# Source code for sympy.functions.elementary.miscellaneous

from __future__ import print_function, division

from sympy.core import S, sympify
from sympy.core.containers import Tuple
from sympy.core.operations import LatticeOp, ShortCircuit
from sympy.core.function import Application, Lambda, ArgumentIndexError
from sympy.core.expr import Expr
from sympy.core.mul import Mul
from sympy.core.numbers import Rational
from sympy.core.power import Pow
from sympy.core.relational import Equality
from sympy.core.singleton import Singleton
from sympy.core.symbol import Dummy
from sympy.core.rules import Transform
from sympy.core.compatibility import as_int, with_metaclass, range
from sympy.core.logic import fuzzy_and, fuzzy_or, _torf
from sympy.functions.elementary.integers import floor
from sympy.logic.boolalg import And

[docs]class IdentityFunction(with_metaclass(Singleton, Lambda)): """ The identity function Examples ======== >>> from sympy import Id, Symbol >>> x = Symbol('x') >>> Id(x) x """ def __new__(cls): from sympy.sets.sets import FiniteSet x = Dummy('x') #construct "by hand" to avoid infinite loop obj = Expr.__new__(cls, Tuple(x), x) obj.nargs = FiniteSet(1) return obj
Id = S.IdentityFunction ############################################################################### ############################# ROOT and SQUARE ROOT FUNCTION ################### ###############################################################################
[docs]def sqrt(arg): """The square root function sqrt(x) -> Returns the principal square root of x. Examples ======== >>> from sympy import sqrt, Symbol >>> x = Symbol('x') >>> sqrt(x) sqrt(x) >>> sqrt(x)**2 x Note that sqrt(x**2) does not simplify to x. >>> sqrt(x**2) sqrt(x**2) This is because the two are not equal to each other in general. For example, consider x == -1: >>> from sympy import Eq >>> Eq(sqrt(x**2), x).subs(x, -1) False This is because sqrt computes the principal square root, so the square may put the argument in a different branch. This identity does hold if x is positive: >>> y = Symbol('y', positive=True) >>> sqrt(y**2) y You can force this simplification by using the powdenest() function with the force option set to True: >>> from sympy import powdenest >>> sqrt(x**2) sqrt(x**2) >>> powdenest(sqrt(x**2), force=True) x To get both branches of the square root you can use the rootof function: >>> from sympy import rootof >>> [rootof(x**2-3,i) for i in (0,1)] [-sqrt(3), sqrt(3)] See Also ======== sympy.polys.rootoftools.rootof, root, real_root References ========== .. [1] http://en.wikipedia.org/wiki/Square_root .. [2] http://en.wikipedia.org/wiki/Principal_value """ # arg = sympify(arg) is handled by Pow return Pow(arg, S.Half)
def cbrt(arg): """This function computes the principial cube root of arg, so it's just a shortcut for arg**Rational(1, 3). Examples ======== >>> from sympy import cbrt, Symbol >>> x = Symbol('x') >>> cbrt(x) x**(1/3) >>> cbrt(x)**3 x Note that cbrt(x**3) does not simplify to x. >>> cbrt(x**3) (x**3)**(1/3) This is because the two are not equal to each other in general. For example, consider x == -1: >>> from sympy import Eq >>> Eq(cbrt(x**3), x).subs(x, -1) False This is because cbrt computes the principal cube root, this identity does hold if x is positive: >>> y = Symbol('y', positive=True) >>> cbrt(y**3) y See Also ======== sympy.polys.rootoftools.rootof, root, real_root References ========== * http://en.wikipedia.org/wiki/Cube_root * http://en.wikipedia.org/wiki/Principal_value """ return Pow(arg, Rational(1, 3))
