/

# Source code for sympy.functions.elementary.miscellaneous

from __future__ import print_function, division

from sympy.core import S, C, sympify
from sympy.core.add import Add
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 _.

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
==========

..  http://en.wikipedia.org/wiki/Directed_complete_partial_order
..  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)