = expr.subranks
mapping = {}
counter = 0
for i, rank in enumerate(ranks):
for j in range(rank):
mapping[counter] = (i, j)
counter += 1
return mapping
def sort_args_by_name(self):
"""
Sort arguments in the tensor product so that their order is lexicographical.
Examples
========
>>> from sympy.tensor.array.expressions.from_matrix_to_array import convert_matrix_to_array
>>> from sympy import MatrixSymbol
>>> from sympy.abc import N
>>> A = MatrixSymbol("A", N, N)
>>> B = MatrixSymbol("B", N, N)
>>> C = MatrixSymbol("C", N, N)
>>> D = MatrixSymbol("D", N, N)
>>> cg = convert_matrix_to_array(C*D*A*B)
>>> cg
ArrayContraction(ArrayTensorProduct(A, D, C, B), (0, 3), (1, 6), (2, 5))
>>> cg.sort_args_by_name()
ArrayContraction(ArrayTensorProduct(A, D, B, C), (0, 3), (1, 4), (2, 7))
"""
expr = self.expr
if not isinstance(expr, ArrayTensorProduct):
return self
args = expr.args
sorted_data = sorted(enumerate(args), key=lambda x: default_sort_key(x[1]))
pos_sorted, args_sorted = zip(*sorted_data)
reordering_map = {i: pos_sorted.index(i) for i, arg in enumerate(args)}
contraction_tuples = self._get_contraction_tuples()
contraction_tuples = [[(reordering_map[j], k) for j, k in i] for i in contraction_tuples]
c_tp = _array_tensor_product(*args_sorted)
new_contr_indices = self._contraction_tuples_to_contraction_indices(
c_tp,
contraction_tuples
)
return _array_contraction(c_tp, *new_contr_indices)
def _get_contraction_links(self):
r"""
Returns a dictionary of links between arguments in the tensor product
being contracted.
See the example for an explanation of the values.
Examples
========
>>> from sympy import MatrixSymbol
>>> from sympy.abc import N
>>> from sympy.tensor.array.expressions.from_matrix_to_array import convert_matrix_to_array
>>> A = MatrixSymbol("A", N, N)
>>> B = MatrixSymbol("B", N, N)
>>> C = MatrixSymbol("C", N, N)
>>> D = MatrixSymbol("D", N, N)
Matrix multiplications are pairwise contractions between neighboring
matrices:
`A_{ij} B_{jk} C_{kl} D_{lm}`
>>> cg = convert_matrix_to_array(A*B*C*D)
>>> cg
ArrayContraction(ArrayTensorProduct(B, C, A, D), (0, 5), (1, 2), (3, 6))
>>> cg._get_contraction_links()
{0: {0: (2, 1), 1: (1, 0)}, 1: {0: (0, 1), 1: (3, 0)}, 2: {1: (0, 0)}, 3: {0: (1, 1)}}
This dictionary is interpreted as follows: argument in position 0 (i.e.
matrix `A`) has its second index (i.e. 1) contracted to `(1, 0)`, that
is argument in position 1 (matrix `B`) on the first index slot of `B`,
this is the contraction provided by the index `j` from `A`.
The argument in position 1 (that is, matrix `B`) has two contractions,
the ones provided by the indices `j` and `k`, respectively the first
and second indices (0 and 1 in the sub-dict). The link `(0, 1)` and
`(2, 0)` respectively. `(0, 1)` is the index slot 1 (the 2nd) of
argument in position 0 (that is, `A_{\ldot j}`), and so on.
"""
args, dlinks = _get_contraction_links([self], self.subranks, *self.contraction_indices)
return dlinks
def as_explicit(self):
expr = self.expr
if hasattr(expr, "as_explicit"):
expr = expr.as_explicit()
return tensorcontraction(expr, *self.contraction_indices)
class Reshape(_CodegenArrayAbstract):
"""
Reshape the dimensions of an array expression.
