Source code for sympy.functions.elementary.miscellaneous

from sympy.core import S, C, sympify
from sympy.core.basic import Basic
from sympy.core.containers import Tuple
from sympy.core.operations import LatticeOp, ShortCircuit
from sympy.core.function import Application, Lambda
from sympy.core.expr import Expr
from sympy.core.singleton import Singleton

class IdentityFunction(Lambda):
    """The identity function

    >>> from sympy import Id, Symbol
    >>> x = Symbol('x')
    >>> Id(x)
    x
    """
    __metaclass__ = Singleton
    __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

###############################################################################
############################# SQUARE ROOT FUNCTION ############################
###############################################################################

[docs]def sqrt(arg): # arg = sympify(arg) is handled by Pow return C.Pow(arg, S.Half) ############################################################################### ############################# MINIMUM and MAXIMUM ############################# ###############################################################################
class MinMaxBase(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 == 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. If value is greter than all of the localzeros, then it is new localzero and it is apending to them. if value is greter than one of the localzeros, then update localzero's set. """ localzeros = set() for v in values: is_newzero = True 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]) break 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
[docs]class Max(MinMaxBase, Application, Basic): """ 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. Example ------- >>> 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 [1] http://en.wikipedia.org/wiki/Directed_complete_partial_order [2] http://en.wikipedia.org/wiki/Lattice_(order) 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)
[docs]class Min(MinMaxBase, Application, Basic): """ Return, if possible, the minimum value of the list. Example ------- >>> 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)