Source code for sympy.core.function

There are three types of functions implemented in SymPy:

    1) defined functions (in the sense that they can be evaluated) like
       exp or sin; they have a name and a body:
           f = exp
    2) undefined function which have a name but no body. Undefined
       functions can be defined using a Function class as follows:
           f = Function('f')
       (the result will be a Function instance)
    3) anonymous function (or lambda function) which have a body (defined
       with dummy variables) but have no name:
           f = Lambda(x, exp(x)*x)
           f = Lambda((x, y), exp(x)*y)
    The fourth type of functions are composites, like (sin + cos)(x); these work in
    SymPy core, but are not yet part of SymPy.


    >>> import sympy
    >>> f = sympy.Function("f")
    >>> from import x
    >>> f(x)
    >>> print(sympy.srepr(f(x).func))
    >>> f(x).args

from __future__ import print_function, division

from .add import Add
from .assumptions import ManagedProperties
from .basic import Basic
from .cache import cacheit
from .compatibility import iterable, is_sequence, as_int, ordered
from .decorators import _sympifyit
from .expr import Expr, AtomicExpr
from .numbers import Rational, Float
from .operations import LatticeOp
from .rules import Transform
from .singleton import S
from .sympify import sympify

from sympy.core.containers import Tuple, Dict
from sympy.core.logic import fuzzy_and
from sympy.core.compatibility import string_types, with_metaclass, range
from sympy.utilities import default_sort_key
from sympy.utilities.misc import filldedent
from sympy.utilities.iterables import uniq
from sympy.core.evaluate import global_evaluate

import sys

import mpmath
import mpmath.libmp as mlib

import inspect
import collections

def _coeff_isneg(a):
    """Return True if the leading Number is negative.


    >>> from sympy.core.function import _coeff_isneg
    >>> from sympy import S, Symbol, oo, pi
    >>> _coeff_isneg(-3*pi)
    >>> _coeff_isneg(S(3))
    >>> _coeff_isneg(-oo)
    >>> _coeff_isneg(Symbol('n', negative=True)) # coeff is 1


    if a.is_Mul:
        a = a.args[0]
    return a.is_Number and a.is_negative

[docs]class PoleError(Exception): pass
class ArgumentIndexError(ValueError): def __str__(self): return ("Invalid operation with argument number %s for Function %s" % (self.args[1], self.args[0])) def _getnargs(cls): if hasattr(cls, 'eval'): if sys.version_info < (3, ): return _getnargs_old(cls.eval) else: return _getnargs_new(cls.eval) else: return None def _getnargs_old(eval_): evalargspec = inspect.getargspec(eval_) if evalargspec.varargs: return None else: evalargs = len(evalargspec.args) - 1 # subtract 1 for cls if evalargspec.defaults: # if there are default args then they are optional; the # fewest args will occur when all defaults are used and # the most when none are used (i.e. all args are given) return tuple(range( evalargs - len(evalargspec.defaults), evalargs + 1)) return evalargs def _getnargs_new(eval_): parameters = inspect.signature(eval_).parameters.items() if [p for n,p in parameters if p.kind == p.VAR_POSITIONAL]: return None else: p_or_k = [p for n,p in parameters if p.kind == p.POSITIONAL_OR_KEYWORD] num_no_default = len(list(filter(lambda p:p.default == p.empty, p_or_k))) num_with_default = len(list(filter(lambda p:p.default != p.empty, p_or_k))) if not num_with_default: return num_no_default return tuple(range(num_no_default, num_no_default+num_with_default+1))
[docs]class FunctionClass(ManagedProperties): """ Base class for function classes. FunctionClass is a subclass of type. Use Function('<function name>' [ , signature ]) to create undefined function classes. """ _new = type.__new__ def __init__(cls, *args, **kwargs): # honor kwarg value or class-defined value before using # the number of arguments in the eval function (if present) nargs = kwargs.pop('nargs', cls.__dict__.get('nargs', _getnargs(cls))) super(FunctionClass, cls).__init__(args, kwargs) # Canonicalize nargs here; change to set in nargs. if is_sequence(nargs): if not nargs: raise ValueError(filldedent(''' Incorrectly specified nargs as %s: if there are no arguments, it should be `nargs = 0`; if there are any number of arguments, it should be `nargs = None`''' % str(nargs))) nargs = tuple(ordered(set(nargs))) elif nargs is not None: nargs = (as_int(nargs),) cls._nargs = nargs @property def __signature__(self): """ Allow Python 3's inspect.signature to give a useful signature for Function subclasses. """ # Python 3 only, but backports (like the one in IPython) still might # call this. try: from inspect import signature except ImportError: return None # TODO: Look at nargs return signature(self.eval) @property def nargs(self): """Return a set of the allowed number of arguments for the function. Examples ======== >>> from sympy.core.function import Function >>> from import x, y >>> f = Function('f') If the function can take any number of arguments, the set of whole numbers is returned: >>> Function('f').nargs Naturals0() If the function was initialized to accept one or more arguments, a corresponding set will be returned: >>> Function('f', nargs=1).nargs {1} >>> Function('f', nargs=(2, 1)).nargs {1, 2} The undefined function, after application, also has the nargs attribute; the actual number of arguments is always available by checking the ``args`` attribute: >>> f = Function('f') >>> f(1).nargs Naturals0() >>> len(f(1).args) 1 """ from sympy.sets.sets import FiniteSet # XXX it would be nice to handle this in __init__ but there are import # problems with trying to import FiniteSet there return FiniteSet(*self._nargs) if self._nargs else S.Naturals0 def __repr__(cls): return cls.__name__
class Application(with_metaclass(FunctionClass, Basic)): """ Base class for applied functions. Instances of Application represent the result of applying an application of any type to any object. """ is_Function = True @cacheit def __new__(cls, *args, **options): from sympy.sets.fancysets import Naturals0 from sympy.sets.sets import FiniteSet args = list(map(sympify, args)) evaluate = options.pop('evaluate', global_evaluate[0]) # WildFunction (and anything else like it) may have nargs defined # and we throw that value away here options.pop('nargs', None) if options: raise ValueError("Unknown options: %s" % options) if evaluate: evaluated = cls.eval(*args) if evaluated is not None: return evaluated obj = super(Application, cls).__new__(cls, *args, **options) # make nargs uniform here try: # things passing through here: # - functions subclassed from Function (e.g. myfunc(1).nargs) # - functions like cos(1).nargs # - AppliedUndef with given nargs like Function('f', nargs=1)(1).nargs # Canonicalize nargs here if is_sequence(obj.nargs): nargs = tuple(ordered(set(obj.nargs))) elif obj.nargs is not None: nargs = (as_int(obj.nargs),) else: nargs = None except AttributeError: # things passing through here: # - WildFunction('f').nargs # - AppliedUndef with no nargs like Function('f')(1).nargs nargs = obj._nargs # note the underscore here # convert to FiniteSet obj.nargs = FiniteSet(*nargs) if nargs else Naturals0() return obj @classmethod def eval(cls, *args): """ Returns a canonical form of cls applied to arguments args. The eval() method is called when the class cls is about to be instantiated and it should return either some simplified instance (possible of some other class), or if the class cls should be unmodified, return None. Examples of eval() for the function "sign" --------------------------------------------- @classmethod def eval(cls, arg): if arg is S.NaN: return S.NaN if arg is S.Zero: return S.Zero if arg.is_positive: return S.One if arg.is_negative: return S.NegativeOne if isinstance(arg, Mul): coeff, terms = arg.as_coeff_Mul(rational=True) if coeff is not S.One: return cls(coeff) * cls(terms) """ return @property def func(self): return self.__class__ def _eval_subs(self, old, new): if (old.is_Function and new.is_Function and callable(old) and callable(new) and old == self.func and len(self.args) in new.nargs): return new(*self.args)