Examples
========
>>> from sympy.tensor.array.expressions import ArraySymbol, Reshape
>>> A = ArraySymbol("A", (6,))
>>> A.shape
(6,)
>>> Reshape(A, (3, 2)).shape
(3, 2)
Check the component-explicit forms:
>>> A.as_explicit()
[A[0], A[1], A[2], A[3], A[4], A[5]]
>>> Reshape(A, (3, 2)).as_explicit()
[[A[0], A[1]], [A[2], A[3]], [A[4], A[5]]]
"""
def __new__(cls, expr, shape):
expr = _sympify(expr)
if not isinstance(shape, Tuple):
shape = Tuple(*shape)
if Equality(Mul.fromiter(expr.shape), Mul.fromiter(shape)) == False:
raise ValueError("shape mismatch")
obj = Expr.__new__(cls, expr, shape)
obj._shape = tuple(shape)
obj._expr = expr
return obj
@property
def shape(self):
return self._shape
@property
def expr(self):
return self._expr
def doit(self, *args, **kwargs):
if kwargs.get("deep", True):
expr = self.expr.doit(*args, **kwargs)
else:
expr = self.expr
if isinstance(expr, (MatrixBase, NDimArray)):
return expr.reshape(*self.shape)
return Reshape(expr, self.shape)
def as_explicit(self):
ee = self.expr
if hasattr(ee, "as_explicit"):
ee = ee.as_explicit()
if isinstance(ee, MatrixBase):
from sympy import Array
ee = Array(ee)
elif isinstance(ee, MatrixExpr):
return self
return ee.reshape(*self.shape)
class _ArgE:
"""
The ``_ArgE`` object contains references to the array expression
(``.element``) and a list containing the information about index
contractions (``.indices``).
Index contractions are numbered and contracted indices show the number of
the contraction. Uncontracted indices have ``None`` value.
For example:
``_ArgE(M, [None, 3])``
This object means that expression ``M`` is part of an array contraction
and has two indices, the first is not contracted (value ``None``),
the second index is contracted to the 4th (i.e. number ``3``) group of the
array contraction object.
"""
indices: list[int | None]
def __init__(self, element, indices: list[int | None] | None = None):
self.element = element
if indices is None:
self.indices = [None for i in range(get_rank(element))]
else:
self.indices = indices
def __str__(self):
return "_ArgE(%s, %s)" % (self.element, self.indices)
__repr__ = __str__
class _IndPos:
"""
Index position, requiring two integers in the constructor:
- arg: the position of the argument in the tensor product,
- rel: the relative position of the index inside the argument.
"""
def __init__(self, arg: int, rel: int):
self.arg = arg
self.rel = rel
def __str__(self):
return "_IndPos(%i, %i)" % (self.arg, self.rel)
__repr__ = __str__
def __iter__(self):
yield from [self.arg, self.rel]
class _EditArrayContraction:
"""
Utility class to help manipulate array contraction objects.
This class takes as input an ``ArrayContraction`` object and turns it into
an editable object.
The field ``args_with_ind`` of this class is a list of ``_ArgE`` objects
which can be used to easily edit the contraction structure of the
expression.
Once editing is finished, the ``ArrayContraction`` object may be recreated
by calling the ``.to_array_contraction()`` method.
"""
def __init__(self, base_array: typing.Union[ArrayContraction, ArrayDiagonal, ArrayTensorProduct]):
expr: Basic
diagonalized: tuple[tuple[int, ...], ...]
contraction_indices: list[tuple[int]]
if isinstance(base_array, ArrayContraction):
mapping = _get_mapping_from_subranks(base_array.subranks)
expr = base_array.expr
contraction_indices = base_array.contraction_indices
diagonalized = ()
elif isinstance(base_array, ArrayDiagonal):
if isinstance(base_array.expr, ArrayContraction):
mapping = _get_mapping_from_subranks(base_array.expr.subranks)
expr = base_array.expr.expr
diagonalized = ArrayContraction._push_indices_down(base_array.expr.contraction_indices, base_array.diagonal_indices)
contraction_indices = base_array.expr.contraction_indices
elif isinstance(base_array.expr, ArrayTensorProduct):
mapping = {}
expr = base_array.expr
diagonalized = base_array.diagonal_indices
contraction_indices = []
else:
mapping = {}
expr = base_array.expr
diagonalized = base_array.diagonal_indices
contraction_indices = []
elif isinstance(base_array, ArrayTensorProduct):
expr = base_array
contraction_indices = []
diagonalized = ()
else:
raise NotImplementedError()
if isinstance(expr, ArrayTensorProduct):
args = list(expr.args)
else:
args = [expr]
args_with_ind: list[_ArgE] = [_ArgE(arg) for arg in args]
for i, contraction_tuple in enumerate(contraction_indices):
for j in contraction_tuple:
arg_pos, rel_pos = mapping[j]
args_with_ind[arg_pos].indices[rel_pos] = i
self.args_with_ind: list[_ArgE] = args_with_ind
self.number_of_contraction_indices: int = len(contraction_indices)
self._track_permutation: list[list[int]] | None = None
mapping = _get_mapping_from_subranks(base_array.subranks)
