Source code for sympy.core.basic

"""Base class for all the objects in SymPy"""
from __future__ import print_function, division
from collections import defaultdict
from itertools import chain

from .assumptions import BasicMeta, ManagedProperties
from .cache import cacheit
from .sympify import _sympify, sympify, SympifyError
from .compatibility import (iterable, Iterator, ordered,
    string_types, with_metaclass, zip_longest, range, PY3, Mapping)
from .singleton import S

from inspect import getmro

def as_Basic(expr):
    """Return expr as a Basic instance using strict sympify
    or raise a TypeError; this is just a wrapper to _sympify,
    raising a TypeError instead of a SympifyError."""
    from sympy.utilities.misc import func_name
        return _sympify(expr)
    except SympifyError:
        raise TypeError(
            'Argument must be a Basic object, not `%s`' % func_name(

[docs]class Basic(with_metaclass(ManagedProperties)): """ Base class for all objects in SymPy. Conventions: 1) Always use ``.args``, when accessing parameters of some instance: >>> from sympy import cot >>> from import x, y >>> cot(x).args (x,) >>> cot(x).args[0] x >>> (x*y).args (x, y) >>> (x*y).args[1] y 2) Never use internal methods or variables (the ones prefixed with ``_``): >>> cot(x)._args # do not use this, use cot(x).args instead (x,) """ __slots__ = ['_mhash', # hash value '_args', # arguments '_assumptions' ] # To be overridden with True in the appropriate subclasses is_number = False is_Atom = False is_Symbol = False is_symbol = False is_Indexed = False is_Dummy = False is_Wild = False is_Function = False is_Add = False is_Mul = False is_Pow = False is_Number = False is_Float = False is_Rational = False is_Integer = False is_NumberSymbol = False is_Order = False is_Derivative = False is_Piecewise = False is_Poly = False is_AlgebraicNumber = False is_Relational = False is_Equality = False is_Boolean = False is_Not = False is_Matrix = False is_Vector = False is_Point = False is_MatAdd = False is_MatMul = False def __new__(cls, *args): obj = object.__new__(cls) obj._assumptions = cls.default_assumptions obj._mhash = None # will be set by __hash__ method. obj._args = args # all items in args must be Basic objects return obj def copy(self): return self.func(*self.args) def __reduce_ex__(self, proto): """ Pickling support.""" return type(self), self.__getnewargs__(), self.__getstate__() def __getnewargs__(self): return self.args def __getstate__(self): return {} def __setstate__(self, state): for k, v in state.items(): setattr(self, k, v) def __hash__(self): # hash cannot be cached using cache_it because infinite recurrence # occurs as hash is needed for setting cache dictionary keys h = self._mhash if h is None: h = hash((type(self).__name__,) + self._hashable_content()) self._mhash = h return h def _hashable_content(self): """Return a tuple of information about self that can be used to compute the hash. If a class defines additional attributes, like ``name`` in Symbol, then this method should be updated accordingly to return such relevant attributes. Defining more than _hashable_content is necessary if __eq__ has been defined by a class. See note about this in Basic.__eq__.""" return self._args @property def assumptions0(self): """ Return object `type` assumptions. For example: Symbol('x', real=True) Symbol('x', integer=True) are different objects. In other words, besides Python type (Symbol in this case), the initial assumptions are also forming their typeinfo. Examples ======== >>> from sympy import Symbol >>> from import x >>> x.assumptions0 {'commutative': True} >>> x = Symbol("x", positive=True) >>> x.assumptions0 {'commutative': True, 'complex': True, 'hermitian': True, 'imaginary': False, 'negative': False, 'nonnegative': True, 'nonpositive': False, 'nonzero': True, 'positive': True, 'real': True, 'zero': False} """ return {}
[docs] def compare(self, other): """ Return -1, 0, 1 if the object is smaller, equal, or greater than other. Not in the mathematical sense. If the object is of a different type from the "other" then their classes are ordered according to the sorted_classes list. Examples ======== >>> from import x, y >>> -1 >>> 0 >>> 1 """ # all redefinitions of __cmp__ method should start with the # following lines: if self is other: return 0 n1 = self.__class__ n2 = other.__class__ c = (n1 > n2) - (n1 < n2) if c: return c # st = self._hashable_content() ot = other._hashable_content() c = (len(st) > len(ot)) - (len(st) < len(ot)) if c: return c for l, r in zip(st, ot): l = Basic(*l) if isinstance(l, frozenset) else l r = Basic(*r) if isinstance(r, frozenset) else r if isinstance(l, Basic): c = else: c = (l > r) - (l < r) if c: return c return 0
@staticmethod def _compare_pretty(a, b): from sympy.series.order import Order if isinstance(a, Order) and not isinstance(b, Order): return 1 if not isinstance(a, Order) and isinstance(b, Order): return -1 if a.is_Rational and b.is_Rational: l = a.p * b.q r = b.p * a.q return (l > r) - (l < r) else: from sympy.core.symbol import Wild p1, p2, p3 = Wild("p1"), Wild("p2"), Wild("p3") r_a = a.match(p1 * p2**p3) if r_a and p3 in r_a: a3 = r_a[p3] r_b = b.match(p1 * p2**p3) if r_b and p3 in r_b: b3 = r_b[p3] c =, b3) if c != 0: return c return, b)
[docs] @classmethod def fromiter(cls, args, **assumptions): """ Create a new object from an iterable. This is a convenience function that allows one to create objects from any iterable, without having to convert to a list or tuple first. Examples ======== >>> from sympy import Tuple >>> Tuple.fromiter(i for i in range(5)) (0, 1, 2, 3, 4) """ return cls(*tuple(args), **assumptions)
[docs] @classmethod def class_key(cls): """Nice order of classes. """ return 5, 0, cls.__name__
[docs] @cacheit def sort_key(self, order=None): """ Return a sort key. Examples ======== >>> from sympy.core import S, I >>> sorted([S(1)/2, I, -I], key=lambda x: x.sort_key()) [1/2, -I, I] >>> S("[x, 1/x, 1/x**2, x**2, x**(1/2), x**(1/4), x**(3/2)]") [x, 1/x, x**(-2), x**2, sqrt(x), x**(1/4), x**(3/2)] >>> sorted(_, key=lambda x: x.sort_key()) [x**(-2), 1/x, x**(1/4), sqrt(x), x, x**(3/2), x**2] """ # XXX: remove this when issue 5169 is fixed def inner_key(arg): if isinstance(arg, Basic): return arg.sort_key(order) else: return arg args = self._sorted_args args = len(args), tuple([inner_key(arg) for arg in args]) return self.class_key(), args, S.One.sort_key(), S.One