[docs]class Function(Application, Expr): """Base class for applied mathematical functions. It also serves as a constructor for undefined function classes. Examples ======== First example shows how to use Function as a constructor for undefined function classes: >>> from sympy import Function, Symbol >>> x = Symbol('x') >>> f = Function('f') >>> g = Function('g')(x) >>> f f >>> f(x) f(x) >>> g g(x) >>> f(x).diff(x) Derivative(f(x), x) >>> g.diff(x) Derivative(g(x), x) In the following example Function is used as a base class for ``my_func`` that represents a mathematical function *my_func*. Suppose that it is well known, that *my_func(0)* is *1* and *my_func* at infinity goes to *0*, so we want those two simplifications to occur automatically. Suppose also that *my_func(x)* is real exactly when *x* is real. Here is an implementation that honours those requirements: >>> from sympy import Function, S, oo, I, sin >>> class my_func(Function): ... ... @classmethod ... def eval(cls, x): ... if x.is_Number: ... if x is S.Zero: ... return S.One ... elif x is S.Infinity: ... return S.Zero ... ... def _eval_is_real(self): ... return self.args[0].is_real ... >>> x = S('x') >>> my_func(0) + sin(0) 1 >>> my_func(oo) 0 >>> my_func(3.54).n() # Not yet implemented for my_func. my_func(3.54) >>> my_func(I).is_real False In order for ``my_func`` to become useful, several other methods would need to be implemented. See source code of some of the already implemented functions for more complete examples. Also, if the function can take more than one argument, then ``nargs`` must be defined, e.g. if ``my_func`` can take one or two arguments then, >>> class my_func(Function): ... nargs = (1, 2) ... >>> """ @property def _diff_wrt(self): """Allow derivatives wrt functions. Examples ======== >>> from sympy import Function, Symbol >>> f = Function('f') >>> x = Symbol('x') >>> f(x)._diff_wrt True """ return True @cacheit def __new__(cls, *args, **options): # Handle calls like Function('f') if cls is Function: return UndefinedFunction(*args, **options) n = len(args) if n not in cls.nargs: # XXX: exception message must be in exactly this format to # make it work with NumPy's functions like vectorize(). See, # for example, # The ideal solution would be just to attach metadata to # the exception and change NumPy to take advantage of this. temp = ('%(name)s takes %(qual)s %(args)s ' 'argument%(plural)s (%(given)s given)') raise TypeError(temp % { 'name': cls, 'qual': 'exactly' if len(cls.nargs) == 1 else 'at least', 'args': min(cls.nargs), 'plural': 's'*(min(cls.nargs) != 1), 'given': n}) evaluate = options.get('evaluate', global_evaluate[0]) result = super(Function, cls).__new__(cls, *args, **options) if not evaluate or not isinstance(result, cls): return result pr = max(cls._should_evalf(a) for a in result.args) pr2 = min(cls._should_evalf(a) for a in result.args) if pr2 > 0: return result.evalf(mlib.libmpf.prec_to_dps(pr)) return result @classmethod def _should_evalf(cls, arg): """ Decide if the function should automatically evalf(). By default (in this implementation), this happens if (and only if) the ARG is a floating point number. This function is used by __new__. Returns the precision to evalf to, or -1 if it shouldn't evalf. """ from sympy.core.symbol import Wild if arg.is_Float: return arg._prec if not arg.is_Add: return -1 # Don't use as_real_imag() here, that's too much work a, b = Wild('a'), Wild('b') m = arg.match(a + b*S.ImaginaryUnit) if not m or not (m[a].is_Float or m[b].is_Float): return -1 l = [m[i]._prec for i in m if m[i].is_Float] l.append(-1) return max(l) @classmethod def class_key(cls): from sympy.sets.fancysets import Naturals0 funcs = { 'exp': 10, 'log': 11, 'sin': 20, 'cos': 21, 'tan': 22, 'cot': 23, 'sinh': 30, 'cosh': 31, 'tanh': 32, 'coth': 33, 'conjugate': 40, 're': 41, 'im': 42, 'arg': 43, } name = cls.__name__ try: i = funcs[name] except KeyError: i = 0 if isinstance(cls.nargs, Naturals0) else 10000 return 4, i, name @property def is_commutative(self): """ Returns whether the functon is commutative. """ if all(getattr(t, 'is_commutative') for t in self.args): return True else: return False def _eval_evalf(self, prec): # Lookup mpmath function based on name fname = self.func.__name__ try: if not hasattr(mpmath, fname): from sympy.utilities.lambdify import MPMATH_TRANSLATIONS fname = MPMATH_TRANSLATIONS[fname] func = getattr(mpmath, fname) except (AttributeError, KeyError): try: return Float(self._imp_(*[i.evalf(prec) for i in self.args]), prec) except (AttributeError, TypeError, ValueError): return # Convert all args to mpf or mpc # Convert the arguments to *higher* precision than requested for the # final result. # XXX + 5 is a guess, it is similar to what is used in Should # we be more intelligent about it? try: args = [arg._to_mpmath(prec + 5) for arg in self.args] def bad(m): from mpmath import mpf, mpc # the precision of an mpf value is the last element # if that is 1 (and m[1] is not 1 which would indicate a # power of 2), then the eval failed; so check that none of # the arguments failed to compute to a finite precision. # Note: An mpc value has two parts, the re and imag tuple; # check each of those parts, too. Anything else is allowed to # pass if isinstance(m, mpf): m = m._mpf_ return m[1] !=1 and m[-1] == 1 elif isinstance(m, mpc): m, n = m._mpc_ return m[1] !=1 and m[-1] == 1 and \ n[1] !=1 and n[-1] == 1 else: return False if any(bad(a) for a in args): raise ValueError # one or more args failed to compute with significance except ValueError: return with mpmath.workprec(prec): v = func(*args) return Expr._from_mpmath(v, prec) def _eval_derivative(self, s): # f(x).diff(s) -> x.diff(s) * f.fdiff(1)(s) i = 0 l = [] for a in self.args: i += 1 da = a.diff(s) if da is S.Zero: continue try: df = self.fdiff(i) except ArgumentIndexError: df = Function.fdiff(self, i) l.append(df * da) return Add(*l) def _eval_is_commutative(self): return fuzzy_and(a.is_commutative for a in self.args) def _eval_is_complex(self): return fuzzy_and(a.is_complex for a in self.args)
[docs] def as_base_exp(self): """ Returns the method as the 2-tuple (base, exponent). """ return self, S.One
def _eval_aseries(self, n, args0, x, logx): """ Compute an asymptotic expansion around args0, in terms of self.args. This function is only used internally by _eval_nseries and should not be called directly; derived classes can overwrite this to implement asymptotic expansions. """ from sympy.utilities.misc import filldedent raise PoleError(filldedent(''' Asymptotic expansion of %s around %s is not implemented.''' % (type(self), args0))) def _eval_nseries(self, x, n, logx): """ This function does compute series for multivariate functions, but the expansion is always in terms of *one* variable. Examples ======== >>> from sympy import atan2 >>> from import x, y >>> atan2(x, y).series(x, n=2) atan2(0, y) + x/y + O(x**2) >>> atan2(x, y).series(y, n=2) -y/x + atan2(x, 0) + O(y**2) This function also computes asymptotic expansions, if necessary and possible: >>> from sympy import loggamma >>> loggamma(1/x)._eval_nseries(x,0,None) -1/x - log(x)/x + log(x)/2 + O(1) """ from sympy import Order from sympy.sets.sets import FiniteSet args = self.args args0 = [t.limit(x, 0) for t in args] if any(t.is_finite is False for t in args0): from sympy import oo, zoo, nan # XXX could use t.as_leading_term(x) here but it's a little # slower a = [t.compute_leading_term(x, logx=logx) for t in args] a0 = [t.limit(x, 0) for t in a] if any([t.has(oo, -oo, zoo, nan) for t in a0]): return self._eval_aseries(n, args0, x, logx) # Careful: the argument goes to oo, but only logarithmically so. We # are supposed to do a power series expansion "around the # logarithmic term". e.g. # f(1+x+log(x)) # -> f(1+logx) + x*f'(1+logx) + O(x**2) # where 'logx' is given in the argument a = [t._eval_nseries(x, n, logx) for t in args] z = [r - r0 for (r, r0) in zip(a, a0)] p = [Dummy() for t in z] q = [] v = None for ai, zi, pi in zip(a0, z, p): if zi.has(x): if v is not None: raise NotImplementedError q.append(ai + pi) v = pi else: q.append(ai) e1 = self.func(*q) if v is None: return e1 s = e1._eval_nseries(v, n, logx) o = s.getO() s = s.removeO() s = s.subs(v, zi).expand() + Order(o.expr.subs(v, zi), x) return s if (self.func.nargs is S.Naturals0 or (self.func.nargs == FiniteSet(1) and args0[0]) or any(c > 1 for c in self.func.nargs)): e = self e1 = e.expand() if e == e1: #for example when e = sin(x+1) or e = sin(cos(x)) #let's try the general algorithm term = e.subs(x, S.Zero) if term.is_finite is False or term is S.NaN: raise PoleError("Cannot expand %s around 0" % (self)) series = term fact = S.One _x = Dummy('x') e = e.subs(x, _x) for i in range(n - 1): i += 1 fact *= Rational(i) e = e.diff(_x) subs = e.subs(_x, S.Zero) if subs is S.NaN: # try to evaluate a limit if we have to subs = e.limit(_x, S.Zero) if subs.is_finite is False: raise PoleError("Cannot expand %s around 0" % (self)) term = subs*(x**i)/fact term = term.expand() series += term return series + Order(x**n, x) return e1.nseries(x, n=n, logx=logx) arg = self.args[0] l = [] g = None # try to predict a number of terms needed nterms = n + 2 cf = Order(arg.as_leading_term(x), x).getn() if cf != 0: nterms = int(nterms / cf) for i in range(nterms): g = self.taylor_term(i, arg, g) g = g.nseries(x, n=n, logx=logx) l.append(g) return Add(*l) + Order(x**n, x)
[docs] def fdiff(self, argindex=1): """ Returns the first derivative of the function. """ if not (1 <= argindex <= len(self.args)): raise ArgumentIndexError(self, argindex) if self.args[argindex - 1].is_Symbol: for i in range(len(self.args)): if i == argindex - 1: continue # See issue 8510 if self.args[argindex - 1] in self.args[i].free_symbols: break else: return Derivative(self, self.args[argindex - 1], evaluate=False) # See issue 4624 and issue 4719 and issue 5600 arg_dummy = Dummy('xi_%i' % argindex) arg_dummy.dummy_index = hash(self.args[argindex - 1]) new_args = [arg for arg in self.args] new_args[argindex-1] = arg_dummy return Subs(Derivative(self.func(*new_args), arg_dummy), arg_dummy, self.args[argindex - 1])