# Trick: add diagonalized indices as negative indices into the editor object:
for i, e in enumerate(diagonalized):
for j in e:
arg_pos, rel_pos = mapping[j]
self.args_with_ind[arg_pos].indices[rel_pos] = -1 - i
def insert_after(self, arg: _ArgE, new_arg: _ArgE):
pos = self.args_with_ind.index(arg)
self.args_with_ind.insert(pos + 1, new_arg)
def get_new_contraction_index(self):
self.number_of_contraction_indices += 1
return self.number_of_contraction_indices - 1
def refresh_indices(self):
updates = {}
for arg_with_ind in self.args_with_ind:
updates.update({i: -1 for i in arg_with_ind.indices if i is not None})
for i, e in enumerate(sorted(updates)):
updates[e] = i
self.number_of_contraction_indices = len(updates)
for arg_with_ind in self.args_with_ind:
arg_with_ind.indices = [updates.get(i, None) for i in arg_with_ind.indices]
def merge_scalars(self):
scalars = []
for arg_with_ind in self.args_with_ind:
if len(arg_with_ind.indices) == 0:
scalars.append(arg_with_ind)
for i in scalars:
self.args_with_ind.remove(i)
scalar = Mul.fromiter([i.element for i in scalars])
if len(self.args_with_ind) == 0:
self.args_with_ind.append(_ArgE(scalar))
else:
from sympy.tensor.array.expressions.from_array_to_matrix import _a2m_tensor_product
self.args_with_ind[0].element = _a2m_tensor_product(scalar, self.args_with_ind[0].element)
def to_array_contraction(self):
# Count the ranks of the arguments:
counter = 0
# Create a collector for the new diagonal indices:
diag_indices = defaultdict(list)
count_index_freq = Counter()
for arg_with_ind in self.args_with_ind:
count_index_freq.update(Counter(arg_with_ind.indices))
free_index_count = count_index_freq[None]
# Construct the inverse permutation:
inv_perm1 = []
inv_perm2 = []
# Keep track of which diagonal indices have already been processed:
done = set()
# Counter for the diagonal indices:
counter4 = 0
for arg_with_ind in self.args_with_ind:
# If some diagonalization axes have been removed, they should be
# permuted in order to keep the permutation.
# Add permutation here
counter2 = 0 # counter for the indices
for i in arg_with_ind.indices:
if i is None:
inv_perm1.append(counter4)
counter2 += 1
counter4 += 1
continue
if i >= 0:
continue
# Reconstruct the diagonal indices:
diag_indices[-1 - i].append(counter + counter2)
if count_index_freq[i] == 1 and i not in done:
inv_perm1.append(free_index_count - 1 - i)
done.add(i)
elif i not in done:
inv_perm2.append(free_index_count - 1 - i)
done.add(i)
counter2 += 1
# Remove negative indices to restore a proper editor object:
arg_with_ind.indices = [i if i is not None and i >= 0 else None for i in arg_with_ind.indices]
counter += len([i for i in arg_with_ind.indices if i is None or i < 0])
inverse_permutation = inv_perm1 + inv_perm2
permutation = _af_invert(inverse_permutation)
# Get the diagonal indices after the detection of HadamardProduct in the expression:
diag_indices_filtered = [tuple(v) for v in diag_indices.values() if len(v) > 1]
self.merge_scalars()
self.refresh_indices()
args = [arg.element for arg in self.args_with_ind]
contraction_indices = self.get_contraction_indices()
expr = _array_contraction(_array_tensor_product(*args), *contraction_indices)
expr2 = _array_diagonal(expr, *diag_indices_filtered)
if self._track_permutation is not None:
permutation2 = _af_invert([j for i in self._track_permutation for j in i])
expr2 = _permute_dims(expr2, permutation2)
expr3 = _permute_dims(expr2, permutation)
return expr3
def get_contraction_indices(self) -> list[list[int]]:
contraction_indices: list[list[int]] = [[] for i in range(self.number_of_contraction_indices)]
current_position: int = 0
for arg_with_ind in self.args_with_ind:
for j in arg_with_ind.indices:
if j is not None:
contraction_indices[j].append(current_position)
current_position += 1
return contraction_indices
def get_mapping_for_index(self, ind) -> list[_IndPos]:
if ind >= self.number_of_contraction_indices:
raise ValueError("index value exceeding the index range")
positions: list[_IndPos] = []
for i, arg_with_ind in enumerate(self.args_with_ind):
for j, arg_ind in enumerate(arg_with_ind.indices):
if ind == arg_ind:
positions.append(_IndPos(i, j))
return positions
def get_contraction_indices_to_ind_rel_pos(self) -> list[list[_IndPos]]:
contraction_indices: list[list[_IndPos]] = [[] for i in range(self.number_of_contraction_indices)]
for i, arg_with_ind in enumerate(self.args_with_ind):
for j, ind in enumerate(arg_with_ind.indices):
if ind is not None:
contraction_indices[ind].append(_IndPos(i, j))
return contraction_indices
def count_args_with_index(self, index: int) -> int:
"""
Count the number of arguments that have the given index.