def __eq__(self, other): """Return a boolean indicating whether a == b on the basis of their symbolic trees. This is the same as == 0 but faster. Notes ===== If a class that overrides __eq__() needs to retain the implementation of __hash__() from a parent class, the interpreter must be told this explicitly by setting __hash__ = <ParentClass>.__hash__. Otherwise the inheritance of __hash__() will be blocked, just as if __hash__ had been explicitly set to None. References ========== from """ if self is other: return True tself = type(self) tother = type(other) if type(self) is not type(other): try: other = _sympify(other) tother = type(other) except SympifyError: return NotImplemented # As long as we have the ordering of classes (sympy.core), # comparing types will be slow in Python 2, because it uses # __cmp__. Until we can remove it # (, we only compare # types in Python 2 directly if they actually have __ne__. if PY3 or type(tself).__ne__ is not type.__ne__: if tself != tother: return False elif tself is not tother: return False return self._hashable_content() == other._hashable_content() def __ne__(self, other): """a != b -> Compare two symbolic trees and see whether they are different this is the same as: != 0 but faster """ return not self == other
[docs] def dummy_eq(self, other, symbol=None): """ Compare two expressions and handle dummy symbols. Examples ======== >>> from sympy import Dummy >>> from import x, y >>> u = Dummy('u') >>> (u**2 + 1).dummy_eq(x**2 + 1) True >>> (u**2 + 1) == (x**2 + 1) False >>> (u**2 + y).dummy_eq(x**2 + y, x) True >>> (u**2 + y).dummy_eq(x**2 + y, y) False """ dummy_symbols = [s for s in self.free_symbols if s.is_Dummy] if not dummy_symbols: return self == other elif len(dummy_symbols) == 1: dummy = dummy_symbols.pop() else: raise ValueError( "only one dummy symbol allowed on the left-hand side") if symbol is None: symbols = other.free_symbols if not symbols: return self == other elif len(symbols) == 1: symbol = symbols.pop() else: raise ValueError("specify a symbol in which expressions should be compared") tmp = dummy.__class__() return self.subs(dummy, tmp) == other.subs(symbol, tmp)
# Note, we always use the default ordering (lex) in __str__ and __repr__, # regardless of the global setting. See issue 5487. def __repr__(self): """Method to return the string representation. Return the expression as a string. """ from sympy.printing import sstr return sstr(self, order=None) def __str__(self): from sympy.printing import sstr return sstr(self, order=None)
[docs] def atoms(self, *types): """Returns the atoms that form the current object. By default, only objects that are truly atomic and can't be divided into smaller pieces are returned: symbols, numbers, and number symbols like I and pi. It is possible to request atoms of any type, however, as demonstrated below. Examples ======== >>> from sympy import I, pi, sin >>> from import x, y >>> (1 + x + 2*sin(y + I*pi)).atoms() {1, 2, I, pi, x, y} If one or more types are given, the results will contain only those types of atoms. >>> from sympy import Number, NumberSymbol, Symbol >>> (1 + x + 2*sin(y + I*pi)).atoms(Symbol) {x, y} >>> (1 + x + 2*sin(y + I*pi)).atoms(Number) {1, 2} >>> (1 + x + 2*sin(y + I*pi)).atoms(Number, NumberSymbol) {1, 2, pi} >>> (1 + x + 2*sin(y + I*pi)).atoms(Number, NumberSymbol, I) {1, 2, I, pi} Note that I (imaginary unit) and zoo (complex infinity) are special types of number symbols and are not part of the NumberSymbol class. The type can be given implicitly, too: >>> (1 + x + 2*sin(y + I*pi)).atoms(x) # x is a Symbol {x, y} Be careful to check your assumptions when using the implicit option since ``S(1).is_Integer = True`` but ``type(S(1))`` is ``One``, a special type of sympy atom, while ``type(S(2))`` is type ``Integer`` and will find all integers in an expression: >>> from sympy import S >>> (1 + x + 2*sin(y + I*pi)).atoms(S(1)) {1} >>> (1 + x + 2*sin(y + I*pi)).atoms(S(2)) {1, 2} Finally, arguments to atoms() can select more than atomic atoms: any sympy type (loaded in core/ can be listed as an argument and those types of "atoms" as found in scanning the arguments of the expression recursively: >>> from sympy import Function, Mul >>> from sympy.core.function import AppliedUndef >>> f = Function('f') >>> (1 + f(x) + 2*sin(y + I*pi)).atoms(Function) {f(x), sin(y + I*pi)} >>> (1 + f(x) + 2*sin(y + I*pi)).atoms(AppliedUndef) {f(x)} >>> (1 + x + 2*sin(y + I*pi)).atoms(Mul) {I*pi, 2*sin(y + I*pi)} """ if types: types = tuple( [t if isinstance(t, type) else type(t) for t in types]) else: types = (Atom,) result = set() for expr in preorder_traversal(self): if isinstance(expr, types): result.add(expr) return result
@property def free_symbols(self): """Return from the atoms of self those which are free symbols. For most expressions, all symbols are free symbols. For some classes this is not true. e.g. Integrals use Symbols for the dummy variables which are bound variables, so Integral has a method to return all symbols except those. Derivative keeps track of symbols with respect to which it will perform a derivative; those are bound variables, too, so it has its own free_symbols method. Any other method that uses bound variables should implement a free_symbols method.""" return set().union(*[a.free_symbols for a in self.args]) @property def expr_free_symbols(self): return set([]) @property def canonical_variables(self): """Return a dictionary mapping any variable defined in ``self.variables`` as underscore-suffixed numbers corresponding to their position in ``self.variables``. Enough underscores are added to ensure that there will be no clash with existing free symbols. Examples ======== >>> from sympy import Lambda >>> from import x >>> Lambda(x, 2*x).canonical_variables {x: 0_} """ from sympy import Symbol if not hasattr(self, 'variables'): return {} u = "_" while any(str(s).endswith(u) for s in self.free_symbols): u += "_" name = '%%i%s' % u V = self.variables return dict(list(zip(V, [Symbol(name % i, **v.assumptions0) for i, v in enumerate(V)])))