def _eval_as_leading_term(self, x): """Stub that should be overridden by new Functions to return the first non-zero term in a series if ever an x-dependent argument whose leading term vanishes as x -> 0 might be encountered. See, for example, cos._eval_as_leading_term. """ from sympy import Order args = [a.as_leading_term(x) for a in self.args] o = Order(1, x) if any(x in a.free_symbols and o.contains(a) for a in args): # Whereas x and any finite number are contained in O(1, x), # expressions like 1/x are not. If any arg simplified to a # vanishing expression as x -> 0 (like x or x**2, but not # 3, 1/x, etc...) then the _eval_as_leading_term is needed # to supply the first non-zero term of the series, # # e.g. expression leading term # ---------- ------------ # cos(1/x) cos(1/x) # cos(cos(x)) cos(1) # cos(x) 1 <- _eval_as_leading_term needed # sin(x) x <- _eval_as_leading_term needed # raise NotImplementedError( '%s has no _eval_as_leading_term routine' % self.func) else: return self.func(*args) def _sage_(self): import sage.all as sage fname = self.func.__name__ func = getattr(sage, fname) args = [arg._sage_() for arg in self.args] return func(*args)
class AppliedUndef(Function): """ Base class for expressions resulting from the application of an undefined function. """ def __new__(cls, *args, **options): args = list(map(sympify, args)) obj = super(AppliedUndef, cls).__new__(cls, *args, **options) return obj def _eval_as_leading_term(self, x): return self def _sage_(self): import sage.all as sage fname = str(self.func) args = [arg._sage_() for arg in self.args] func = sage.function(fname)(*args) return func class UndefinedFunction(FunctionClass): """ The (meta)class of undefined functions. """ def __new__(mcl, name, bases=(AppliedUndef,), __dict__=None, **kwargs): __dict__ = __dict__ or {} __dict__.update(kwargs) __dict__['__module__'] = None # For pickling ret = super(UndefinedFunction, mcl).__new__(mcl, name, bases, __dict__) return ret def __instancecheck__(cls, instance): return cls in type(instance).__mro__ UndefinedFunction.__eq__ = lambda s, o: (isinstance(o, s.__class__) and (s.class_key() == o.class_key()))
[docs]class WildFunction(Function, AtomicExpr): """ A WildFunction function matches any function (with its arguments). Examples ======== >>> from sympy import WildFunction, Function, cos >>> from import x, y >>> F = WildFunction('F') >>> f = Function('f') >>> F.nargs Naturals0() >>> x.match(F) >>> F.match(F) {F_: F_} >>> f(x).match(F) {F_: f(x)} >>> cos(x).match(F) {F_: cos(x)} >>> f(x, y).match(F) {F_: f(x, y)} To match functions with a given number of arguments, set ``nargs`` to the desired value at instantiation: >>> F = WildFunction('F', nargs=2) >>> F.nargs {2} >>> f(x).match(F) >>> f(x, y).match(F) {F_: f(x, y)} To match functions with a range of arguments, set ``nargs`` to a tuple containing the desired number of arguments, e.g. if ``nargs = (1, 2)`` then functions with 1 or 2 arguments will be matched. >>> F = WildFunction('F', nargs=(1, 2)) >>> F.nargs {1, 2} >>> f(x).match(F) {F_: f(x)} >>> f(x, y).match(F) {F_: f(x, y)} >>> f(x, y, 1).match(F) """ include = set() def __init__(cls, name, **assumptions): from sympy.sets.sets import Set, FiniteSet = name nargs = assumptions.pop('nargs', S.Naturals0) if not isinstance(nargs, Set): # Canonicalize nargs here. See also FunctionClass. if is_sequence(nargs): nargs = tuple(ordered(set(nargs))) elif nargs is not None: nargs = (as_int(nargs),) nargs = FiniteSet(*nargs) cls.nargs = nargs def matches(self, expr, repl_dict={}, old=False): if not isinstance(expr, (AppliedUndef, Function)): return None if len(expr.args) not in self.nargs: return None repl_dict = repl_dict.copy() repl_dict[self] = expr return repl_dict
[docs]class Derivative(Expr): """ Carries out differentiation of the given expression with respect to symbols. expr must define ._eval_derivative(symbol) method that returns the differentiation result. This function only needs to consider the non-trivial case where expr contains symbol and it should call the diff() method internally (not _eval_derivative); Derivative should be the only one to call _eval_derivative. Simplification of high-order derivatives: Because there can be a significant amount of simplification that can be done when multiple differentiations are performed, results will be automatically simplified in a fairly conservative fashion unless the keyword ``simplify`` is set to False. >>> from sympy import sqrt, diff >>> from import x >>> e = sqrt((x + 1)**2 + x) >>> diff(e, x, 5, simplify=False).count_ops() 136 >>> diff(e, x, 5).count_ops() 30 Ordering of variables: If evaluate is set to True and the expression can not be evaluated, the list of differentiation symbols will be sorted, that is, the expression is assumed to have continuous derivatives up to the order asked. This sorting assumes that derivatives wrt Symbols commute, derivatives wrt non-Symbols commute, but Symbol and non-Symbol derivatives don't commute with each other. Derivative wrt non-Symbols: This class also allows derivatives wrt non-Symbols that have _diff_wrt set to True, such as Function and Derivative. When a derivative wrt a non- Symbol is attempted, the non-Symbol is temporarily converted to a Symbol while the differentiation is performed. Note that this may seem strange, that Derivative allows things like f(g(x)).diff(g(x)), or even f(cos(x)).diff(cos(x)). The motivation for allowing this syntax is to make it easier to work with variational calculus (i.e., the Euler-Lagrange method). The best way to understand this is that the action of derivative with respect to a non-Symbol is defined by the above description: the object is substituted for a Symbol and the derivative is taken with respect to that. This action is only allowed for objects for which this can be done unambiguously, for example Function and Derivative objects. Note that this leads to what may appear to be mathematically inconsistent results. For example:: >>> from sympy import cos, sin, sqrt >>> from import x >>> (2*cos(x)).diff(cos(x)) 2 >>> (2*sqrt(1 - sin(x)**2)).diff(cos(x)) 0 This appears wrong because in fact 2*cos(x) and 2*sqrt(1 - sin(x)**2) are identically equal. However this is the wrong way to think of this. Think of it instead as if we have something like this:: >>> from import c, s >>> def F(u): ... return 2*u ... >>> def G(u): ... return 2*sqrt(1 - u**2) ... >>> F(cos(x)) 2*cos(x) >>> G(sin(x)) 2*sqrt(-sin(x)**2 + 1) >>> F(c).diff(c) 2 >>> F(c).diff(c) 2 >>> G(s).diff(c) 0 >>> G(sin(x)).diff(cos(x)) 0 Here, the Symbols c and s act just like the functions cos(x) and sin(x), respectively. Think of 2*cos(x) as f(c).subs(c, cos(x)) (or f(c) *at* c = cos(x)) and 2*sqrt(1 - sin(x)**2) as g(s).subs(s, sin(x)) (or g(s) *at* s = sin(x)), where f(u) == 2*u and g(u) == 2*sqrt(1 - u**2). Here, we define the function first and evaluate it at the function, but we can actually unambiguously do this in reverse in SymPy, because expr.subs(Function, Symbol) is well-defined: just structurally replace the function everywhere it appears in the expression. This is the same notational convenience used in the Euler-Lagrange method when one says F(t, f(t), f'(t)).diff(f(t)). What is actually meant is that the expression in question is represented by some F(t, u, v) at u = f(t) and v = f'(t), and F(t, f(t), f'(t)).diff(f(t)) simply means F(t, u, v).diff(u) at u = f(t). We do not allow derivatives to be taken with respect to expressions where this is not so well defined. For example, we do not allow expr.diff(x*y) because there are multiple ways of structurally defining where x*y appears in an expression, some of which may surprise the reader (for example, a very strict definition would have that (x*y*z).diff(x*y) == 0). >>> from import x, y, z >>> (x*y*z).diff(x*y) Traceback (most recent call last): ... ValueError: Can't differentiate wrt the variable: x*y, 1 Note that this definition also fits in nicely with the definition of the chain rule. Note how the chain rule in SymPy is defined using unevaluated Subs objects:: >>> from sympy import symbols, Function >>> f, g = symbols('f g', cls=Function) >>> f(2*g(x)).diff(x) 2*Derivative(g(x), x)*Subs(Derivative(f(_xi_1), _xi_1), (_xi_1,), (2*g(x),)) >>> f(g(x)).diff(x) Derivative(g(x), x)*Subs(Derivative(f(_xi_1), _xi_1), (_xi_1,), (g(x),)) Finally, note that, to be consistent with variational calculus, and to ensure that the definition of substituting a Function for a Symbol in an expression is well-defined, derivatives of functions are assumed to not be related to the function. In other words, we have:: >>> from