"""
Differential operators, for all sympy expressions
For multivector-customized differential operators, see :class:`galgebra.mv.Dop`.
"""
import copy
import numbers
import warnings
from typing import List, Tuple, Any, Iterable
from sympy import Symbol, S, Add, simplify, diff, Expr, Dummy
from . import printer
from . import metric
from .printer import ZERO_STR
def _consolidate_terms(terms):
"""
Remove zero coefs and consolidate coefs with repeated pdiffs.
"""
new_coefs = []
new_pdiffs = []
for coef, pd in terms:
if coef != S.Zero:
if pd in new_pdiffs:
index = new_pdiffs.index(pd)
new_coefs[index] += coef
else:
new_coefs.append(coef)
new_pdiffs.append(pd)
return tuple(zip(new_coefs, new_pdiffs))
def _merge_terms(terms1, terms2):
""" Concatenate and consolidate two sets of already-consolidated terms """
pdiffs1 = [pdiff for _, pdiff in terms1]
pdiffs2 = [pdiff for _, pdiff in terms2]
pdiffs = pdiffs1 + [x for x in pdiffs2 if x not in pdiffs1]
coefs = len(pdiffs) * [S.Zero]
for coef, pdiff in terms1:
index = pdiffs.index(pdiff)
coefs[index] += coef
for coef, pdiff in terms2:
index = pdiffs.index(pdiff)
coefs[index] += coef
# remove zeros
return [(coef, pdiff) for coef, pdiff in zip(coefs, pdiffs) if coef != S.Zero]
def _eval_derivative_n_times_terms(terms, x, n):
for i in range(n):
new_terms = []
for k, term in enumerate(terms):
dc = _basic_diff(term[0], x)
pd = _basic_diff(term[1], x)
# print 'D0, term, dc, pd =', D0, term, dc, pd
if dc != 0:
new_terms.append((dc, term[1]))
if pd != 0:
new_terms.append((term[0], pd))
terms = new_terms
return _consolidate_terms(terms)
################ Scalar Partial Differential Operator Class ############
class _BaseDop(printer.GaPrintable):
""" Base class for differential operators - used to avoid accidental promotion """
pass
[docs]
class Sdop(_BaseDop):
"""
Scalar differential operator is of the form (Einstein summation)
.. math:: D = c_{i}*D_{i}
where the :math:`c_{i}`'s are scalar coefficient (they could be functions)
and the :math:`D_{i}`'s are partial differential operators (:class:`Pdop`).
Attributes
----------
terms : tuple of tuple
the structure :math:`((c_{1},D_{1}),(c_{2},D_{2}), ...)`
"""
def TSimplify(self):
return Sdop([
(metric.Simp.apply(coef), pdiff) for coef, pdiff in self.terms
])
[docs]
@staticmethod
def consolidate_coefs(sdop):
"""
Remove zero coefs and consolidate coefs with repeated pdiffs.
"""
if isinstance(sdop, Sdop):
return Sdop(_consolidate_terms(sdop.terms))
else:
return _consolidate_terms(sdop)
def simplify(self, modes=simplify):
return Sdop([
(metric.apply_function_list(modes, coef), pdiff)
for coef, pdiff in self.terms
])
def _with_sorted_terms(self):
new_terms = sorted(self.terms, key=lambda term: Pdop.sort_key(term[1]))
return Sdop(new_terms)
def _sympystr(self, print_obj):
if len(self.terms) == 0:
return ZERO_STR
self = self._with_sorted_terms()
s = ''
for coef, pdop in self.terms:
coef_str = print_obj._print(coef)
pd_str = print_obj._print(pdop)
if coef == S.One:
s += pd_str
elif coef == S.NegativeOne:
s += '-' + pd_str
else:
if isinstance(coef, Add):
s += '(' + coef_str + ')*' + pd_str
else:
s += coef_str + '*' + pd_str
s += ' + '
s = s.replace('+ -', '- ')
s = s[:-3]
return s
def _latex(self, print_obj):
if len(self.terms) == 0:
return ZERO_STR
self = self._with_sorted_terms()
s = ''
for coef, pdop in self.terms:
coef_str = print_obj._print(coef)
pd_str = print_obj._print(pdop)
if coef == S.One:
if pd_str == '':
s += '1'
else:
s += pd_str
elif coef == S.NegativeOne:
if pd_str == '':
s += '-1'
else:
s += '-' + pd_str
else:
if isinstance(coef, Add):
s += r'\left ( ' + coef_str + r'\right ) ' + pd_str
else:
s += coef_str + ' ' + pd_str
s += ' + '
s = s.replace('+ -', '- ')
return s[:-3]
def __init_from_symbol(self, symbol: Symbol) -> None:
self.terms = ((S.One, Pdop(symbol)),)
def __init_from_coef_and_pdiffs(self, coefs: List[Any], pdiffs: List['Pdop']) -> None:
if not isinstance(coefs, list) or not isinstance(pdiffs, list):
raise TypeError("coefs and pdiffs must be lists")
if len(coefs) != len(pdiffs):
raise ValueError('In Sdop.__init__ coefficent list and Pdop list must be same length.')