[docs] def rcall(self, *args): """Apply on the argument recursively through the expression tree. This method is used to simulate a common abuse of notation for operators. For instance in SymPy the the following will not work: ``(x+Lambda(y, 2*y))(z) == x+2*z``, however you can use >>> from sympy import Lambda >>> from import x, y, z >>> (x + Lambda(y, 2*y)).rcall(z) x + 2*z """ return Basic._recursive_call(self, args)
@staticmethod def _recursive_call(expr_to_call, on_args): """Helper for rcall method. """ from sympy import Symbol def the_call_method_is_overridden(expr): for cls in getmro(type(expr)): if '__call__' in cls.__dict__: return cls != Basic if callable(expr_to_call) and the_call_method_is_overridden(expr_to_call): if isinstance(expr_to_call, Symbol): # XXX When you call a Symbol it is return expr_to_call # transformed into an UndefFunction else: return expr_to_call(*on_args) elif expr_to_call.args: args = [Basic._recursive_call( sub, on_args) for sub in expr_to_call.args] return type(expr_to_call)(*args) else: return expr_to_call def is_hypergeometric(self, k): from sympy.simplify import hypersimp return hypersimp(self, k) is not None @property def is_comparable(self): """Return True if self can be computed to a real number (or already is a real number) with precision, else False. Examples ======== >>> from sympy import exp_polar, pi, I >>> (I*exp_polar(I*pi/2)).is_comparable True >>> (I*exp_polar(I*pi*2)).is_comparable False A False result does not mean that `self` cannot be rewritten into a form that would be comparable. For example, the difference computed below is zero but without simplification it does not evaluate to a zero with precision: >>> e = 2**pi*(1 + 2**pi) >>> dif = e - e.expand() >>> dif.is_comparable False >>> dif.n(2)._prec 1 """ is_real = self.is_real if is_real is False: return False if not self.is_number: return False # don't re-eval numbers that are already evaluated since # this will create spurious precision n, i = [p.evalf(2) if not p.is_Number else p for p in self.as_real_imag()] if not (i.is_Number and n.is_Number): return False if i: # if _prec = 1 we can't decide and if not, # the answer is False because numbers with # imaginary parts can't be compared # so return False return False else: return n._prec != 1 @property def func(self): """ The top-level function in an expression. The following should hold for all objects:: >> x == x.func(*x.args) Examples ======== >>> from import x >>> a = 2*x >>> a.func <class 'sympy.core.mul.Mul'> >>> a.args (2, x) >>> a.func(*a.args) 2*x >>> a == a.func(*a.args) True """ return self.__class__ @property def args(self): """Returns a tuple of arguments of 'self'. Examples ======== >>> from sympy import cot >>> from import x, y >>> cot(x).args (x,) >>> cot(x).args[0] x >>> (x*y).args (x, y) >>> (x*y).args[1] y Notes ===== Never use self._args, always use self.args. Only use _args in __new__ when creating a new function. Don't override .args() from Basic (so that it's easy to change the interface in the future if needed). """ return self._args @property def _sorted_args(self): """ The same as ``args``. Derived classes which don't fix an order on their arguments should override this method to produce the sorted representation. """ return self.args
[docs] def as_poly(self, *gens, **args): """Converts ``self`` to a polynomial or returns ``None``. >>> from sympy import sin >>> from import x, y >>> print((x**2 + x*y).as_poly()) Poly(x**2 + x*y, x, y, domain='ZZ') >>> print((x**2 + x*y).as_poly(x, y)) Poly(x**2 + x*y, x, y, domain='ZZ') >>> print((x**2 + sin(y)).as_poly(x, y)) None """ from sympy.polys import Poly, PolynomialError try: poly = Poly(self, *gens, **args) if not poly.is_Poly: return None else: return poly except PolynomialError: return None
[docs] def as_content_primitive(self, radical=False, clear=True): """A stub to allow Basic args (like Tuple) to be skipped when computing the content and primitive components of an expression. See Also ======== sympy.core.expr.Expr.as_content_primitive """ return S.One, self
[docs] def subs(self, *args, **kwargs): """ Substitutes old for new in an expression after sympifying args. `args` is either: - two arguments, e.g. foo.subs(old, new) - one iterable argument, e.g. foo.subs(iterable). The iterable may be o an iterable container with (old, new) pairs. In this case the replacements are processed in the order given with successive patterns possibly affecting replacements already made. o a dict or set whose key/value items correspond to old/new pairs. In this case the old/new pairs will be sorted by op count and in case of a tie, by number of args and the default_sort_key. The resulting sorted list is then processed as an iterable container (see previous). If the keyword ``simultaneous`` is True, the subexpressions will not be evaluated until all the substitutions have been made. Examples ======== >>> from sympy import pi, exp, limit, oo >>> from import x, y >>> (1 + x*y).subs(x, pi) pi*y + 1 >>> (1 + x*y).subs({x:pi, y:2}) 1 + 2*pi >>> (1 + x*y).subs([(x, pi), (y, 2)]) 1 + 2*pi >>> reps = [(y, x**2), (x, 2)] >>> (x + y).subs(reps) 6 >>> (x + y).subs(reversed(reps)) x**2 + 2 >>> (x**2 + x**4).subs(x**2, y) y**2 + y To replace only the x**2 but not the x**4, use xreplace: >>> (x**2 + x**4).xreplace({x**2: y}) x**4 + y To delay evaluation until all substitutions have been made, set the keyword ``simultaneous`` to True: >>> (x/y).subs([(x, 0), (y, 0)]) 0 >>> (x/y).subs([(x, 0), (y, 0)], simultaneous=True) nan This has the added feature of not allowing subsequent substitutions to affect those already made: >>> ((x + y)/y).subs({x + y: y, y: x + y}) 1 >>> ((x + y)/y).subs({x + y: y, y: x + y}, simultaneous=True) y/(x + y) In order to obtain a canonical result, unordered iterables are sorted by count_op length, number of arguments and by the default_sort_key to break any ties. All other iterables are left unsorted. >>> from sympy import sqrt, sin, cos >>> from import a, b, c, d, e >>> A = (sqrt(sin(2*x)), a) >>> B = (sin(2*x), b) >>> C = (cos(2*x), c) >>> D = (x, d) >>> E = (exp(x), e) >>> expr = sqrt(sin(2*x))*sin(exp(x)*x)*cos(2*x) + sin(2*x) >>> expr.subs(dict([A, B, C, D, E])) a*c*sin(d*e) + b The resulting expression represents a literal replacement of the old arguments with the new arguments. This may not reflect the limiting behavior of the expression: >>> (x**3 - 3*x).subs({x: oo}) nan >>> limit(x**3 - 3*x, x, oo) oo If the substitution will be followed by numerical evaluation, it is better to pass the substitution to evalf as >>> (1/x).evalf(subs={x: 3.0}, n=21) 0.333333333333333333333 rather than >>> (1/x).subs({x: 3.0}).evalf(21) 0.333333333333333314830 as the former will ensure that the desired level of precision is obtained. See Also ======== replace: replacement capable of doing wildcard-like matching, parsing of match, and conditional replacements xreplace: exact node replacement in expr tree; also capable of using matching rules evalf: calculates the given formula to a desired level of precision """ from sympy.core.containers import Dict from sympy.utilities import default_sort_key from sympy import Dummy, Symbol unordered = False if len(args) == 1: sequence = args[0] if isinstance(sequence, set): unordered = True elif isinstance(sequence, (Dict, Mapping)): unordered = True sequence = sequence.items() elif not iterable(sequence): from sympy.utilities.misc import filldedent raise ValueError(filldedent(""" When a single argument is passed to subs it should be a dictionary of old: new pairs or an iterable of (old, new) tuples.""")) elif len(args) == 2: sequence = [args] else: raise ValueError("subs accepts either 1 or 2 arguments") sequence = list(sequence) for i in range(len(sequence)): s = list(sequence[i]) for j, si in enumerate(s): try: si = sympify(si, strict=True) except SympifyError: if type(si) is str: si = Symbol(si) else: # if it can't be sympified, skip it sequence[i] = None break s[j] = si else: sequence[i] = None if _aresame(*s) else tuple(s) sequence = list(filter(None, sequence)) if unordered: sequence = dict(sequence) if not all(k.is_Atom for k in sequence): d = {} for o, n in sequence.items(): try: ops = o.count_ops(), len(o.args) except TypeError: ops = (0, 0) d.setdefault(ops, []).append((o, n)) newseq = [] for k in sorted(d.keys(), reverse=True): newseq.extend( sorted([v[0] for v in d[k]], key=default_sort_key)) sequence = [(k, sequence[k]) for k in newseq] del newseq, d else: sequence = sorted([(k, v) for (k, v) in sequence.items()], key=default_sort_key) if kwargs.pop('simultaneous', False): # XXX should this be the default for dict subs? reps = {} rv = self kwargs['hack2'] = True m = Dummy() for old, new in sequence: d = Dummy(commutative=new.is_commutative) # using d*m so Subs will be used on dummy variables # in things like Derivative(f(x, y), x) in which x # is both free and bound rv = rv._subs(old, d*m, **kwargs) if not isinstance(rv, Basic): break reps[d] = new reps[m] = S.One # get rid of m return rv.xreplace(reps) else: rv = self for old, new in sequence: rv = rv._subs(old, new, **kwargs) if not isinstance(rv, Basic): break return rv
@cacheit def _subs(self, old, new, **hints): """Substitutes an expression old -> new. If self is not equal to old then _eval_subs is called. If _eval_subs doesn't want to make any special replacement then a None is received which indicates that the fallback should be applied wherein a search for replacements is made amongst the arguments of self. >>> from sympy import Add >>> from import x, y, z Examples ======== Add's _eval_subs knows how to target x + y in the following so it makes the change: >>> (x + y + z).subs(x + y, 1) z + 1 Add's _eval_subs doesn't need to know how to find x + y in the following: >>> Add._eval_subs(z*(x + y) + 3, x + y, 1) is None True The returned None will cause the fallback routine to traverse the args and pass the z*(x + y) arg to Mul where the change will take place and the substitution will succeed: >>> (z*(x + y) + 3).subs(x + y, 1) z + 3 ** Developers Notes ** An _eval_subs routine for a class should be written if: 1) any arguments are not instances of Basic (e.g. bool, tuple); 2) some arguments should not be targeted (as in integration variables); 3) if there is something other than a literal replacement that should be attempted (as in Piecewise where the condition may be updated without doing a replacement). If it is overridden, here are some special cases that might arise: 1) If it turns out that no special change was made and all the original sub-arguments should be checked for replacements then None should be returned. 2) If it is necessary to do substitutions on a portion of the expression then _subs should be called. _subs will handle the case of any sub-expression being equal to old (which usually would not be the case) while its fallback will handle the recursion into the sub-arguments. For example, after Add's _eval_subs removes some matching terms it must process the remaining terms so it calls _subs on each of the un-matched terms and then adds them onto the terms previously obtained. 3) If the initial expression should remain unchanged then the original expression should be returned. (Whenever an expression is returned, modified or not, no further substitution of old -> new is attempted.) Sum's _eval_subs routine uses this strategy when a substitution is attempted on any of its summation variables. """ def fallback(self, old, new): """ Try to replace old with new in any of self's arguments. """ hit = False args = list(self.args) for i, arg in enumerate(args): if not hasattr(arg, '_eval_subs'): continue arg = arg._subs(old, new, **hints) if not _aresame(arg, args[i]): hit = True args[i] = arg if hit: rv = self.func(*args) hack2 = hints.get('hack2', False) if hack2 and self.is_Mul and not rv.is_Mul: # 2-arg hack coeff = S.One nonnumber = [] for i in args: if i.is_Number: coeff *= i else: nonnumber.append(i) nonnumber = self.func(*nonnumber) if coeff is S.One: return nonnumber else: return self.func(coeff, nonnumber, evaluate=False) return rv return self if _aresame(self, old): return new rv = self._eval_subs(old, new) if rv is None: rv = fallback(self, old, new) return rv def _eval_subs(self, old, new): """Override this stub if you want to do anything more than attempt a replacement of old with new in the arguments of self. See also: _subs """ return None