sympy import diff >>> diff(f(x), x).diff(f(x)) 0 The same is true for derivatives of different orders:: >>> diff(f(x), x, 2).diff(diff(f(x), x, 1)) 0 >>> diff(f(x), x, 1).diff(diff(f(x), x, 2)) 0 Note, any class can allow derivatives to be taken with respect to itself. See the docstring of Expr._diff_wrt. Examples ======== Some basic examples: >>> from sympy import Derivative, Symbol, Function >>> f = Function('f') >>> g = Function('g') >>> x = Symbol('x') >>> y = Symbol('y') >>> Derivative(x**2, x, evaluate=True) 2*x >>> Derivative(Derivative(f(x,y), x), y) Derivative(f(x, y), x, y) >>> Derivative(f(x), x, 3) Derivative(f(x), x, x, x) >>> Derivative(f(x, y), y, x, evaluate=True) Derivative(f(x, y), x, y) Now some derivatives wrt functions: >>> Derivative(f(x)**2, f(x), evaluate=True) 2*f(x) >>> Derivative(f(g(x)), x, evaluate=True) Derivative(g(x), x)*Subs(Derivative(f(_xi_1), _xi_1), (_xi_1,), (g(x),)) """ is_Derivative = True @property def _diff_wrt(self): """Allow derivatives wrt Derivatives if it contains a function. Examples ======== >>> from sympy import Function, Symbol, Derivative >>> f = Function('f') >>> x = Symbol('x') >>> Derivative(f(x),x)._diff_wrt True >>> Derivative(x**2,x)._diff_wrt False """ if self.expr.is_Function: return True else: return False def __new__(cls, expr, *variables, **assumptions): expr = sympify(expr) # There are no variables, we differentiate wrt all of the free symbols # in expr. if not variables: variables = expr.free_symbols if len(variables) != 1: if expr.is_number: return S.Zero from sympy.utilities.misc import filldedent if len(variables) == 0: raise ValueError(filldedent(''' Since there are no variables in the expression, the variable(s) of differentiation must be supplied to differentiate %s''' % expr)) else: raise ValueError(filldedent(''' Since there is more than one variable in the expression, the variable(s) of differentiation must be supplied to differentiate %s''' % expr)) # Standardize the variables by sympifying them and making appending a # count of 1 if there is only one variable: diff(e,x)->diff(e,x,1). variables = list(sympify(variables)) if not variables[-1].is_Integer or len(variables) == 1: variables.append(S.One) # Split the list of variables into a list of the variables we are diff # wrt, where each element of the list has the form (s, count) where # s is the entity to diff wrt and count is the order of the # derivative. variable_count = [] all_zero = True i = 0 while i < len(variables) - 1: # process up to final Integer v, count = variables[i: i + 2] iwas = i if v._diff_wrt: # We need to test the more specific case of count being an # Integer first. if count.is_Integer: count = int(count) i += 2 elif count._diff_wrt: count = 1 i += 1 if i == iwas: # didn't get an update because of bad input from sympy.utilities.misc import filldedent last_digit = int(str(count)[-1]) ordinal = 'st' if last_digit == 1 else 'nd' if last_digit == 2 else 'rd' if last_digit == 3 else 'th' raise ValueError(filldedent(''' Can\'t calculate %s%s derivative wrt %s.''' % (count, ordinal, v))) if all_zero and not count == 0: all_zero = False if count: variable_count.append((v, count)) # We make a special case for 0th derivative, because there is no # good way to unambiguously print this. if all_zero: return expr # Pop evaluate because it is not really an assumption and we will need # to track it carefully below. evaluate = assumptions.pop('evaluate', False) # Look for a quick exit if there are symbols that don't appear in # expression at all. Note, this cannnot check non-symbols like # functions and Derivatives as those can be created by intermediate # derivatives. if evaluate and all(isinstance(sc[0], Symbol) for sc in variable_count): symbol_set = set(sc[0] for sc in variable_count) if symbol_set.difference(expr.free_symbols): return S.Zero # We make a generator so as to only generate a variable when necessary. # If a high order of derivative is requested and the expr becomes 0 # after a few differentiations, then we won't need the other variables. variablegen = (v for v, count in variable_count for i in range(count)) # If we can't compute the derivative of expr (but we wanted to) and # expr is itself not a Derivative, finish building an unevaluated # derivative class by calling Expr.__new__. if (not (hasattr(expr, '_eval_derivative') and evaluate) and (not isinstance(expr, Derivative))): variables = list(variablegen) # If we wanted to evaluate, we sort the variables into standard # order for later comparisons. This is too aggressive if evaluate # is False, so we don't do it in that case. if evaluate: #TODO: check if assumption of discontinuous derivatives exist variables = cls._sort_variables(variables) # Here we *don't* need to reinject evaluate into assumptions # because we are done with it and it is not an assumption that # Expr knows about. obj = Expr.__new__(cls, expr, *variables, **assumptions) return obj # Compute the derivative now by repeatedly calling the # _eval_derivative method of expr for each variable. When this method # returns None, the derivative couldn't be computed wrt that variable # and we save the variable for later. unhandled_variables = [] # Once we encouter a non_symbol that is unhandled, we stop taking # derivatives entirely. This is because derivatives wrt functions # don't commute with derivatives wrt symbols and we can't safely # continue. unhandled_non_symbol = False nderivs = 0 # how many derivatives were performed for v in variablegen: is_symbol = v.is_symbol if unhandled_non_symbol: obj = None else: if not is_symbol: new_v = Dummy('xi_%i' % i) new_v.dummy_index = hash(v) expr = expr.xreplace({v: new_v}) old_v = v v = new_v obj = expr._eval_derivative(v) nderivs += 1 if not is_symbol: if obj is not None: if not old_v.is_symbol and obj.is_Derivative: # Derivative evaluated at a point that is not a # symbol obj = Subs(obj, v, old_v) else: obj = obj.xreplace({v: old_v}) v = old_v if obj is None: unhandled_variables.append(v) if not is_symbol: unhandled_non_symbol = True elif obj is S.Zero: return S.Zero else: expr = obj if unhandled_variables: unhandled_variables = cls._sort_variables(unhandled_variables) expr = Expr.__new__(cls, expr, *unhandled_variables, **assumptions) else: # We got a Derivative at the end of it all, and we rebuild it by # sorting its variables. if isinstance(expr, Derivative): expr = cls( expr.args[0], *cls._sort_variables(expr.args[1:]) ) if nderivs > 1 and assumptions.get('simplify', True): from sympy.core.exprtools import factor_terms from sympy.simplify.simplify import signsimp expr = factor_terms(signsimp(expr)) return expr @classmethod def _sort_variables(cls, vars): """Sort variables, but disallow sorting of non-symbols. When taking derivatives, the following rules usually hold: * Derivative wrt different symbols commute. * Derivative wrt different non-symbols commute. * Derivatives wrt symbols and non-symbols don't commute. Examples ======== >>> from sympy import Derivative, Function, symbols >>> vsort = Derivative._sort_variables >>> x, y, z = symbols('x y z') >>> f, g, h = symbols('f g h', cls=Function) >>> vsort((x,y,z)) [x, y, z] >>> vsort((h(x),g(x),f(x))) [f(x), g(x), h(x)] >>> vsort((z,y,x,h(x),g(x),f(x))) [x, y, z, f(x), g(x), h(x)] >>> vsort((x,f(x),y,f(y))) [x, f(x), y, f(y)] >>> vsort((y,x,g(x),f(x),z,h(x),y,x)) [x, y, f(x), g(x), z, h(x), x, y] >>> vsort((z,y,f(x),x,f(x),g(x))) [y, z, f(x), x, f(x), g(x)] >>> vsort((z,y,f(x),x,f(x),g(x),z,z,y,x)) [y, z, f(x), x, f(x), g(x), x, y, z, z] """ sorted_vars = [] symbol_part = [] non_symbol_part = [] for v in vars: if not v.is_symbol: if len(symbol_part) > 0: sorted_vars.extend(sorted(symbol_part, key=default_sort_key)) symbol_part = [] non_symbol_part.append(v) else: if len(non_symbol_part) > 0: sorted_vars.extend(sorted(non_symbol_part, key=default_sort_key)) non_symbol_part = [] symbol_part.append(v) if len(non_symbol_part) > 0: sorted_vars.extend(sorted(non_symbol_part, key=default_sort_key)) if len(symbol_part) > 0: sorted_vars.extend(sorted(symbol_part, key=default_sort_key)) return sorted_vars def _eval_is_commutative(self): return self.expr.is_commutative def _eval_derivative(self, v): # If the variable s we are diff wrt is not in self.variables, we # assume that we might be able to take the derivative. if v not in self.variables: obj = self.expr.diff(v) if obj is S.Zero: return S.Zero if isinstance(obj, Derivative): return obj.func(obj.expr, *(self.variables + obj.variables)) # The derivative wrt s could have simplified things such that the # derivative wrt things in self.variables can now be done. Thus, # we set evaluate=True to see if there are any other derivatives # that can be done. The most common case is when obj is a simple # number so that the derivative wrt anything else will vanish. return self.func(obj, *self.variables, evaluate=True) # In this case s was in self.variables so the derivatve wrt s has # already been attempted and was not computed, either because it # couldn't be or evaluate=False originally. return self.func(self.expr, *(self.variables + (v, )), evaluate=False) def doit(self, **hints): expr = self.expr if hints.get('deep', True): expr = expr.doit(**hints) hints['evaluate'] = True return self.func(expr, *self.variables, **hints) @_sympifyit('z0', NotImplementedError)