[docs]def root(arg, n, k=0): """root(x, n, k) -> Returns the k-th n-th root of x, defaulting to the principle root (k=0). Examples ======== >>> from sympy import root, Rational >>> from sympy.abc import x, n >>> root(x, 2) sqrt(x) >>> root(x, 3) x**(1/3) >>> root(x, n) x**(1/n) >>> root(x, -Rational(2, 3)) x**(-3/2) To get the k-th n-th root, specify k: >>> root(-2, 3, 2) -(-1)**(2/3)*2**(1/3) To get all n n-th roots you can use the rootof function. The following examples show the roots of unity for n equal 2, 3 and 4: >>> from sympy import rootof, I >>> [rootof(x**2 - 1, i) for i in range(2)] [-1, 1] >>> [rootof(x**3 - 1,i) for i in range(3)] [1, -1/2 - sqrt(3)*I/2, -1/2 + sqrt(3)*I/2] >>> [rootof(x**4 - 1,i) for i in range(4)] [-1, 1, -I, I] SymPy, like other symbolic algebra systems, returns the complex root of negative numbers. This is the principal root and differs from the text-book result that one might be expecting. For example, the cube root of -8 does not come back as -2: >>> root(-8, 3) 2*(-1)**(1/3) The real_root function can be used to either make the principle result real (or simply to return the real root directly): >>> from sympy import real_root >>> real_root(_) -2 >>> real_root(-32, 5) -2 Alternatively, the n//2-th n-th root of a negative number can be computed with root: >>> root(-32, 5, 5//2) -2 See Also ======== sympy.polys.rootoftools.rootof sympy.core.power.integer_nthroot sqrt, real_root References ========== * http://en.wikipedia.org/wiki/Square_root * http://en.wikipedia.org/wiki/Real_root * http://en.wikipedia.org/wiki/Root_of_unity * http://en.wikipedia.org/wiki/Principal_value * http://mathworld.wolfram.com/CubeRoot.html """ n = sympify(n) if k: return Pow(arg, S.One/n)*S.NegativeOne**(2*k/n) return Pow(arg, 1/n)
def real_root(arg, n=None): """Return the real nth-root of arg if possible. If n is omitted then all instances of (-n)**(1/odd) will be changed to -n**(1/odd); this will only create a real root of a principle root -- the presence of other factors may cause the result to not be real. Examples ======== >>> from sympy import root, real_root, Rational >>> from sympy.abc import x, n >>> real_root(-8, 3) -2 >>> root(-8, 3) 2*(-1)**(1/3) >>> real_root(_) -2 If one creates a non-principle root and applies real_root, the result will not be real (so use with caution): >>> root(-8, 3, 2) -2*(-1)**(2/3) >>> real_root(_) -2*(-1)**(2/3) See Also ======== sympy.polys.rootoftools.rootof sympy.core.power.integer_nthroot root, sqrt """ from sympy import im, Piecewise if n is not None: try: n = as_int(n) arg = sympify(arg) if arg.is_positive or arg.is_negative: rv = root(arg, n) else: raise ValueError except ValueError: return root(arg, n)*Piecewise( (S.One, ~Equality(im(arg), 0)), (Pow(S.NegativeOne, S.One/n)**(2*floor(n/2)), And( Equality(n % 2, 1), arg < 0)), (S.One, True)) else: rv = sympify(arg) n1pow = Transform(lambda x: -(-x.base)**x.exp, lambda x: x.is_Pow and x.base.is_negative and x.exp.is_Rational and x.exp.p == 1 and x.exp.q % 2) return rv.xreplace(n1pow) ############################################################################### ############################# MINIMUM and MAXIMUM ############################# ############################################################################### class MinMaxBase(Expr, LatticeOp): def __new__(cls, *args, **assumptions): if not args: raise ValueError("The Max/Min functions must have arguments.") args = (sympify(arg) for arg in args) # first standard filter, for cls.zero and cls.identity # also reshape Max(a, Max(b, c)) to Max(a, b, c) try: _args = frozenset(cls._new_args_filter(args)) except ShortCircuit: return cls.zero # second filter # variant I: remove ones which can be removed # args = cls._collapse_arguments(set(_args), **assumptions) # variant II: find local zeros args = cls._find_localzeros(set(_args), **assumptions) if not args: return cls.identity elif len(args) == 1: return args.pop() else: # base creation # XXX should _args be made canonical with sorting? _args = frozenset(args) obj = Expr.