You can not select more than 25 topics
Topics must start with a letter or number, can include dashes ('-') and can be up to 35 characters long.
257 lines
8.6 KiB
257 lines
8.6 KiB
from sympy import permutedims
|
|
from sympy.core.numbers import Number
|
|
from sympy.core.singleton import S
|
|
from sympy.core.symbol import Symbol
|
|
from sympy.core.sympify import sympify
|
|
from sympy.tensor.tensor import Tensor, TensExpr, TensAdd, TensMul
|
|
|
|
|
|
class PartialDerivative(TensExpr):
|
|
"""
|
|
Partial derivative for tensor expressions.
|
|
|
|
Examples
|
|
========
|
|
|
|
>>> from sympy.tensor.tensor import TensorIndexType, TensorHead
|
|
>>> from sympy.tensor.toperators import PartialDerivative
|
|
>>> from sympy import symbols
|
|
>>> L = TensorIndexType("L")
|
|
>>> A = TensorHead("A", [L])
|
|
>>> B = TensorHead("B", [L])
|
|
>>> i, j, k = symbols("i j k")
|
|
|
|
>>> expr = PartialDerivative(A(i), A(j))
|
|
>>> expr
|
|
PartialDerivative(A(i), A(j))
|
|
|
|
The ``PartialDerivative`` object behaves like a tensorial expression:
|
|
|
|
>>> expr.get_indices()
|
|
[i, -j]
|
|
|
|
Notice that the deriving variables have opposite valence than the
|
|
printed one: ``A(j)`` is printed as covariant, but the index of the
|
|
derivative is actually contravariant, i.e. ``-j``.
|
|
|
|
Indices can be contracted:
|
|
|
|
>>> expr = PartialDerivative(A(i), A(i))
|
|
>>> expr
|
|
PartialDerivative(A(L_0), A(L_0))
|
|
>>> expr.get_indices()
|
|
[L_0, -L_0]
|
|
|
|
The method ``.get_indices()`` always returns all indices (even the
|
|
contracted ones). If only uncontracted indices are needed, call
|
|
``.get_free_indices()``:
|
|
|
|
>>> expr.get_free_indices()
|
|
[]
|
|
|
|
Nested partial derivatives are flattened:
|
|
|
|
>>> expr = PartialDerivative(PartialDerivative(A(i), A(j)), A(k))
|
|
>>> expr
|
|
PartialDerivative(A(i), A(j), A(k))
|
|
>>> expr.get_indices()
|
|
[i, -j, -k]
|
|
|
|
Replace a derivative with array values:
|
|
|
|
>>> from sympy.abc import x, y
|
|
>>> from sympy import sin, log
|
|
>>> compA = [sin(x), log(x)*y**3]
|
|
>>> compB = [x, y]
|
|
>>> expr = PartialDerivative(A(i), B(j))
|
|
>>> expr.replace_with_arrays({A(i): compA, B(i): compB})
|
|
[[cos(x), 0], [y**3/x, 3*y**2*log(x)]]
|
|
|
|
The returned array is indexed by `(i, -j)`.
|
|
|
|
Be careful that other SymPy modules put the indices of the deriving
|
|
variables before the indices of the derivand in the derivative result.
|
|
For example:
|
|
|
|
>>> expr.get_free_indices()
|
|
[i, -j]
|
|
|
|
>>> from sympy import Matrix, Array
|
|
>>> Matrix(compA).diff(Matrix(compB)).reshape(2, 2)
|
|
[[cos(x), y**3/x], [0, 3*y**2*log(x)]]
|
|
>>> Array(compA).diff(Array(compB))
|
|
[[cos(x), y**3/x], [0, 3*y**2*log(x)]]
|
|
|
|
These are the transpose of the result of ``PartialDerivative``,
|
|
as the matrix and the array modules put the index `-j` before `i` in the
|
|
derivative result. An array read with index order `(-j, i)` is indeed the
|
|
transpose of the same array read with index order `(i, -j)`. By specifying
|
|
the index order to ``.replace_with_arrays`` one can get a compatible
|
|
expression:
|
|
|
|
>>> expr.replace_with_arrays({A(i): compA, B(i): compB}, [-j, i])
|
|
[[cos(x), y**3/x], [0, 3*y**2*log(x)]]
|
|
"""
|
|
|
|
def __new__(cls, expr, *variables):
|
|
|
|
# Flatten:
|
|
if isinstance(expr, PartialDerivative):
|
|
variables = expr.variables + variables
|
|
expr = expr.expr
|
|
|
|
args, indices, free, dum = cls._contract_indices_for_derivative(
|
|
S(expr), variables)
|
|
|
|
obj = TensExpr.__new__(cls, *args)
|
|
|
|
obj._indices = indices
|
|
obj._free = free
|
|
obj._dum = dum
|
|
return obj
|
|
|
|
@property
|
|
def coeff(self):
|
|
return S.One
|
|
|
|
@property
|
|
def nocoeff(self):
|
|
return self
|
|
|
|
@classmethod
|
|
def _contract_indices_for_derivative(cls, expr, variables):
|
|
variables_opposite_valence = []
|
|
|
|
for i in variables:
|
|
if isinstance(i, Tensor):
|
|
i_free_indices = i.get_free_indices()
|
|
variables_opposite_valence.append(
|
|
i.xreplace({k: -k for k in i_free_indices}))
|