[docs] def doit_numerically(self, z0): """ Evaluate the derivative at z numerically. When we can represent derivatives at a point, this should be folded into the normal evalf. For now, we need a special method. """ import mpmath from sympy.core.expr import Expr if len(self.free_symbols) != 1 or len(self.variables) != 1: raise NotImplementedError('partials and higher order derivatives') z = list(self.free_symbols)[0] def eval(x): f0 = self.expr.subs(z, Expr._from_mpmath(x, f0 = f0.evalf(mlib.libmpf.prec_to_dps( return f0._to_mpmath( return Expr._from_mpmath(mpmath.diff(eval, z0._to_mpmath(,
@property def expr(self): return self._args[0] @property def variables(self): return self._args[1:] @property def free_symbols(self): return self.expr.free_symbols def _eval_subs(self, old, new): if old in self.variables and not new._diff_wrt: # issue 4719 return Subs(self, old, new) # If both are Derivatives with the same expr, check if old is # equivalent to self or if old is a subderivative of self. if old.is_Derivative and old.expr == self.expr: # Check if canonnical order of variables is equal. old_vars = collections.Counter(old.variables) self_vars = collections.Counter(self.variables) if old_vars == self_vars: return new # collections.Counter doesn't have __le__ def _subset(a, b): return all(a[i] <= b[i] for i in a) if _subset(old_vars, self_vars): return Derivative(new, *(self_vars - old_vars).elements()) return Derivative(*(x._subs(old, new) for x in self.args)) def _eval_lseries(self, x, logx): dx = self.variables for term in self.expr.lseries(x, logx=logx): yield self.func(term, *dx) def _eval_nseries(self, x, n, logx): arg = self.expr.nseries(x, n=n, logx=logx) o = arg.getO() dx = self.variables rv = [self.func(a, *dx) for a in Add.make_args(arg.removeO())] if o: rv.append(o/x) return Add(*rv) def _eval_as_leading_term(self, x): series_gen = self.expr.lseries(x) d = S.Zero for leading_term in series_gen: d = diff(leading_term, *self.variables) if d != 0: break return d def _sage_(self): import sage.all as sage args = [arg._sage_() for arg in self.args] return sage.derivative(*args)
[docs] def as_finite_difference(self, points=1, x0=None, wrt=None): """ Expresses a Derivative instance as a finite difference. Parameters ========== points : sequence or coefficient, optional If sequence: discrete values (length >= order+1) of the independent variable used for generating the finite difference weights. If it is a coefficient, it will be used as the step-size for generating an equidistant sequence of length order+1 centered around ``x0``. Default: 1 (step-size 1) x0 : number or Symbol, optional the value of the independent variable (``wrt``) at which the derivative is to be approximated. Default: same as ``wrt``. wrt : Symbol, optional "with respect to" the variable for which the (partial) derivative is to be approximated for. If not provided it is required that the derivative is ordinary. Default: ``None``. Examples ======== >>> from sympy import symbols, Function, exp, sqrt, Symbol >>> x, h = symbols('x h') >>> f = Function('f') >>> f(x).diff(x).as_finite_difference() -f(x - 1/2) + f(x + 1/2) The default step size and number of points are 1 and ``order + 1`` respectively. We can change the step size by passing a symbol as a parameter: >>> f(x).diff(x).as_finite_difference(h) -f(-h/2 + x)/h + f(h/2 + x)/h We can also specify the discretized values to be used in a sequence: >>> f(x).diff(x).as_finite_difference([x, x+h, x+2*h]) -3*f(x)/(2*h) + 2*f(h + x)/h - f(2*h + x)/(2*h) The algorithm is not restricted to use equidistant spacing, nor do we need to make the approximation around ``x0``, but we can get an expression estimating the derivative at an offset: >>> e, sq2 = exp(1), sqrt(2) >>> xl = [x-h, x+h, x+e*h] >>> f(x).diff(x, 1).as_finite_difference(xl, x+h*sq2) # doctest: +ELLIPSIS 2*h*((h + sqrt(2)*h)/(2*h) - (-sqrt(2)*h + h)/(2*h))*f(E*h + x)/... Partial derivatives are also supported: >>> y = Symbol('y') >>> d2fdxdy=f(x,y).diff(x,y) >>> d2fdxdy.as_finite_difference(wrt=x) -Derivative(f(x - 1/2, y), y) + Derivative(f(x + 1/2, y), y) We can apply ``as_finite_difference`` to ``Derivative`` instances in compound expressions using ``replace``: >>> (1 + 42**f(x).diff(x)).replace(lambda arg: arg.is_Derivative, ... lambda arg: arg.as_finite_difference()) 42**(-f(x - 1/2) + f(x + 1/2)) + 1 See also ======== sympy.calculus.finite_diff.apply_finite_diff sympy.calculus.finite_diff.differentiate_finite sympy.calculus.finite_diff.finite_diff_weights """ from ..calculus.finite_diff import _as_finite_diff return _as_finite_diff(self, points, x0, wrt)
[docs]class Lambda(Expr): """ Lambda(x, expr) represents a lambda function similar to Python's 'lambda x: expr'. A function of several variables is written as Lambda((x, y, ...), expr). A simple example: >>> from sympy import Lambda >>> from import x >>> f = Lambda(x, x**2) >>> f(4) 16 For multivariate functions, use: >>> from import y, z, t >>> f2 = Lambda((x, y, z, t), x + y**z + t**z) >>> f2(1, 2, 3, 4) 73 A handy shortcut for lots of arguments: >>> p = x, y, z >>> f = Lambda(p, x + y*z) >>> f(*p) x + y*z """ is_Function = True def __new__(cls, variables, expr): from sympy.sets.sets import FiniteSet v = list(variables) if iterable(variables) else [variables] for i in v: if not getattr(i, 'is_Symbol', False): raise TypeError('variable is not a symbol: %s' % i) if len(v) == 1 and v[0] == expr: return S.IdentityFunction obj = Expr.__new__(cls, Tuple(*v), sympify(expr)) obj.nargs = FiniteSet(len(v)) return obj @property def variables(self): """The variables used in the internal representation of the function""" return self._args[0] @property def expr(self): """The return value of the function""" return self._args[1] @property def free_symbols(self): return self.expr.free_symbols - set(self.variables) def __call__(self, *args): n = len(args) if n not in self.nargs: # Lambda only ever has 1 value in nargs # XXX: exception message must be in exactly this format to # make it work with NumPy's functions like vectorize(). See, # for example, # The ideal solution would be just to attach metadata to # the exception and change NumPy to take advantage of this. ## XXX does this apply to Lambda? If not, remove this comment. temp = ('%(name)s takes exactly %(args)s ' 'argument%(plural)s (%(given)s given)') raise TypeError(temp % { 'name': self, 'args': list(self.nargs)[0], 'plural': 's'*(list(self.nargs)[0] != 1), 'given': n}) return self.expr.xreplace(dict(list(zip(self.variables, args)))) def __eq__(self, other): if not isinstance(other, Lambda): return False if self.nargs != other.nargs: return False selfexpr = self.args[1] otherexpr = other.args[1] otherexpr = otherexpr.xreplace(dict(list(zip(other.args[0], self.args[0])))) return selfexpr == otherexpr def __ne__(self, other): return not(self == other) def __hash__(self): return super(Lambda, self).__hash__() def _hashable_content(self): return (self.expr.xreplace(self.canonical_variables),) @property def is_identity(self): """Return ``True`` if this ``Lambda`` is an identity function. """ if len(self.args) == 2: return self.args[0] == self.args[1] else: return None
[docs]class Subs(Expr): """ Represents unevaluated substitutions of an expression. ``Subs(expr, x, x0)`` receives 3 arguments: an expression, a variable or list of distinct variables and a point or list of evaluation points corresponding to those variables. ``Subs`` objects are generally useful to represent unevaluated derivatives calculated at a point. The variables may be expressions, but they are subjected to the limitations of subs(), so it is usually a good practice to use only symbols for variables, since in that case there can be no ambiguity. There's no automatic expansion - use the method .doit() to effect all possible substitutions of the object and also of objects inside the expression. When evaluating derivatives at a point that is not a symbol, a Subs object is returned. One is also able to calculate derivatives of Subs objects - in this case the expression is always expanded (for the unevaluated form, use Derivative()). A simple example: >>> from sympy import Subs, Function, sin >>> from import x, y, z >>> f = Function('f') >>> e = Subs(f(x).diff(x), x, y) >>> e.subs(y, 0) Subs(Derivative(f(x), x), (x,), (0,)) >>> e.subs(f, sin).doit() cos(y) An example with several variables: >>> Subs(f(x)*sin(y) + z, (x, y), (0, 1)) Subs(z + f(x)*sin(y), (x, y), (0, 1)) >>> _.doit() z + f(0)*sin(1) """ def __new__(cls, expr, variables, point, **assumptions): from sympy import Symbol if not is_sequence(variables, Tuple): variables = [variables] variables = list(sympify(variables)) if list(uniq(variables)) != variables: repeated = [ v for v in set(variables) if variables.count(v) > 1 ] raise ValueError('cannot substitute expressions %s more than ' 'once.' % repeated) point = Tuple(*(point if is_sequence(point, Tuple) else [point])) if len(point) != len(variables): raise ValueError('Number of point values must be the same as ' 'the number of variables.') expr = sympify(expr) # use symbols with names equal to the point value (with preppended _) # to give a variable-independent expression pre = "_" pts = sorted(set(point), key=default_sort_key) from sympy.printing import StrPrinter class CustomStrPrinter(StrPrinter): def _print_Dummy(self, expr): return str(expr) + str(expr.dummy_index) def mystr(expr, **settings): p = CustomStrPrinter(settings) return p.doprint(expr) while 1: s_pts = {p: Symbol(pre + mystr(p)) for p in pts} reps = [(v, s_pts[p]) for v, p in zip(variables, point)] # if any underscore-preppended symbol is already a free symbol # and is a variable with a different point value, then there # is a clash, e.g. _0 clashes in Subs(_0 + _1, (_0, _1), (1, 0)) # because the new symbol that would be created is _1 but _1 # is already mapped to 0 so __0 and __1 are used for the new # symbols if any(r in expr.free_symbols and r in variables and Symbol(pre + mystr(point[variables.index(r)])) != r for _, r in reps): pre += "_" continue break obj = Expr.__new__(cls, expr, Tuple(*variables), point) obj._expr = expr.subs(reps) return obj def _eval_is_commutative(self): return self.expr.is_commutative def doit(self): return self.expr.doit().subs(list(zip(self.variables, self.point))) def evalf(self, prec=None, **options): return self.doit().evalf(prec, **options) n = evalf @property def variables(self): """The variables to be evaluated""" return self._args[1] @property def expr(self): """The expression on which the substitution operates""" return self._args[0] @property def point(self): """The values for which the variables are to be substituted""" return self._args[2] @property def free_symbols(self): return (self.expr.free_symbols - set(self.variables) | set(self.point.free_symbols)) def _has(self, pattern): if pattern in self.variables and pattern not in self.point: return False return super(Subs, self)._has(pattern) def __eq__(self, other): if not isinstance(other, Subs): return False return self._expr == other._expr def __ne__(self, other): return not(self == other) def __hash__(self): return super(Subs, self).__hash__() def _hashable_content(self): return (self._expr.xreplace(self.canonical_variables),) def _eval_subs(self, old, new): if old in self.variables: if old in self.point: newpoint = tuple(new if i == old else i for i in self.point) return self.func(self.expr, self.variables, newpoint) return self def _eval_derivative(self, s): if s not in self.free_symbols: return S.Zero return Add((Subs(self.expr.diff(s), self.variables, self.point).doit() if s not in self.variables else S.Zero), *[p.diff(s) * Subs(self.expr.diff(v), self.variables, self.point).doit() for v, p in zip(self.variables, self.point)]) def _eval_nseries(self, x, n, logx): if x in self.point: # x is the variable being substituted into apos = self.point.index(x) other = self.variables[apos] arg = self.expr.nseries(other, n=n, logx=logx) o = arg.getO() subs_args = [self.func(a, *self.args[1:]) for a in arg.removeO().args] return Add(*subs_args) + o.subs(other, x) arg = self.expr.nseries(x, n=n, logx=logx) o = arg.getO() subs_args = [self.func(a, *self.args[1:]) for a in arg.removeO().args] return Add(*subs_args) + o def _eval_as_leading_term(self, x): if x in self.point: ipos = self.point.index(x) xvar = self.variables[ipos] return self.expr.as_leading_term(xvar) if x in self.variables: # if `x` is a dummy variable, it means it won't exist after the # substitution has been performed: return self # The variable is independent of the substitution: return self.expr.as_leading_term(x)
[docs]def diff(f, *symbols, **kwargs): """ Differentiate f with respect to symbols. This is just a wrapper to unify .diff() and the Derivative class; its interface is similar to that of integrate(). You can use the same shortcuts for multiple variables as with Derivative. For example, diff(f(x), x, x, x) and diff(f(x), x, 3) both return the third derivative of f(x). You can pass evaluate=False to get an unevaluated Derivative class. Note that if there are 0 symbols (such as diff(f(x), x, 0), then the result will be the function (the zeroth derivative), even if evaluate=False. Examples ======== >>> from sympy import sin, cos, Function, diff >>> from import x, y >>> f = Function('f') >>> diff(sin(x), x) cos(x) >>> diff(f(x), x, x, x) Derivative(f(x), x, x, x) >>> diff(f(x), x, 3) Derivative(f(x), x, x, x) >>> diff(sin(x)*cos(y), x, 2, y, 2) sin(x)*cos(y) >>> type(diff(sin(x), x)) cos >>> type(diff(sin(x), x, evaluate=False)) <class 'sympy.core.function.Derivative'> >>> type(diff(sin(x), x, 0)) sin >>> type(diff(sin(x), x, 0, evaluate=False)) sin >>> diff(sin(x)) cos(x) >>> diff(sin(x*y)) Traceback (most recent call last): ... ValueError: specify differentiation variables to differentiate sin(x*y) Note that ``diff(sin(x))`` syntax is meant only for convenience in interactive sessions and should be avoided in library code. References ========== See Also ======== Derivative sympy.geometry.util.idiff: computes the derivative implicitly """ kwargs.setdefault('evaluate', True) try: return f._eval_diff(*symbols, **kwargs) except AttributeError: pass return Derivative(f, *symbols, **kwargs)
[docs]def expand(e, deep=True, modulus=None, power_base=True, power_exp=True, mul=True, log=True, multinomial=True, basic=True, **hints): """ Expand an expression using methods given as hints. Hints evaluated unless explicitly set to False are: ``basic``, ``log``, ``multinomial``, ``mul``, ``power_base``, and ``power_exp`` The following hints are supported but not applied unless set to True: ``complex``, ``func``, and ``trig``. In addition, the following meta-hints are supported by some or all of the other hints: ``frac``, ``numer``, ``denom``, ``modulus``, and ``force``. ``deep`` is supported by all hints. Additionally, subclasses of Expr may define their own hints or meta-hints. The ``basic`` hint is used for any special rewriting of an object that should be done automatically (along with the other hints like ``mul``) when expand is called. This is a catch-all hint to handle any sort of expansion that may not be described by the existing hint names. To use this hint an object should override the ``_eval_expand_basic`` method. Objects may also define their own expand methods, which are not run by default. See the API section below. If ``deep`` is set to ``True`` (the default), things like arguments of functions are recursively expanded. Use ``deep=False`` to only expand on the top level. If the ``force`` hint is used, assumptions about variables will be ignored in making the expansion. Hints ===== These hints are run by default mul --- Distributes multiplication over addition: >>> from sympy import cos, exp, sin >>> from import x, y, z >>> (y*(x + z)).expand(mul=True) x*y + y*z multinomial ----------- Expand (x + y + ...)**n where n is a positive integer. >>> ((x + y + z)**2).expand(multinomial=True) x**2 + 2*x*y + 2*x*z + y**2 + 2*y*z + z**2 power_exp --------- Expand addition in exponents into multiplied bases. >>> exp(x + y).expand(power_exp=True) exp(x)*exp(y) >>> (2**(x + y)).expand(power_exp=True) 2**x*2**y power_base ---------- Split powers of multiplied bases. This only happens by default if assumptions allow, or if the ``force`` meta-hint is used: >>> ((x*y)**z).expand(power_base=True) (x*y)**z >>> ((x*y)**z).expand(power_base=True, force=True) x**z*y**z >>> ((2*y)**z).expand(power_base=True) 2**z*y**z Note that in some cases where this expansion always holds, SymPy performs it automatically: >>> (x*y)**2 x**2*y**2 log --- Pull out power of an argument as a coefficient and split logs products into sums of logs. Note that these only work if the arguments of the log function have the proper assumptions--the arguments must be positive and the exponents must be real--or else the ``force`` hint must be True: >>> from sympy import log, symbols >>> log(x**2*y).expand(log=True) log(x**2*y) >>> log(x**2*y).expand(log=True, force=True) 2*log(x) + log(y) >>> x, y = symbols('x,y', positive=True) >>> log(x**2*y).expand(log=True) 2*log(x) + log(y) basic ----- This hint is intended primarily as a way for custom subclasses to enable expansion by default. These hints are not run by default: complex ------- Split an expression into real and imaginary parts. >>> x, y = symbols('x,y') >>> (x + y).expand(complex=True) re(x) + re(y) + I*im(x) + I*im(y) >>> cos(x).expand(complex=True) -I*sin(re(x))*sinh(im(x)) + cos(re(x))*cosh(im(x)) Note that this is just a wrapper around ``as_real_imag()``. Most