[docs] def xreplace(self, rule): """ Replace occurrences of objects within the expression. Parameters ========== rule : dict-like Expresses a replacement rule Returns ======= xreplace : the result of the replacement Examples ======== >>> from sympy import symbols, pi, exp >>> x, y, z = symbols('x y z') >>> (1 + x*y).xreplace({x: pi}) pi*y + 1 >>> (1 + x*y).xreplace({x: pi, y: 2}) 1 + 2*pi Replacements occur only if an entire node in the expression tree is matched: >>> (x*y + z).xreplace({x*y: pi}) z + pi >>> (x*y*z).xreplace({x*y: pi}) x*y*z >>> (2*x).xreplace({2*x: y, x: z}) y >>> (2*2*x).xreplace({2*x: y, x: z}) 4*z >>> (x + y + 2).xreplace({x + y: 2}) x + y + 2 >>> (x + 2 + exp(x + 2)).xreplace({x + 2: y}) x + exp(y) + 2 xreplace doesn't differentiate between free and bound symbols. In the following, subs(x, y) would not change x since it is a bound symbol, but xreplace does: >>> from sympy import Integral >>> Integral(x, (x, 1, 2*x)).xreplace({x: y}) Integral(y, (y, 1, 2*y)) Trying to replace x with an expression raises an error: >>> Integral(x, (x, 1, 2*x)).xreplace({x: 2*y}) # doctest: +SKIP ValueError: Invalid limits given: ((2*y, 1, 4*y),) See Also ======== replace: replacement capable of doing wildcard-like matching, parsing of match, and conditional replacements subs: substitution of subexpressions as defined by the objects themselves. """ value, _ = self._xreplace(rule) return value
def _xreplace(self, rule): """ Helper for xreplace. Tracks whether a replacement actually occurred. """ if self in rule: return rule[self], True elif rule: args = [] changed = False for a in self.args: try: a_xr = a._xreplace(rule) args.append(a_xr[0]) changed |= a_xr[1] except AttributeError: args.append(a) args = tuple(args) if changed: return self.func(*args), True return self, False
[docs] @cacheit def has(self, *patterns): """ Test whether any subexpression matches any of the patterns. Examples ======== >>> from sympy import sin >>> from import x, y, z >>> (x**2 + sin(x*y)).has(z) False >>> (x**2 + sin(x*y)).has(x, y, z) True >>> x.has(x) True Note ``has`` is a structural algorithm with no knowledge of mathematics. Consider the following half-open interval: >>> from sympy.sets import Interval >>> i = Interval.Lopen(0, 5); i Interval.Lopen(0, 5) >>> i.args (0, 5, True, False) >>> i.has(4) # there is no "4" in the arguments False >>> i.has(0) # there *is* a "0" in the arguments True Instead, use ``contains`` to determine whether a number is in the interval or not: >>> i.contains(4) True >>> i.contains(0) False Note that ``expr.has(*patterns)`` is exactly equivalent to ``any(expr.has(p) for p in patterns)``. In particular, ``False`` is returned when the list of patterns is empty. >>> x.has() False """ return any(self._has(pattern) for pattern in patterns)
def _has(self, pattern): """Helper for .has()""" from sympy.core.function import UndefinedFunction, Function if isinstance(pattern, UndefinedFunction): return any(f.func == pattern or f == pattern for f in self.atoms(Function, UndefinedFunction)) pattern = sympify(pattern) if isinstance(pattern, BasicMeta): return any(isinstance(arg, pattern) for arg in preorder_traversal(self)) try: match = pattern._has_matcher() return any(match(arg) for arg in preorder_traversal(self)) except AttributeError: return any(arg == pattern for arg in preorder_traversal(self)) def _has_matcher(self): """Helper for .has()""" return lambda other: self == other
[docs] def replace(self, query, value, map=False, simultaneous=True, exact=False): """ Replace matching subexpressions of ``self`` with ``value``. If ``map = True`` then also return the mapping {old: new} where ``old`` was a sub-expression found with query and ``new`` is the replacement value for it. If the expression itself doesn't match the query, then the returned value will be ``self.xreplace(map)`` otherwise it should be ``self.subs(ordered(map.items()))``. Traverses an expression tree and performs replacement of matching subexpressions from the bottom to the top of the tree. The default approach is to do the replacement in a simultaneous fashion so changes made are targeted only once. If this is not desired or causes problems, ``simultaneous`` can be set to False. In addition, if an expression containing more than one Wild symbol is being used to match subexpressions and the ``exact`` flag is True, then the match will only succeed if non-zero values are received for each Wild that appears in the match pattern. The list of possible combinations of queries and replacement values is listed below: Examples ======== Initial setup >>> from sympy import log, sin, cos, tan, Wild, Mul, Add >>> from import x, y >>> f = log(sin(x)) + tan(sin(x**2)) 1.1. type -> type obj.replace(type, newtype) When object of type ``type`` is found, replace it with the result of passing its argument(s) to ``newtype``. >>> f.replace(sin, cos) log(cos(x)) + tan(cos(x**2)) >>> sin(x).replace(sin, cos, map=True) (cos(x), {sin(x): cos(x)}) >>> (x*y).replace(Mul, Add) x + y 1.2. type -> func obj.replace(type, func) When object of type ``type`` is found, apply ``func`` to its argument(s). ``func`` must be written to handle the number of arguments of ``type``. >>> f.replace(sin, lambda arg: sin(2*arg)) log(sin(2*x)) + tan(sin(2*x**2)) >>> (x*y).replace(Mul, lambda *args: sin(2*Mul(*args))) sin(2*x*y) 2.1. pattern -> expr obj.replace(pattern(wild), expr(wild)) Replace subexpressions matching ``pattern`` with the expression written in terms of the Wild symbols in ``pattern``. >>> a = Wild('a') >>> f.replace(sin(a), tan(a)) log(tan(x)) + tan(tan(x**2)) >>> f.replace(sin(a), tan(a/2)) log(tan(x/2)) + tan(tan(x**2/2)) >>> f.replace(sin(a), a) log(x) + tan(x**2) >>> (x*y).replace(a*x, a) y When the default value of False is used with patterns that have more than one Wild symbol, non-intuitive results may be obtained: >>> b = Wild('b') >>> (2*x).replace(a*x + b, b - a) 2/x For this reason, the ``exact`` option