Python inequalities.solve_univariate_inequality函数代码示例

def continuous_domain(f, symbol, domain):
    """
    Returns the intervals in the given domain for which the function is continuous.
    This method is limited by the ability to determine the various
    singularities and discontinuities of the given function.

    Examples
    ========
    >>> from sympy import Symbol, S, tan, log, pi, sqrt
    >>> from sympy.sets import Interval
    >>> from sympy.calculus.util import continuous_domain
    >>> x = Symbol('x')
    >>> continuous_domain(1/x, x, S.Reals)
    (-oo, 0) U (0, oo)
    >>> continuous_domain(tan(x), x, Interval(0, pi))
    [0, pi/2) U (pi/2, pi]
    >>> continuous_domain(sqrt(x - 2), x, Interval(-5, 5))
    [2, 5]
    >>> continuous_domain(log(2*x - 1), x, S.Reals)
    (1/2, oo)

    """
    from sympy.solvers.inequalities import solve_univariate_inequality
    from sympy.solvers.solveset import solveset, _has_rational_power

    if domain.is_subset(S.Reals):
        constrained_interval = domain
        for atom in f.atoms(Pow):
            predicate, denom = _has_rational_power(atom, symbol)
            constraint = S.EmptySet
            if predicate and denom == 2:
                constraint = solve_univariate_inequality(atom.base >= 0,
                                                         symbol).as_set()
                constrained_interval = Intersection(constraint,
                                                    constrained_interval)

        for atom in f.atoms(log):
            constraint = solve_univariate_inequality(atom.args[0] > 0,
                                                     symbol).as_set()
            constrained_interval = Intersection(constraint,
                                                constrained_interval)

        domain = constrained_interval

    try:
        sings = S.EmptySet
        for atom in f.atoms(Pow):
            predicate, denom = _has_rational_power(atom, symbol)
            if predicate and denom == 2:
                sings = solveset(1/f, symbol, domain)
                break
        else:
            sings = Intersection(solveset(1/f, symbol), domain)

    except:
        raise NotImplementedError("Methods for determining the continuous domains"
                                  " of this function has not been developed.")

    return domain - sings

    def as_set(self):
        """
        Rewrites univariate inequality in terms of real sets

        Examples
        ========

        >>> from sympy import Symbol, Eq
        >>> x = Symbol('x', real=True)
        >>> (x > 0).as_set()
        Interval.open(0, oo)
        >>> Eq(x, 0).as_set()
        {0}

        """
        from sympy.solvers.inequalities import solve_univariate_inequality
        syms = self.free_symbols

        if len(syms) == 1:
            sym = syms.pop()
        else:
            raise NotImplementedError("Sorry, Relational.as_set procedure"
                                      " is not yet implemented for"
                                      " multivariate expressions")

        return solve_univariate_inequality(self, sym, relational=False)

 def _eval_as_set(self):
     # self is univariate and periodicity(self, x) in (0, None)
     from sympy.solvers.inequalities import solve_univariate_inequality
     syms = self.free_symbols
     assert len(syms) == 1
     x = syms.pop()
     return solve_univariate_inequality(self, x, relational=False)

def _solve_abs(f, symbol):
    """ Helper function to solve equation involving absolute value function """
    p, q, r = Wild('p'), Wild('q'), Wild('r')
    pattern_match = f.match(p*Abs(q) + r) or {}
    if not pattern_match.get(p, S.Zero).is_zero:
        f_p, f_q, f_r = pattern_match[p], pattern_match[q], pattern_match[r]
        q_pos_cond = solve_univariate_inequality(f_q >= 0, symbol,
                                                 relational=False)
        q_neg_cond = solve_univariate_inequality(f_q < 0, symbol,
                                                 relational=False)

        sols_q_pos = solveset_real(f_p*f_q + f_r,
                                           symbol).intersect(q_pos_cond)
        sols_q_neg = solveset_real(f_p*(-f_q) + f_r,
                                           symbol).intersect(q_neg_cond)
        return Union(sols_q_pos, sols_q_neg)
    else:
        return ConditionSet(symbol, Eq(f, 0), S.Complexes)