objects that wish to redefine ``_eval_expand_complex()`` should consider redefining ``as_real_imag()`` instead. func ---- Expand other functions. >>> from sympy import gamma >>> gamma(x + 1).expand(func=True) x*gamma(x) trig ---- Do trigonometric expansions. >>> cos(x + y).expand(trig=True) -sin(x)*sin(y) + cos(x)*cos(y) >>> sin(2*x).expand(trig=True) 2*sin(x)*cos(x) Note that the forms of ``sin(n*x)`` and ``cos(n*x)`` in terms of ``sin(x)`` and ``cos(x)`` are not unique, due to the identity `\sin^2(x) + \cos^2(x) = 1`. The current implementation uses the form obtained from Chebyshev polynomials, but this may change. See `this MathWorld article <>`_ for more information. Notes ===== - You can shut off unwanted methods:: >>> (exp(x + y)*(x + y)).expand() x*exp(x)*exp(y) + y*exp(x)*exp(y) >>> (exp(x + y)*(x + y)).expand(power_exp=False) x*exp(x + y) + y*exp(x + y) >>> (exp(x + y)*(x + y)).expand(mul=False) (x + y)*exp(x)*exp(y) - Use deep=False to only expand on the top level:: >>> exp(x + exp(x + y)).expand() exp(x)*exp(exp(x)*exp(y)) >>> exp(x + exp(x + y)).expand(deep=False) exp(x)*exp(exp(x + y)) - Hints are applied in an arbitrary, but consistent order (in the current implementation, they are applied in alphabetical order, except multinomial comes before mul, but this may change). Because of this, some hints may prevent expansion by other hints if they are applied first. For example, ``mul`` may distribute multiplications and prevent ``log`` and ``power_base`` from expanding them. Also, if ``mul`` is applied before ``multinomial`, the expression might not be fully distributed. The solution is to use the various ``expand_hint`` helper functions or to use ``hint=False`` to this function to finely control which hints are applied. Here are some examples:: >>> from sympy import expand, expand_mul, expand_power_base >>> x, y, z = symbols('x,y,z', positive=True) >>> expand(log(x*(y + z))) log(x) + log(y + z) Here, we see that ``log`` was applied before ``mul``. To get the mul expanded form, either of the following will work:: >>> expand_mul(log(x*(y + z))) log(x*y + x*z) >>> expand(log(x*(y + z)), log=False) log(x*y + x*z) A similar thing can happen with the ``power_base`` hint:: >>> expand((x*(y + z))**x) (x*y + x*z)**x To get the ``power_base`` expanded form, either of the following will work:: >>> expand((x*(y + z))**x, mul=False) x**x*(y + z)**x >>> expand_power_base((x*(y + z))**x) x**x*(y + z)**x >>> expand((x + y)*y/x) y + y**2/x The parts of a rational expression can be targeted:: >>> expand((x + y)*y/x/(x + 1), frac=True) (x*y + y**2)/(x**2 + x) >>> expand((x + y)*y/x/(x + 1), numer=True) (x*y + y**2)/(x*(x + 1)) >>> expand((x + y)*y/x/(x + 1), denom=True) y*(x + y)/(x**2 + x) - The ``modulus`` meta-hint can be used to reduce the coefficients of an expression post-expansion:: >>> expand((3*x + 1)**2) 9*x**2 + 6*x + 1 >>> expand((3*x + 1)**2, modulus=5) 4*x**2 + x + 1 - Either ``expand()`` the function or ``.expand()`` the method can be used. Both are equivalent:: >>> expand((x + 1)**2) x**2 + 2*x + 1 >>> ((x + 1)**2).expand() x**2 + 2*x + 1 API === Objects can define their own expand hints by defining ``_eval_expand_hint()``. The function should take the form:: def _eval_expand_hint(self, **hints): # Only apply the method to the top-level expression ... See also the example below. Objects should define ``_eval_expand_hint()`` methods only if ``hint`` applies to that specific object. The generic ``_eval_expand_hint()`` method defined in Expr will handle the no-op case. Each hint should be responsible for expanding that hint only. Furthermore, the expansion should be applied to the top-level expression only. ``expand()`` takes care of the recursion that happens when ``deep=True``. You should only call ``_eval_expand_hint()`` methods directly if you are 100% sure that the object has the method, as otherwise you are liable to get unexpected ``AttributeError``s. Note, again, that you do not need to recursively apply the hint to args of your object: this is handled automatically by ``expand()``. ``_eval_expand_hint()`` should generally not be used at all outside of an ``_eval_expand_hint()`` method. If you want to apply a specific expansion from within another method, use the public ``expand()`` function, method, or ``expand_hint()`` functions. In order for expand to work, objects must be rebuildable by their args, i.e., ``obj.func(*obj.args) == obj`` must hold. Expand methods are passed ``**hints`` so that expand hints may use 'metahints'--hints that control how different expand methods are applied. For example, the ``force=True`` hint described above that causes ``expand(log=True)`` to ignore assumptions is such a metahint. The ``deep`` meta-hint is handled exclusively by ``expand()`` and is not passed to ``_eval_expand_hint()`` methods. Note that expansion hints should generally be methods that perform some kind of 'expansion'. For hints that simply rewrite an expression, use the .rewrite() API. Examples ======== >>> from sympy import Expr, sympify >>> class MyClass(Expr): ... def __new__(cls, *args): ... args = sympify(args) ... return Expr.__new__(cls, *args) ... ... def _eval_expand_double(self, **hints): ... ''' ... Doubles the args of MyClass. ... ... If there more than four args, doubling is not performed, ... unless force=True is also used (False by default). ... ''' ... force = hints.pop('force', False) ... if not force and len(self.args) > 4: ... return self ... return self.func(*(self.args + self.args)) ... >>> a = MyClass(1, 2, MyClass(3, 4)) >>> a MyClass(1, 2, MyClass(3, 4)) >>> a.expand(double=True) MyClass(1, 2, MyClass(3, 4, 3, 4), 1, 2, MyClass(3, 4, 3, 4)) >>> a.expand(double=True, deep=False) MyClass(1, 2, MyClass(3, 4), 1, 2, MyClass(3, 4)) >>> b = MyClass(1, 2, 3, 4, 5) >>> b.expand(double=True) MyClass(1, 2, 3, 4, 5) >>> b.expand(double=True, force=True) MyClass(1, 2, 3, 4, 5, 1, 2, 3, 4, 5) See Also ======== expand_log, expand_mul, expand_multinomial, expand_complex, expand_trig, expand_power_base, expand_power_exp, expand_func, hyperexpand """ # don't modify this; modify the Expr.expand method hints['power_base'] = power_base hints['power_exp'] = power_exp hints['mul'] = mul hints['log'] = log hints['multinomial'] = multinomial hints['basic'] = basic return sympify(e).expand(deep=deep, modulus=modulus, **hints) # This is a special application of two hints
def _mexpand(expr, recursive=False): # expand multinomials and then expand products; this may not always # be sufficient to give a fully expanded expression (see # test_issue_8247_8354 in test_arit) if expr is None: return was = None while was != expr: was, expr = expr, expand_mul(expand_multinomial(expr)) if not recursive: break return expr # These are simple wrappers around single hints.
[docs]def expand_mul(expr, deep=True): """ Wrapper around expand that only uses the mul hint. See the expand docstring for more information. Examples ======== >>> from sympy import symbols, expand_mul, exp, log >>> x, y = symbols('x,y', positive=True) >>> expand_mul(exp(x+y)*(x+y)*log(x*y**2)) x*exp(x + y)*log(x*y**2) + y*exp(x + y)*log(x*y**2) """ return sympify(expr).expand(deep=deep, mul=True, power_exp=False, power_base=False, basic=False, multinomial=False, log=False)
[docs]def expand_multinomial(expr, deep=True): """ Wrapper around expand that only uses the multinomial hint. See the expand docstring for more information. Examples ======== >>> from sympy import symbols, expand_multinomial, exp >>> x, y = symbols('x y', positive=True) >>> expand_multinomial((x + exp(x + 1))**2) x**2 + 2*x*exp(x + 1) + exp(2*x + 2) """ return sympify(expr).expand(deep=deep, mul=False, power_exp=False, power_base=False, basic=False, multinomial=True, log=False)
[docs]def expand_log(expr, deep=True, force=False): """ Wrapper around expand that only uses the log hint. See the expand docstring for more information. Examples ======== >>> from sympy import symbols, expand_log, exp, log >>> x, y = symbols('x,y', positive=True) >>> expand_log(exp(x+y)*(x+y)*log(x*y**2)) (x + y)*(log(x) + 2*log(y))*exp(x + y) """ return sympify(expr).expand(deep=deep, log=True, mul=False, power_exp=False, power_base=False, multinomial=False, basic=False, force=force)
[docs]def expand_func(expr, deep=True): """ Wrapper around expand that only uses the func hint. See the expand docstring for more information. Examples ======== >>> from sympy import expand_func, gamma >>> from import x >>> expand_func(gamma(x + 2)) x*(x + 1)*gamma(x) """ return sympify(expr).expand(deep=deep, func=True, basic=False, log=False, mul=False, power_exp=False, power_base=False, multinomial=False)
[docs]def expand_trig(expr, deep=True): """ Wrapper around expand that only uses the trig hint. See the expand docstring for more information. Examples ======== >>> from sympy import expand_trig, sin >>> from import x, y >>> expand_trig(sin(x+y)*(x+y)) (x + y)*(sin(x)*cos(y) + sin(y)*cos(x)) """ return sympify(expr).expand(deep=deep, trig=True, basic=False, log=False, mul=False, power_exp=False, power_base=False, multinomial=False)