can be used to make the replacement only when the match gives non-zero values for all Wild symbols: >>> (2*x + y).replace(a*x + b, b - a, exact=True) y - 2 >>> (2*x).replace(a*x + b, b - a, exact=True) 2*x 2.2. pattern -> func obj.replace(pattern(wild), lambda wild: expr(wild)) All behavior is the same as in 2.1 but now a function in terms of pattern variables is used rather than an expression: >>> f.replace(sin(a), lambda a: sin(2*a)) log(sin(2*x)) + tan(sin(2*x**2)) 3.1. func -> func obj.replace(filter, func) Replace subexpression ``e`` with ``func(e)`` if ``filter(e)`` is True. >>> g = 2*sin(x**3) >>> g.replace(lambda expr: expr.is_Number, lambda expr: expr**2) 4*sin(x**9) The expression itself is also targeted by the query but is done in such a fashion that changes are not made twice. >>> e = x*(x*y + 1) >>> e.replace(lambda x: x.is_Mul, lambda x: 2*x) 2*x*(2*x*y + 1) See Also ======== subs: substitution of subexpressions as defined by the objects themselves. xreplace: exact node replacement in expr tree; also capable of using matching rules """ from sympy.core.symbol import Dummy from sympy.simplify.simplify import bottom_up try: query = sympify(query) except SympifyError: pass try: value = sympify(value) except SympifyError: pass if isinstance(query, type): _query = lambda expr: isinstance(expr, query) if isinstance(value, type): _value = lambda expr, result: value(*expr.args) elif callable(value): _value = lambda expr, result: value(*expr.args) else: raise TypeError( "given a type, replace() expects another " "type or a callable") elif isinstance(query, Basic): _query = lambda expr: expr.match(query) # XXX remove the exact flag and make multi-symbol # patterns use exact=True semantics; to do this the query must # be tested to find out how many Wild symbols are present. # See # ?fromgroups=#!topic/sympy/zPzo5FtRiqI # for a method of inspecting a function to know how many # parameters it has. if isinstance(value, Basic): if exact: _value = lambda expr, result: (value.subs(result) if all(val for val in result.values()) else expr) else: _value = lambda expr, result: value.subs(result) elif callable(value): # match dictionary keys get the trailing underscore stripped # from them and are then passed as keywords to the callable; # if ``exact`` is True, only accept match if there are no null # values amongst those matched. if exact: _value = lambda expr, result: (value(**dict([( str(key)[:-1], val) for key, val in result.items()])) if all(val for val in result.values()) else expr) else: _value = lambda expr, result: value(**dict([( str(key)[:-1], val) for key, val in result.items()])) else: raise TypeError( "given an expression, replace() expects " "another expression or a callable") elif callable(query): _query = query if callable(value): _value = lambda expr, result: value(expr) else: raise TypeError( "given a callable, replace() expects " "another callable") else: raise TypeError( "first argument to replace() must be a " "type, an expression or a callable") mapping = {} # changes that took place mask = [] # the dummies that were used as change placeholders def rec_replace(expr): result = _query(expr) if result or result == {}: new = _value(expr, result) if new is not None and new != expr: mapping[expr] = new if simultaneous: # don't let this expression be changed during rebuilding com = getattr(new, 'is_commutative', True) if com is None: com = True d = Dummy(commutative=com) mask.append((d, new)) expr = d else: expr = new return expr rv = bottom_up(self, rec_replace, atoms=True) # restore original expressions for Dummy symbols if simultaneous: mask = list(reversed(mask)) for o, n in mask: r = {o: n} rv = rv.xreplace(r) if not map: return rv else: if simultaneous: # restore subexpressions in mapping for o, n in mask: r = {o: n} mapping = {k.xreplace(r): v.xreplace(r) for k, v in mapping.items()} return rv, mapping
[docs] def find(self, query, group=False): """Find all subexpressions matching a query. """ query = _make_find_query(query) results = list(filter(query, preorder_traversal(self))) if not group: return set(results) else: groups = {} for result in results: if result in groups: groups[result] += 1 else: groups[result] = 1 return groups
[docs] def count(self, query): """Count the number of matching subexpressions. """ query = _make_find_query(query) return sum(bool(query(sub)) for sub in preorder_traversal(self))
[docs] def matches(self, expr, repl_dict={}, old=False): """ Helper method for match() that looks for a match between Wild symbols in self and expressions in expr. Examples ======== >>> from sympy import symbols, Wild, Basic >>> a, b, c = symbols('a b c') >>> x = Wild('x') >>> Basic(a + x, x).matches(Basic(a + b, c)) is None True >>> Basic(a + x, x).matches(Basic(a + b + c, b + c)) {x_: b + c} """ expr = sympify(expr) if not isinstance(expr, self.__class__): return None if self == expr: return repl_dict if len(self.args) != len(expr.args): return None d = repl_dict.copy() for arg, other_arg in zip(self.args, expr.args): if arg == other_arg: continue d = arg.xreplace(d).matches(other_arg, d, old=old) if d is None: return None return d
[docs] def match(self, pattern, old=False): """ Pattern matching. Wild symbols match all. Return ``None`` when expression (self) does not match with pattern. Otherwise return a dictionary such that:: pattern.xreplace(self.match(pattern)) == self Examples ======== >>> from sympy import Wild >>> from import x, y >>> p = Wild("p") >>> q = Wild("q") >>> r = Wild("r") >>> e = (x+y)**(x+y) >>> e.match(p**p) {p_: x + y} >>> e.match(p**q) {p_: x + y, q_: x + y} >>> e = (2*x)**2 >>> e.match(p*q**r) {p_: 4, q_: x, r_: 2} >>> (p*q**r).xreplace(e.match(p*q**r)) 4*x**2 The ``old`` flag will give the old-style pattern matching where expressions and patterns are essentially solved to give the match. Both of the following give None unless ``old=True``: >>> (x - 2).match(p - x, old=True) {p_: 2*x - 2} >>> (2/x).match(p*x, old=True) {p_: 2/x**2} """ pattern = sympify(pattern) return pattern.matches(self, old=old)
[docs] def count_ops(self, visual=None): """wrapper for count_ops that returns the operation count.""" from sympy import count_ops return count_ops(self, visual)