def _solve_abs(f, symbol):
    """ Helper function to solve equation involving absolute value function """
    from sympy.solvers.inequalities import solve_univariate_inequality
    assert f.has(Abs)
    p, q, r = Wild('p'), Wild('q'), Wild('r')
    pattern_match = f.match(p*Abs(q) + r)
    if not pattern_match[p].is_zero:
        f_p, f_q, f_r = pattern_match[p], pattern_match[q], pattern_match[r]
        q_pos_cond = solve_univariate_inequality(f_q >= 0, symbol,
                                                 relational=False)
        q_neg_cond = solve_univariate_inequality(f_q < 0, symbol,
                                                 relational=False)

        sols_q_pos = solveset_real(f_p*f_q + f_r,
                                           symbol).intersect(q_pos_cond)
        sols_q_neg = solveset_real(f_p*(-f_q) + f_r,
                                           symbol).intersect(q_neg_cond)
        return Union(sols_q_pos, sols_q_neg)
    else:
        raise NotImplementedError

def _solve_abs(f, symbol, domain):
    """ Helper function to solve equation involving absolute value function """
    if not domain.is_subset(S.Reals):
        raise ValueError(filldedent('''
            Absolute values cannot be inverted in the
            complex domain.'''))
    p, q, r = Wild('p'), Wild('q'), Wild('r')
    pattern_match = f.match(p*Abs(q) + r) or {}
    if not pattern_match.get(p, S.Zero).is_zero:
        f_p, f_q, f_r = pattern_match[p], pattern_match[q], pattern_match[r]
        q_pos_cond = solve_univariate_inequality(f_q >= 0, symbol,
                                                 relational=False)
        q_neg_cond = solve_univariate_inequality(f_q < 0, symbol,
                                                 relational=False)

        sols_q_pos = solveset_real(f_p*f_q + f_r,
                                           symbol).intersect(q_pos_cond)
        sols_q_neg = solveset_real(f_p*(-f_q) + f_r,
                                           symbol).intersect(q_neg_cond)
        return Union(sols_q_pos, sols_q_neg)
    else:
        return ConditionSet(symbol, Eq(f, 0), domain)

#.........这里部分代码省略.........
    solveset_complex: solver for complex domain

    Examples
    ========

    >>> from sympy import exp, sin, Symbol, pprint, S
    >>> from sympy.solvers.solveset import solveset, solveset_real

    * The default domain is complex. Not specifying a domain will lead
      to the solving of the equation in the complex domain (and this
      is not affected by the assumptions on the symbol):

    >>> x = Symbol('x')
    >>> pprint(solveset(exp(x) - 1, x), use_unicode=False)
    {2*n*I*pi | n in Integers()}

    >>> x = Symbol('x', real=True)
    >>> pprint(solveset(exp(x) - 1, x), use_unicode=False)
    {2*n*I*pi | n in Integers()}

    * If you want to use `solveset` to solve the equation in the
      real domain, provide a real domain. (Using `solveset\_real`
      does this automatically.)

    >>> R = S.Reals
    >>> x = Symbol('x')
    >>> solveset(exp(x) - 1, x, R)
    {0}
    >>> solveset_real(exp(x) - 1, x)
    {0}

    The solution is mostly unaffected by assumptions on the symbol,
    but there may be some slight difference:

    >>> pprint(solveset(sin(x)/x,x), use_unicode=False)
    ({2*n*pi | n in Integers()} \ {0}) U ({2*n*pi + pi | n in Integers()} \ {0})

    >>> p = Symbol('p', positive=True)
    >>> pprint(solveset(sin(p)/p, p), use_unicode=False)
    {2*n*pi | n in Integers()} U {2*n*pi + pi | n in Integers()}

    * Inequalities can be solved over the real domain only. Use of a complex
      domain leads to a NotImplementedError.

    >>> solveset(exp(x) > 1, x, R)
    (0, oo)

    """
    f = sympify(f)

    if f is S.true:
        return domain

    if f is S.false:
        return S.EmptySet

    if not isinstance(f, (Expr, Number)):
        raise ValueError("%s is not a valid SymPy expression" % (f))

    free_symbols = f.free_symbols

    if not free_symbols:
        b = Eq(f, 0)
        if b is S.true:
            return domain
        elif b is S.false:
            return S.EmptySet
        else:
            raise NotImplementedError(filldedent('''
                relationship between value and 0 is unknown: %s''' % b))

    if symbol is None:
        if len(free_symbols) == 1:
            symbol = free_symbols.pop()
        else:
            raise ValueError(filldedent('''
                The independent variable must be specified for a
                multivariate equation.'''))
    elif not getattr(symbol, 'is_Symbol', False):
        raise ValueError('A Symbol must be given, not type %s: %s' %
            (type(symbol), symbol))