"""
counter: int = 0
for arg_with_ind in self.args_with_ind:
if index in arg_with_ind.indices:
counter += 1
return counter
def get_args_with_index(self, index: int) -> list[_ArgE]:
"""
Get a list of arguments having the given index.
"""
ret: list[_ArgE] = [i for i in self.args_with_ind if index in i.indices]
return ret
@property
def number_of_diagonal_indices(self):
data = set()
for arg in self.args_with_ind:
data.update({i for i in arg.indices if i is not None and i < 0})
return len(data)
def track_permutation_start(self):
permutation = []
perm_diag = []
counter = 0
counter2 = -1
for arg_with_ind in self.args_with_ind:
perm = []
for i in arg_with_ind.indices:
if i is not None:
if i < 0:
perm_diag.append(counter2)
counter2 -= 1
continue
perm.append(counter)
counter += 1
permutation.append(perm)
max_ind = max(max(i) if i else -1 for i in permutation) if permutation else -1
perm_diag = [max_ind - i for i in perm_diag]
self._track_permutation = permutation + [perm_diag]
def track_permutation_merge(self, destination: _ArgE, from_element: _ArgE):
index_destination = self.args_with_ind.index(destination)
index_element = self.args_with_ind.index(from_element)
self._track_permutation[index_destination].extend(self._track_permutation[index_element]) # type: ignore
self._track_permutation.pop(index_element) # type: ignore
def get_absolute_free_range(self, arg: _ArgE) -> typing.Tuple[int, int]:
"""
Return the range of the free indices of the arg as absolute positions
among all free indices.
"""
counter = 0
for arg_with_ind in self.args_with_ind:
number_free_indices = len([i for i in arg_with_ind.indices if i is None])
if arg_with_ind == arg:
return counter, counter + number_free_indices
counter += number_free_indices
raise IndexError("argument not found")
def get_absolute_range(self, arg: _ArgE) -> typing.Tuple[int, int]:
"""
Return the absolute range of indices for arg, disregarding dummy
indices.
"""
counter = 0
for arg_with_ind in self.args_with_ind:
number_indices = len(arg_with_ind.indices)
if arg_with_ind == arg:
return counter, counter + number_indices
counter += number_indices
raise IndexError("argument not found")
def get_rank(expr):
if isinstance(expr, (MatrixExpr, MatrixElement)):
return 2
if isinstance(expr, _CodegenArrayAbstract):
return len(expr.shape)
if isinstance(expr, NDimArray):
return expr.rank()
if isinstance(expr, Indexed):
return expr.rank
if isinstance(expr, IndexedBase):
shape = expr.shape
if shape is None:
return -1
else:
return len(shape)
if hasattr(expr, "shape"):
return len(expr.shape)
return 0
def _get_subrank(expr):
if isinstance(expr, _CodegenArrayAbstract):
return expr.subrank()
return get_rank(expr)
def _get_subranks(expr):
if isinstance(expr, _CodegenArrayAbstract):
return expr.subranks
else:
return [get_rank(expr)]
def get_shape(expr):
if hasattr(expr, "shape"):
return expr.shape
return ()
def nest_permutation(expr):
if isinstance(expr, PermuteDims):
return expr.nest_permutation()
else:
return expr
def _array_tensor_product(*args, **kwargs):
return ArrayTensorProduct(*args, canonicalize=True, **kwargs)
def _array_contraction(expr, *contraction_indices, **kwargs):
return ArrayContraction(expr, *contraction_indices, canonicalize=True, **kwargs)
def _array_diagonal(expr, *diagonal_indices, **kwargs):
return ArrayDiagonal(expr, *diagonal_indices, canonicalize=True, **kwargs)
def _permute_dims(expr, permutation, **kwargs):
return PermuteDims(expr, permutation, canonicalize=True, **kwargs)
def _array_add(*args, **kwargs):
return ArrayAdd(*args, canonicalize=True, **kwargs)
def _get_array_element_or_slice(expr, indices):
return ArrayElement(expr, indices)
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