self.terms = tuple(zip(coefs, pdiffs))
def __init_from_terms(self, terms: Iterable[Tuple[Any, 'Pdop']]) -> None:
self.terms = tuple(terms)
def __init__(self, *args):
if len(args) == 1:
if isinstance(args[0], Symbol):
self.__init_from_symbol(*args)
elif isinstance(args[0], (list, tuple)):
self.__init_from_terms(*args)
else:
raise TypeError(
"A symbol or sequence is required (got type {})"
.format(type(args[0]).__name__))
elif len(args) == 2:
self.__init_from_coef_and_pdiffs(*args)
else:
raise TypeError(
"Sdop() takes from 1 to 2 positional arguments but {} were "
"given".format(len(args)))
def __call__(self, arg):
# Ensure that we return the right type even when there are no terms - we
# do this by adding `0 * d(arg)/d(nonexistant)`, which must be zero, but
# will be a zero of the right type.
dummy_var = Dummy('nonexistant')
terms = self.terms or ((S.Zero, Pdop(dummy_var)),)
return sum([coef * pdiff(arg) for coef, pdiff in terms])
def __neg__(self):
return Sdop([(-coef, pdiff) for coef, pdiff in self.terms])
@staticmethod
def Add(sdop1, sdop2):
if isinstance(sdop1, Sdop) and isinstance(sdop2, Sdop):
return Sdop(_merge_terms(sdop1.terms, sdop2.terms))
else:
# convert values to multiplicative operators
if not isinstance(sdop2, _BaseDop):
sdop2 = Sdop([(sdop2, Pdop({}))])
elif not isinstance(sdop1, _BaseDop):
sdop1 = Sdop([(sdop1, Pdop({}))])
else:
return NotImplemented
return Sdop.Add(sdop1, sdop2)
def __eq__(self, other):
if isinstance(other, Sdop):
diff = self - other
return len(diff.terms) == 0
else:
return NotImplemented
def __add__(self, sdop):
return Sdop.Add(self, sdop)
def __radd__(self, sdop):
return Sdop.Add(sdop, self)
def __sub__(self, sdop):
return Sdop.Add(self, -sdop)
def __rsub__(self, sdop):
return Sdop.Add(-self, sdop)
def __mul__(self, sdopr):
# alias for applying the operator
return self.__call__(sdopr)
def __rmul__(self, sdop):
return Sdop([(sdop * coef, pdiff) for coef, pdiff in self.terms])
def _eval_derivative_n_times(self, x, n):
return Sdop(_eval_derivative_n_times_terms(self.terms, x, n))
#################### Partial Derivative Operator Class #################
def _basic_diff(f, x, n=1):
""" Simple wrapper for `diff` that works for our types too """
if isinstance(f, (Expr, Symbol, numbers.Number)): # f is sympy expression
return diff(f, x, n)
elif hasattr(f, '_eval_derivative_n_times'):
# one of our types
return f._eval_derivative_n_times(x, n)
else:
raise ValueError('In_basic_diff type(arg) = ' + str(type(f)) + ' not allowed.')
[docs]
class Pdop(_BaseDop):
r"""
Partial derivative operatorp.
The partial derivatives are of the form
.. math::
\partial_{i_{1}...i_{n}} =
\frac{\partial^{i_{1}+...+i_{n}}}{\partial{x_{1}^{i_{1}}}...\partial{x_{n}^{i_{n}}}}.
If :math:`i_{j} = 0` then the partial derivative does not contain the
:math:`x^{i_{j}}` coordinate.
Attributes
----------
pdiffs : dict
A dictionary where coordinates are keys and key value are the number of
times one differentiates with respect to the key.
order : int
Total number of differentiations.
When this is zero (i.e. when :attr:`pdiffs` is ``{}``) then this object
is the identity operator, and returns its operand unchanged.
"""
def sort_key(self, order=None):
return (
# lower order derivatives first
self.order,
# sorted by symbol after that, after expansion
sorted([
x.sort_key(order)
for x, k in self.pdiffs.items()
for i in range(k)
])
)
def __eq__(self, A):
if isinstance(A, Pdop) and self.pdiffs == A.pdiffs:
return True
else:
if len(self.pdiffs) == 0 and A == S.One:
return True
return False
def __init__(self, __arg):
"""
The partial differential operator is a partial derivative with
respect to a set of real symbols (variables).
"""
# galgebra 0.4.5
if __arg is None:
warnings.warn(
"`Pdop(None)` is deprecated, use `Pdop({})` instead",
DeprecationWarning, stacklevel=2)
__arg = {}
if isinstance(__arg, dict): # Pdop defined by dictionary
self.pdiffs = __arg
elif isinstance(__arg, Symbol): # First order derivative with respect to symbol
self.pdiffs = {__arg: 1}
else:
raise TypeError('A dictionary or symbol is required, got {!r}'.format(__arg))
self.order = sum(self.pdiffs.values())
def _eval_derivative_n_times(self, x, n) -> 'Pdop': # pdiff(self)
# d is partial derivative
pdiffs = copy.copy(self.pdiffs)
if x in pdiffs:
pdiffs[x] += n
else:
pdiffs[x] = n
return Pdop(pdiffs)
def __call__(self, arg):
"""
Calculate nth order partial derivative (order defined by
self) of expression
"""
for x, n in self.pdiffs.items():
arg = _basic_diff(arg, x, n)
return arg
def __mul__(self, other): # functional product of self and arg (self*arg)
return self(other)
def __rmul__(self, other): # functional product of arg and self (arg*self)
assert not isinstance(other, Pdop)
return Sdop([(other, self)])
def _sympystr(self, print_obj):
if self.order == 0:
return 'D{}'
s = 'D'
for x in self.pdiffs:
s += '{' + print_obj._print(x) + '}'
n = self.pdiffs[x]
if n > 1:
s += '^' + print_obj._print(n)
return s
def _latex(self, print_obj):
if self.order == 0:
return ''
s = r'\frac{\partial'
if self.order > 1:
s += '^{' + print_obj._print(self.order) + '}'
s += '}{'
keys = list(self.pdiffs.keys())
keys.sort(key=lambda x: x.sort_key())
for key in keys:
i = self.pdiffs[key]
s += r'\partial ' + print_obj._print(key)
if i > 1:
s += '^{' + print_obj._print(i) + '}'
s += '}'
return s