|
elif isinstance(i, Symbol):
|
|
variables_opposite_valence.append(i)
|
|
|
|
args, indices, free, dum = TensMul._tensMul_contract_indices(
|
|
[expr] + variables_opposite_valence, replace_indices=True)
|
|
|
|
for i in range(1, len(args)):
|
|
args_i = args[i]
|
|
if isinstance(args_i, Tensor):
|
|
i_indices = args[i].get_free_indices()
|
|
args[i] = args[i].xreplace({k: -k for k in i_indices})
|
|
|
|
return args, indices, free, dum
|
|
|
|
def doit(self, **hints):
|
|
args, indices, free, dum = self._contract_indices_for_derivative(self.expr, self.variables)
|
|
|
|
obj = self.func(*args)
|
|
obj._indices = indices
|
|
obj._free = free
|
|
obj._dum = dum
|
|
|
|
return obj
|
|
|
|
def _expand_partial_derivative(self):
|
|
args, indices, free, dum = self._contract_indices_for_derivative(self.expr, self.variables)
|
|
|
|
obj = self.func(*args)
|
|
obj._indices = indices
|
|
obj._free = free
|
|
obj._dum = dum
|
|
|
|
result = obj
|
|
|
|
if not args[0].free_symbols:
|
|
return S.Zero
|
|
elif isinstance(obj.expr, TensAdd):
|
|
# take care of sums of multi PDs
|
|
result = obj.expr.func(*[
|
|
self.func(a, *obj.variables)._expand_partial_derivative()
|
|
for a in result.expr.args])
|
|
elif isinstance(obj.expr, TensMul):
|
|
# take care of products of multi PDs
|
|
if len(obj.variables) == 1:
|
|
# derivative with respect to single variable
|
|
terms = []
|
|
mulargs = list(obj.expr.args)
|
|
for ind in range(len(mulargs)):
|
|
if not isinstance(sympify(mulargs[ind]), Number):
|
|
# a number coefficient is not considered for
|
|
# expansion of PartialDerivative
|
|
d = self.func(mulargs[ind], *obj.variables)._expand_partial_derivative()
|
|
terms.append(TensMul(*(mulargs[:ind]
|
|
+ [d]
|
|
+ mulargs[(ind + 1):])))
|
|
result = TensAdd.fromiter(terms)
|
|
else:
|
|
# derivative with respect to multiple variables
|
|
# decompose:
|
|
# partial(expr, (u, v))
|
|
# = partial(partial(expr, u).doit(), v).doit()
|
|
result = obj.expr # init with expr
|
|
for v in obj.variables:
|
|
result = self.func(result, v)._expand_partial_derivative()
|
|
# then throw PD on it
|
|
|
|
return result
|
|
|
|
def _perform_derivative(self):
|
|
result = self.expr
|
|
for v in self.variables:
|
|
if isinstance(result, TensExpr):
|
|
result = result._eval_partial_derivative(v)
|
|
else:
|
|
if v._diff_wrt:
|
|
result = result._eval_derivative(v)
|
|
else:
|
|
result = S.Zero
|
|
return result
|
|
|
|
def get_indices(self):
|
|
return self._indices
|
|
|
|
def get_free_indices(self):
|
|
free = sorted(self._free, key=lambda x: x[1])
|
|
return [i[0] for i in free]
|
|
|
|
def _replace_indices(self, repl):
|
|
expr = self.expr.xreplace(repl)
|
|
mirrored = {-k: -v for k, v in repl.items()}
|
|
variables = [i.xreplace(mirrored) for i in self.variables]
|
|
return self.func(expr, *variables)
|
|
|
|
@property
|
|
def expr(self):
|
|
return self.args[0]
|
|
|
|
@property
|
|
def variables(self):
|
|
return self.args[1:]
|
|
|
|
def _extract_data(self, replacement_dict):
|
|
from .array import derive_by_array, tensorcontraction
|
|
indices, array = self.expr._extract_data(replacement_dict)
|
|
for variable in self.variables:
|
|
var_indices, var_array = variable._extract_data(replacement_dict)
|
|
var_indices = [-i for i in var_indices]
|
|
coeff_array, var_array = zip(*[i.as_coeff_Mul() for i in var_array])
|
|
dim_before = len(array.shape)
|
|
array = derive_by_array(array, var_array)
|
|
dim_after = len(array.shape)
|
|
dim_increase = dim_after - dim_before
|
|
array = permutedims(array, [i + dim_increase for i in range(dim_before)] + list(range(dim_increase)))
|
|
array = array.as_mutable()
|
|
varindex = var_indices[0]
|
|
# Remove coefficients of base vector:
|
|
coeff_index = [0] + [slice(None) for i in range(len(indices))]
|
|
for i, coeff in enumerate(coeff_array):
|
|
coeff_index[0] = i
|
|
array[tuple(coeff_index)] /= coeff
|
|
if -varindex in indices:
|
|
pos = indices.index(-varindex)
|
|
array = tensorcontraction(array, (0, pos+1))
|
|
indices.pop(pos)
|
|
else:
|
|
indices.append(varindex)
|
|
return indices, array
|