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# Source code for sympy.functions.elementary.miscellaneous

from __future__ import print_function, division

from sympy.core import S, C, sympify
from sympy.core.basic import Basic
from sympy.core.containers import Tuple
from sympy.core.numbers import Rational
from sympy.core.operations import LatticeOp, ShortCircuit
from sympy.core.function import Application, Lambda, ArgumentIndexError
from sympy.core.expr import Expr
from sympy.core.singleton import Singleton
from sympy.core.rules import Transform
from sympy.core.compatibility import as_int, with_metaclass, xrange
from sympy.core.logic import fuzzy_and

[docs]class IdentityFunction(with_metaclass(Singleton, Lambda)): """ The identity function Examples ======== >>> from sympy import Id, Symbol >>> x = Symbol('x') >>> Id(x) x """ __slots__ = [] nargs = 1 def __new__(cls): x = C.Dummy('x') #construct "by hand" to avoid infinite loop return Expr.__new__(cls, Tuple(x), x)
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 ========== * http://en.wikipedia.org/wiki/Square_root * http://en.wikipedia.org/wiki/Principal_value """ # arg = sympify(arg) is handled by Pow return C.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 C.Pow(arg, C.Rational(1, 3))
[docs]def root(arg, n): """The n-th root function (a shortcut for arg**(1/n)) root(x, n) -> Returns the principal n-th root of x. 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 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 (0,1) ] [-1, 1] >>> [ RootOf(x**3-1,i) for i in (0,1,2) ] [1, -1/2 - sqrt(3)*I/2, -1/2 + sqrt(3)*I/2] >>> [ RootOf(x**4-1,i) for i in (0,1,2,3) ] [-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 such a result real or simply return the real root in the first place: >>> from sympy import real_root >>> real_root(_) -2 >>> real_root(-32, 5) -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) return C.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 -1**(1/odd) will be changed to -1. 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 See Also ======== sympy.polys.rootoftools.RootOf sympy.core.power.integer_nthroot root, sqrt """ if n is not None: n = as_int(n) rv = C.Pow(arg, Rational(1, n)) if n % 2 == 0: return rv else: rv = sympify(arg) n1pow = Transform(lambda x: S.NegativeOne, lambda x: x.is_Pow and x.base is S.NegativeOne 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) _args = frozenset(args) if not _args: return cls.identity elif len(_args) == 1: return set(_args).pop() else: # base creation 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 (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.iter_basic_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 elif cls._is_connected(v, z): is_newzero = False if cls._is_asneeded(v, z): 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. """ if (x == y) or isinstance(x > y, bool) or isinstance(x < y, bool): return True if x.is_Number and y.is_Number: return True return False @classmethod def _is_asneeded(cls, x, y): """ Check if x and y satisfy relation condition. The relation condition for Max function is x > y, for Min function is x < y. They are defined in children Max and Min classes through the method _rel(cls, x, y) """ if (x == y): return False if x.is_Number and y.is_Number: if cls._rel(x, y): return True xy = cls._rel(x, y) if isinstance(xy, bool): if xy: return True return False yx = cls._rel_inversed(x, y) if isinstance(yx, bool): if yx: return False # never occurs? return True 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) @property def is_real(self): return fuzzy_and(arg.is_real for arg in self.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. 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) #doctest: +SKIP 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 @classmethod def _rel(cls, x, y): """ Check if x > y. """ return (x > y) @classmethod def _rel_inversed(cls, x, y): """ Check if x < y. """ return (x < y) def fdiff( self, argindex ): from sympy.functions.special.delta_functions 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 xrange(n) if i != argindex]) return Heaviside( self.args[argindex] - Max(*newargs) ) else: raise ArgumentIndexError(self, argindex)
[docs]class Min(MinMaxBase, Application): """ Return, if possible, the minimum value of the list. 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 @classmethod def _rel(cls, x, y): """ Check if x < y. """ return (x < y) @classmethod def _rel_inversed(cls, x, y): """ Check if x > y. """ return (x > y) def fdiff( self, argindex ): from sympy.functions.special.delta_functions 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 xrange(n) if i != argindex]) return Heaviside( Min(*newargs) - self.args[argindex] ) else: raise ArgumentIndexError(self, argindex)