[docs] def doit(self, **hints): """Evaluate objects that are not evaluated by default like limits, integrals, sums and products. All objects of this kind will be evaluated recursively, unless some species were excluded via 'hints' or unless the 'deep' hint was set to 'False'. >>> from sympy import Integral >>> from import x >>> 2*Integral(x, x) 2*Integral(x, x) >>> (2*Integral(x, x)).doit() x**2 >>> (2*Integral(x, x)).doit(deep=False) 2*Integral(x, x) """ if hints.get('deep', True): terms = [term.doit(**hints) if isinstance(term, Basic) else term for term in self.args] return self.func(*terms) else: return self
def _eval_rewrite(self, pattern, rule, **hints): if self.is_Atom: if hasattr(self, rule): return getattr(self, rule)() return self if hints.get('deep', True): args = [a._eval_rewrite(pattern, rule, **hints) if isinstance(a, Basic) else a for a in self.args] else: args = self.args if pattern is None or isinstance(self, pattern): if hasattr(self, rule): rewritten = getattr(self, rule)(*args) if rewritten is not None: return rewritten return self.func(*args) def _accept_eval_derivative(self, s): # This method needs to be overridden by array-like objects return s._visit_eval_derivative_scalar(self) def _visit_eval_derivative_scalar(self, base): # Base is a scalar # Types are (base: scalar, self: scalar) return base._eval_derivative(self) def _visit_eval_derivative_array(self, base): # Types are (base: array/matrix, self: scalar) # Base is some kind of array/matrix, # it should have `.applyfunc(lambda x: x.diff(self)` implemented: return base._eval_derivative(self) def _eval_derivative_n_times(self, s, n): # This is the default evaluator for derivatives (as called by `diff` # and `Derivative`), it will attempt a loop to derive the expression # `n` times by calling the corresponding `_eval_derivative` method, # while leaving the derivative unevaluated if `n` is symbolic. This # method should be overridden if the object has a closed form for its # symbolic n-th derivative. from sympy import Integer if isinstance(n, (int, Integer)): obj = self for i in range(n): obj2 = obj._accept_eval_derivative(s) if obj == obj2: break obj = obj2 return obj else: return None
[docs] def rewrite(self, *args, **hints): """ Rewrite functions in terms of other functions. Rewrites expression containing applications of functions of one kind in terms of functions of different kind. For example you can rewrite trigonometric functions as complex exponentials or combinatorial functions as gamma function. As a pattern this function accepts a list of functions to to rewrite (instances of DefinedFunction class). As rule you can use string or a destination function instance (in this case rewrite() will use the str() function). There is also the possibility to pass hints on how to rewrite the given expressions. For now there is only one such hint defined called 'deep'. When 'deep' is set to False it will forbid functions to rewrite their contents. Examples ======== >>> from sympy import sin, exp >>> from import x Unspecified pattern: >>> sin(x).rewrite(exp) -I*(exp(I*x) - exp(-I*x))/2 Pattern as a single function: >>> sin(x).rewrite(sin, exp) -I*(exp(I*x) - exp(-I*x))/2 Pattern as a list of functions: >>> sin(x).rewrite([sin, ], exp) -I*(exp(I*x) - exp(-I*x))/2 """ if not args: return self else: pattern = args[:-1] if isinstance(args[-1], string_types): rule = '_eval_rewrite_as_' + args[-1] else: try: rule = '_eval_rewrite_as_' + args[-1].__name__ except: rule = '_eval_rewrite_as_' + args[-1].__class__.__name__ if not pattern: return self._eval_rewrite(None, rule, **hints) else: if iterable(pattern[0]): pattern = pattern[0] pattern = [p for p in pattern if self.has(p)] if pattern: return self._eval_rewrite(tuple(pattern), rule, **hints) else: return self
_constructor_postprocessor_mapping = {} @classmethod def _exec_constructor_postprocessors(cls, obj): # WARNING: This API is experimental. # This is an experimental API that introduces constructor # postprosessors for SymPy Core elements. If an argument of a SymPy # expression has a `_constructor_postprocessor_mapping` attribute, it will # be interpreted as a dictionary containing lists of postprocessing # functions for matching expression node names. clsname = obj.__class__.__name__ postprocessors = defaultdict(list) for i in obj.args: try: if i in Basic._constructor_postprocessor_mapping: for k, v in Basic._constructor_postprocessor_mapping[i].items(): postprocessors[k].extend([j for j in v if j not in postprocessors[k]]) else: postprocessor_mappings = ( Basic._constructor_postprocessor_mapping[cls].items() for cls in type(i).mro() if cls in Basic._constructor_postprocessor_mapping ) for k, v in chain.from_iterable(postprocessor_mappings): postprocessors[k].extend([j for j in v if j not in postprocessors[k]]) except TypeError: pass for f in postprocessors.get(clsname, []): obj = f(obj) if len(postprocessors) > 0 and obj not in Basic._constructor_postprocessor_mapping: Basic._constructor_postprocessor_mapping[obj] = postprocessors return obj
[docs]class Atom(Basic): """ A parent class for atomic things. An atom is an expression with no subexpressions. Examples ======== Symbol, Number, Rational, Integer, ... But not: Add, Mul, Pow, ... """ is_Atom = True __slots__ = [] def matches(self, expr, repl_dict={}, old=False): if self == expr: return repl_dict def xreplace(self, rule, hack2=False): return rule.get(self, self) def doit(self, **hints): return self @classmethod def class_key(cls): return 2, 0, cls.__name__ @cacheit def sort_key(self, order=None): return self.class_key(), (1, (str(self),)), S.One.sort_key(), S.One def _eval_simplify(self, ratio, measure, rational, inverse): return self @property def _sorted_args(self): # this is here as a safeguard against accidentally using _sorted_args # on Atoms -- they cannot be rebuilt as atom.func(*atom._sorted_args) # since there are no args. So the calling routine should be checking # to see that this property is not called for Atoms. raise AttributeError('Atoms have no args. It might be necessary' ' to make a check for Atoms in the calling code.')