__new__(cls, _args, **assumptions) obj._argset = _args return obj @classmethod def _new_args_filter(cls, arg_sequence): """ Generator filtering args. first standard filter, for cls.zero and cls.identity. Also reshape Max(a, Max(b, c)) to Max(a, b, c), and check arguments for comparability """ for arg in arg_sequence: # pre-filter, checking comparability of arguments if (not isinstance(arg, Expr)) or (arg.is_real is False) or (arg is S.ComplexInfinity): raise ValueError("The argument '%s' is not comparable." % arg) if arg == cls.zero: raise ShortCircuit(arg) elif arg == cls.identity: continue elif arg.func == cls: for x in arg.args: yield x else: yield arg @classmethod def _find_localzeros(cls, values, **options): """ Sequentially allocate values to localzeros. When a value is identified as being more extreme than another member it replaces that member; if this is never true, then the value is simply appended to the localzeros. """ localzeros = set() for v in values: is_newzero = True localzeros_ = list(localzeros) for z in localzeros_: if id(v) == id(z): is_newzero = False else: con = cls._is_connected(v, z) if con: is_newzero = False if con is True or con == cls: localzeros.remove(z) localzeros.update([v]) if is_newzero: localzeros.update([v]) return localzeros @classmethod def _is_connected(cls, x, y): """ Check if x and y are connected somehow. """ from sympy.core.exprtools import factor_terms def hit(v, t, f): if not v.is_Relational: return t if v else f for i in range(2): if x == y: return True r = hit(x >= y, Max, Min) if r is not None: return r r = hit(y <= x, Max, Min) if r is not None: return r r = hit(x <= y, Min, Max) if r is not None: return r r = hit(y >= x, Min, Max) if r is not None: return r # simplification can be expensive, so be conservative # in what is attempted x = factor_terms(x - y) y = S.Zero return False 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 evalf(self, prec=None, **options): return self.func(*[a.evalf(prec, **options) for a in self.args]) n = evalf _eval_is_algebraic = lambda s: _torf(i.is_algebraic for i in s.args) _eval_is_antihermitian = lambda s: _torf(i.is_antihermitian for i in s.args) _eval_is_commutative = lambda s: _torf(i.is_commutative for i in s.args) _eval_is_complex = lambda s: _torf(i.is_complex for i in s.args) _eval_is_composite = lambda s: _torf(i.is_composite for i in s.args) _eval_is_even = lambda s: _torf(i.is_even for i in s.args) _eval_is_finite = lambda s: _torf(i.is_finite for i in s.args) _eval_is_hermitian = lambda s: _torf(i.is_hermitian for i in s.args) _eval_is_imaginary = lambda s: _torf(i.is_imaginary for i in s.args) _eval_is_infinite = lambda s: _torf(i.is_infinite for i in s.args) _eval_is_integer = lambda s: _torf(i.is_integer for i in s.args) _eval_is_irrational = lambda s: _torf(i.is_irrational for i in s.args) _eval_is_negative = lambda s: _torf(i.is_negative for i in s.args) _eval_is_noninteger = lambda s: _torf(i.is_noninteger for i in s.args) _eval_is_nonnegative = lambda s: _torf(i.is_nonnegative for i in s.args) _eval_is_nonpositive = lambda s: _torf(i.is_nonpositive for i in s.args) _eval_is_nonzero = lambda s: _torf(i.is_nonzero for i in s.args) _eval_is_odd = lambda s: _torf(i.is_odd for i in s.args) _eval_is_polar = lambda s: _torf(i.is_polar for i in s.args) _eval_is_positive = lambda s: _torf(i.is_positive for i in s.args) _eval_is_prime = lambda s: _torf(i.is_prime for i in s.args) _eval_is_rational = lambda s: _torf(i.is_rational for i in s.args) _eval_is_real = lambda s: _torf(i.is_real for i in s.args) _eval_is_transcendental = lambda s: _torf(i.is_transcendental for i in s.args) _eval_is_zero = lambda s: _torf(i.is_zero for i in s.args)