[docs]def expand_complex(expr, deep=True): """ Wrapper around expand that only uses the complex hint. See the expand docstring for more information. Examples ======== >>> from sympy import expand_complex, exp, sqrt, I >>> from import z >>> expand_complex(exp(z)) I*exp(re(z))*sin(im(z)) + exp(re(z))*cos(im(z)) >>> expand_complex(sqrt(I)) sqrt(2)/2 + sqrt(2)*I/2 See Also ======== Expr.as_real_imag """ return sympify(expr).expand(deep=deep, complex=True, basic=False, log=False, mul=False, power_exp=False, power_base=False, multinomial=False)
[docs]def expand_power_base(expr, deep=True, force=False): """ Wrapper around expand that only uses the power_base hint. See the expand docstring for more information. A wrapper to expand(power_base=True) which separates a power with a base that is a Mul into a product of powers, without performing any other expansions, provided that assumptions about the power's base and exponent allow. deep=False (default is True) will only apply to the top-level expression. force=True (default is False) will cause the expansion to ignore assumptions about the base and exponent. When False, the expansion will only happen if the base is non-negative or the exponent is an integer. >>> from import x, y, z >>> from sympy import expand_power_base, sin, cos, exp >>> (x*y)**2 x**2*y**2 >>> (2*x)**y (2*x)**y >>> expand_power_base(_) 2**y*x**y >>> expand_power_base((x*y)**z) (x*y)**z >>> expand_power_base((x*y)**z, force=True) x**z*y**z >>> expand_power_base(sin((x*y)**z), deep=False) sin((x*y)**z) >>> expand_power_base(sin((x*y)**z), force=True) sin(x**z*y**z) >>> expand_power_base((2*sin(x))**y + (2*cos(x))**y) 2**y*sin(x)**y + 2**y*cos(x)**y >>> expand_power_base((2*exp(y))**x) 2**x*exp(y)**x >>> expand_power_base((2*cos(x))**y) 2**y*cos(x)**y Notice that sums are left untouched. If this is not the desired behavior, apply full ``expand()`` to the expression: >>> expand_power_base(((x+y)*z)**2) z**2*(x + y)**2 >>> (((x+y)*z)**2).expand() x**2*z**2 + 2*x*y*z**2 + y**2*z**2 >>> expand_power_base((2*y)**(1+z)) 2**(z + 1)*y**(z + 1) >>> ((2*y)**(1+z)).expand() 2*2**z*y*y**z """ return sympify(expr).expand(deep=deep, log=False, mul=False, power_exp=False, power_base=True, multinomial=False, basic=False, force=force)
[docs]def expand_power_exp(expr, deep=True): """ Wrapper around expand that only uses the power_exp hint. See the expand docstring for more information. Examples ======== >>> from sympy import expand_power_exp >>> from import x, y >>> expand_power_exp(x**(y + 2)) x**2*x**y """ return sympify(expr).expand(deep=deep, complex=False, basic=False, log=False, mul=False, power_exp=True, power_base=False, multinomial=False)
[docs]def count_ops(expr, visual=False): """ Return a representation (integer or expression) of the operations in expr. If ``visual`` is ``False`` (default) then the sum of the coefficients of the visual expression will be returned. If ``visual`` is ``True`` then the number of each type of operation is shown with the core class types (or their virtual equivalent) multiplied by the number of times they occur. If expr is an iterable, the sum of the op counts of the items will be returned. Examples ======== >>> from import a, b, x, y >>> from sympy import sin, count_ops Although there isn't a SUB object, minus signs are interpreted as either negations or subtractions: >>> (x - y).count_ops(visual=True) SUB >>> (-x).count_ops(visual=True) NEG Here, there are two Adds and a Pow: >>> (1 + a + b**2).count_ops(visual=True) 2*ADD + POW In the following, an Add, Mul, Pow and two functions: >>> (sin(x)*x + sin(x)**2).count_ops(visual=True) ADD + MUL + POW + 2*SIN for a total of 5: >>> (sin(x)*x + sin(x)**2).count_ops(visual=False) 5 Note that "what you type" is not always what you get. The expression 1/x/y is translated by sympy into 1/(x*y) so it gives a DIV and MUL rather than two DIVs: >>> (1/x/y).count_ops(visual=True) DIV + MUL The visual option can be used to demonstrate the difference in operations for expressions in different forms. Here, the Horner representation is compared with the expanded form of a polynomial: >>> eq=x*(1 + x*(2 + x*(3 + x))) >>> count_ops(eq.expand(), visual=True) - count_ops(eq, visual=True) -MUL + 3*POW The count_ops function also handles iterables: >>> count_ops([x, sin(x), None, True, x + 2], visual=False) 2 >>> count_ops([x, sin(x), None, True, x + 2], visual=True) ADD + SIN >>> count_ops({x: sin(x), x + 2: y + 1}, visual=True) 2*ADD + SIN """ from sympy import Integral, Symbol from sympy.simplify.radsimp import fraction from sympy.logic.boolalg import BooleanFunction expr = sympify(expr) if isinstance(expr, Expr): ops = [] args = [expr] NEG = Symbol('NEG') DIV = Symbol('DIV') SUB = Symbol('SUB') ADD = Symbol('ADD') while args: a = args.pop() # XXX: This is a hack to support non-Basic args if isinstance(a, string_types): continue if a.is_Rational: #-1/3 = NEG + DIV if a is not S.One: if a.p < 0: ops.append(NEG) if a.q != 1: ops.append(DIV) continue elif a.is_Mul: if _coeff_isneg(a): ops.append(NEG) if a.args[0] is S.NegativeOne: a = a.as_two_terms()[1] else: a = -a n, d = fraction(a) if n.is_Integer: ops.append(DIV) if n < 0: ops.append(NEG) args.append(d) continue # won't be -Mul but could be Add elif d is not S.One: if not d.is_Integer: args.append(d) ops.append(DIV) args.append(n) continue # could be -Mul elif a.is_Add: aargs = list(a.args) negs = 0 for i, ai in enumerate(aargs): if _coeff_isneg(ai): negs += 1 args.append(-ai) if i > 0: ops.append(SUB) else: args.append(ai) if i > 0: ops.append(ADD) if negs == len(aargs): # -x - y = NEG + SUB ops.append(NEG) elif _coeff_isneg(aargs[0]): # -x + y = SUB, but already recorded ADD ops.append(SUB - ADD) continue if a.is_Pow and a.exp is S.NegativeOne: ops.append(DIV) args.append(a.base) # won't be -Mul but could be Add continue if (a.is_Mul or a.is_Pow or a.is_Function or isinstance(a, Derivative) or isinstance(a, Integral)): o = Symbol(a.func.__name__.upper()) # count the args if (a.is_Mul or isinstance(a, LatticeOp)): ops.append(o*(len(a.args) - 1)) else: ops.append(o) if not a.is_Symbol: args.extend(a.args) elif type(expr) is dict: ops = [count_ops(k, visual=visual) + count_ops(v, visual=visual) for k, v in expr.items()] elif iterable(expr): ops = [count_ops(i, visual=visual) for i in expr] elif isinstance(expr, BooleanFunction): ops = [] for arg in expr.args: ops.append(count_ops(arg, visual=True)) o = Symbol(expr.func.__name__.upper()) ops.append(o) elif not isinstance(expr, Basic): ops = [] else: # it's Basic not isinstance(expr, Expr): if not isinstance(expr, Basic): raise TypeError("Invalid type of expr") else: ops = [] args = [expr] while args: a = args.pop() # XXX: This is a hack to support non-Basic args if isinstance(a, string_types): continue if a.args: o = Symbol(a.func.__name__.upper()) if a.is_Boolean: ops.append(o*(len(a.args)-1)) else: ops.append(o) args.extend(a.args) if not ops: if visual: return S.Zero return 0 ops = Add(*ops) if visual: return ops if ops.is_Number: return int(ops) return sum(int((a.args or [1])[0]) for a in Add.make_args(ops))
[docs]def nfloat(expr, n=15, exponent=False): """Make all Rationals in expr Floats except those in exponents (unless the exponents flag is set to True). Examples ======== >>> from sympy.core.function import nfloat >>> from import x, y >>> from sympy import cos, pi, sqrt >>> nfloat(x**4 + x/2 + cos(pi/3) + 1 + sqrt(y)) x**4 + 0.5*x + sqrt(y) + 1.5 >>> nfloat(x**4 + sqrt(y), exponent=True) x**4.0 + y**0.5 """ from sympy.core.power import Pow from sympy.polys.rootoftools import RootOf if iterable(expr, exclude=string_types): if isinstance(expr, (dict, Dict)): return type(expr)([(k, nfloat(v, n, exponent)) for k, v in list(expr.items())]) return type(expr)([nfloat(a, n, exponent) for a in expr]) rv = sympify(expr) if rv.is_Number: return Float(rv, n) elif rv.is_number: # evalf doesn't always set the precision rv = rv.n(n) if rv.is_Number: rv = Float(rv.n(n), n) else: pass # pure_complex(rv) is likely True return rv # watch out for RootOf instances that don't like to have # their exponents replaced with Dummies and also sometimes have # problems with evaluating at low precision (issue 6393) rv = rv.xreplace({ro: ro.n(n) for ro in rv.atoms(RootOf)}) if not exponent: reps = [(p, Pow(p.base, Dummy())) for p in rv.atoms(Pow)] rv = rv.xreplace(dict(reps)) rv = rv.n(n) if not exponent: rv = rv.xreplace({d.exp: p.exp for p, d in reps}) else: # Pow._eval_evalf special cases Integer exponents so if # exponent is suppose to be handled we have to do so here rv = rv.xreplace(Transform( lambda x: Pow(x.base, Float(x.exp, n)), lambda x: x.is_Pow and x.exp.is_Integer)) return rv.xreplace(Transform( lambda x: x.func(*nfloat(x.args, n, exponent)), lambda x: isinstance(x, Function)))
from sympy.core.symbol import Dummy, Symbol