def _aresame(a, b): """Return True if a and b are structurally the same, else False. Examples ======== To SymPy, 2.0 == 2: >>> from sympy import S >>> 2.0 == S(2) True Since a simple 'same or not' result is sometimes useful, this routine was written to provide that query: >>> from sympy.core.basic import _aresame >>> _aresame(S(2.0), S(2)) False """ from .function import AppliedUndef, UndefinedFunction as UndefFunc for i, j in zip_longest(preorder_traversal(a), preorder_traversal(b)): if i != j or type(i) != type(j): if ((isinstance(i, UndefFunc) and isinstance(j, UndefFunc)) or (isinstance(i, AppliedUndef) and isinstance(j, AppliedUndef))): if i.class_key() != j.class_key(): return False else: return False else: return True def _atomic(e): """Return atom-like quantities as far as substitution is concerned: Derivatives, Functions and Symbols. Don't return any 'atoms' that are inside such quantities unless they also appear outside, too. Examples ======== >>> from sympy import Derivative, Function, cos >>> from import x, y >>> from sympy.core.basic import _atomic >>> f = Function('f') >>> _atomic(x + y) {x, y} >>> _atomic(x + f(y)) {x, f(y)} >>> _atomic(Derivative(f(x), x) + cos(x) + y) {y, cos(x), Derivative(f(x), x)} """ from sympy import Derivative, Function, Symbol pot = preorder_traversal(e) seen = set() try: free = e.free_symbols except AttributeError: return {e} atoms = set() for p in pot: if p in seen: pot.skip() continue seen.add(p) if isinstance(p, Symbol) and p in free: atoms.add(p) elif isinstance(p, (Derivative, Function)): pot.skip() atoms.add(p) return atoms class preorder_traversal(Iterator): """ Do a pre-order traversal of a tree. This iterator recursively yields nodes that it has visited in a pre-order fashion. That is, it yields the current node then descends through the tree breadth-first to yield all of a node's children's pre-order traversal. For an expression, the order of the traversal depends on the order of .args, which in many cases can be arbitrary. Parameters ========== node : sympy expression The expression to traverse. keys : (default None) sort key(s) The key(s) used to sort args of Basic objects. When None, args of Basic objects are processed in arbitrary order. If key is defined, it will be passed along to ordered() as the only key(s) to use to sort the arguments; if ``key`` is simply True then the default keys of ordered will be used. Yields ====== subtree : sympy expression All of the subtrees in the tree. Examples ======== >>> from sympy import symbols >>> from sympy.core.basic import preorder_traversal >>> x, y, z = symbols('x y z') The nodes are returned in the order that they are encountered unless key is given; simply passing key=True will guarantee that the traversal is unique. >>> list(preorder_traversal((x + y)*z, keys=None)) # doctest: +SKIP [z*(x + y), z, x + y, y, x] >>> list(preorder_traversal((x + y)*z, keys=True)) [z*(x + y), z, x + y, x, y] """ def __init__(self, node, keys=None): self._skip_flag = False self._pt = self._preorder_traversal(node, keys) def _preorder_traversal(self, node, keys): yield node if self._skip_flag: self._skip_flag = False return if isinstance(node, Basic): if not keys and hasattr(node, '_argset'): # LatticeOp keeps args as a set. We should use this if we # don't care about the order, to prevent unnecessary sorting. args = node._argset else: args = node.args if keys: if keys != True: args = ordered(args, keys, default=False) else: args = ordered(args) for arg in args: for subtree in self._preorder_traversal(arg, keys): yield subtree elif iterable(node): for item in node: for subtree in self._preorder_traversal(item, keys): yield subtree def skip(self): """ Skip yielding current node's (last yielded node's) subtrees. Examples ======== >>> from sympy.core import symbols >>> from sympy.core.basic import preorder_traversal >>> x, y, z = symbols('x y z') >>> pt = preorder_traversal((x+y*z)*z) >>> for i in pt: ... print(i) ... if i == x+y*z: ... pt.skip() z*(x + y*z) z x + y*z """ self._skip_flag = True def __next__(self): return next(self._pt) def __iter__(self): return self def _make_find_query(query): """Convert the argument of Basic.find() into a callable""" try: query = sympify(query) except SympifyError: pass if isinstance(query, type): return lambda expr: isinstance(expr, query) elif isinstance(query, Basic): return lambda expr: expr.match(query) is not None return query