[docs]class Max(MinMaxBase, Application): """ Return, if possible, the maximum value of the list. When number of arguments is equal one, then return this argument. When number of arguments is equal two, then return, if possible, the value from (a, b) that is >= the other. In common case, when the length of list greater than 2, the task is more complicated. Return only the arguments, which are greater than others, if it is possible to determine directional relation. If is not possible to determine such a relation, return a partially evaluated result. Assumptions are used to make the decision too. Also, only comparable arguments are permitted. It is named Max and not max to avoid conflicts with the built-in function max. Examples ======== >>> from sympy import Max, Symbol, oo >>> from sympy.abc import x, y >>> p = Symbol('p', positive=True) >>> n = Symbol('n', negative=True) >>> Max(x, -2) #doctest: +SKIP Max(x, -2) >>> Max(x, -2).subs(x, 3) 3 >>> Max(p, -2) p >>> Max(x, y) Max(x, y) >>> Max(x, y) == Max(y, x) True >>> Max(x, Max(y, z)) #doctest: +SKIP Max(x, y, z) >>> Max(n, 8, p, 7, -oo) #doctest: +SKIP Max(8, p) >>> Max (1, x, oo) oo * Algorithm The task can be considered as searching of supremums in the directed complete partial orders [1]_. The source values are sequentially allocated by the isolated subsets in which supremums are searched and result as Max arguments. If the resulted supremum is single, then it is returned. The isolated subsets are the sets of values which are only the comparable with each other in the current set. E.g. natural numbers are comparable with each other, but not comparable with the x symbol. Another example: the symbol x with negative assumption is comparable with a natural number. Also there are "least" elements, which are comparable with all others, and have a zero property (maximum or minimum for all elements). E.g. oo. In case of it the allocation operation is terminated and only this value is returned. Assumption: - if A > B > C then A > C - if A == B then B can be removed References ========== .. [1] http://en.wikipedia.org/wiki/Directed_complete_partial_order .. [2] http://en.wikipedia.org/wiki/Lattice_%28order%29 See Also ======== Min : find minimum values """ zero = S.Infinity identity = S.NegativeInfinity def fdiff( self, argindex ): from sympy import Heaviside n = len(self.args) if 0 < argindex and argindex <= n: argindex -= 1 if n == 2: return Heaviside(self.args[argindex] - self.args[1 - argindex]) newargs = tuple([self.args[i] for i in range(n) if i != argindex]) return Heaviside(self.args[argindex] - Max(*newargs)) else: raise ArgumentIndexError(self, argindex) def _eval_rewrite_as_Heaviside(self, *args): from sympy import Heaviside return Add(*[j*Mul(*[Heaviside(j - i) for i in args if i!=j]) \ for j in args]) def _eval_is_positive(self): return fuzzy_or(a.is_positive for a in self.args) def _eval_is_nonnegative(self): return fuzzy_or(a.is_nonnegative for a in self.args) def _eval_is_negative(self): return fuzzy_and(a.is_negative for a in self.args)
[docs]class Min(MinMaxBase, Application): """ Return, if possible, the minimum value of the list. It is named Min and not min to avoid conflicts with the built-in function min. Examples ======== >>> from sympy import Min, Symbol, oo >>> from sympy.abc import x, y >>> p = Symbol('p', positive=True) >>> n = Symbol('n', negative=True) >>> Min(x, -2) #doctest: +SKIP Min(x, -2) >>> Min(x, -2).subs(x, 3) -2 >>> Min(p, -3) -3 >>> Min(x, y) #doctest: +SKIP Min(x, y) >>> Min(n, 8, p, -7, p, oo) #doctest: +SKIP Min(n, -7) See Also ======== Max : find maximum values """ zero = S.NegativeInfinity identity = S.Infinity def fdiff( self, argindex ): from sympy import Heaviside n = len(self.args) if 0 < argindex and argindex <= n: argindex -= 1 if n == 2: return Heaviside( self.args[1-argindex] - self.args[argindex] ) newargs = tuple([ self.args[i] for i in range(n) if i != argindex]) return Heaviside( Min(*newargs) - self.args[argindex] ) else: raise ArgumentIndexError(self, argindex) def _eval_rewrite_as_Heaviside(self, *args): from sympy import Heaviside return Add(*[j*Mul(*[Heaviside(i-j) for i in args if i!=j]) \ for j in args]) def _eval_is_positive(self): return fuzzy_and(a.is_positive for a in self.args) def _eval_is_nonnegative(self): return fuzzy_and(a.is_nonnegative for a in self.args) def _eval_is_negative(self): return fuzzy_or(a.is_negative for a in self.args)