    if isinstance(f, Eq):
        from sympy.core import Add
        f = Add(f.lhs, - f.rhs, evaluate=False)
    elif f.is_Relational:
        if not domain.is_subset(S.Reals):
            raise NotImplementedError(filldedent('''
                Inequalities in the complex domain are
                not supported. Try the real domain by
                setting domain=S.Reals'''))
        try:
            result = solve_univariate_inequality(
            f, symbol, relational=False) - _invalid_solutions(
            f, symbol, domain)
        except NotImplementedError:
            result = ConditionSet(symbol, f, domain)
        return result

    return _solveset(f, symbol, domain, _check=True)

def solveset(f, symbol=None):
    """Solves a given inequality or equation with set as output

    Parameters
    ==========

    f : Expr or a relational.
        The target equation or inequality
    symbol : Symbol
        The variable for which the equation is solved

    Returns
    =======

    Set
        A set of values for `symbol` for which `f` is True or is equal to
        zero. An `EmptySet` is returned if no solution is found.

    `solveset` claims to be complete in the solution set that it returns.

    Raises
    ======

    NotImplementedError
        The algorithms for to find the solution of the given equation are
        not yet implemented.
    ValueError
        The input is not valid.
    RuntimeError
        It is a bug, please report to the github issue tracker.


    `solveset` uses two underlying functions `solveset_real` and
    `solveset_complex` to solve equations. They are
    the solvers for real and complex domain respectively. The domain of
    the solver is decided by the assumption on the variable for which the
    equation is being solved.


    See Also
    ========

    solveset_real: solver for real domain
    solveset_complex: solver for complex domain

    Examples
    ========

    >>> from sympy import exp, Symbol, Eq, pprint
    >>> from sympy.solvers.solveset import solveset
    >>> from sympy.abc import x

    Symbols in Sympy are complex by default. A complex variable
    will lead to the solving of the equation in complex domain
    >>> pprint(solveset(exp(x) - 1, x), use_unicode=False)
    {2*n*I*pi | n in Integers()}

    If you want to solve equation in real domain by the `solveset`
    interface, then specify the variable to real. Alternatively use
    `solveset_real`.
    >>> x = Symbol('x', real=True)
    >>> solveset(exp(x) - 1, x)
    {0}
    >>> solveset(Eq(exp(x), 1), x)
    {0}

    Inequalities are always solved in the real domain irrespective of
    the assumption on the variable for which the inequality is solved.
    >>> solveset(exp(x) > 1, x)
    (0, oo)

    """

    from sympy.solvers.inequalities import solve_univariate_inequality

    if symbol is None:
        free_symbols = f.free_symbols
        if len(free_symbols) == 1:
            symbol = free_symbols.pop()
        else:
            raise ValueError(filldedent('''
                The independent variable must be specified for a
                multivariate equation.'''))
    elif not symbol.is_Symbol:
        raise ValueError('A Symbol must be given, not type %s: %s' % (type(symbol), symbol))

    real = (symbol.is_real is True)

    f = sympify(f)

    if isinstance(f, Eq):
        f = f.lhs - f.rhs

    if f.is_Relational:
        if real is False:
            warnings.warn(filldedent('''
                The variable you are solving for is complex
                but will assumed to be real since solving complex
                inequalities is not supported.
            '''))
#.........这里部分代码省略.........

def solveset(f, symbol=None, domain=S.Complexes):
    """Solves a given inequality or equation with set as output

    Parameters
    ==========

    f : Expr or a relational.
        The target equation or inequality
    symbol : Symbol
        The variable for which the equation is solved
    domain : Set
        The domain over which the equation is solved

    Returns
    =======

    Set
        A set of values for `symbol` for which `f` is True or is equal to
        zero. An `EmptySet` is returned if no solution is found.
        A `ConditionSet` is returned as unsolved object if algorithms
        to evaluatee complete solution are not yet implemented.

    `solveset` claims to be complete in the solution set that it returns.

    Raises
    ======

    NotImplementedError
        The algorithms to solve inequalities in complex domain  are
        not yet implemented.
    ValueError
        The input is not valid.
    RuntimeError
        It is a bug, please report to the github issue tracker.


    `solveset` uses two underlying functions `solveset_real` and
    `solveset_complex` to solve equations. They are the solvers for real and
    complex domain respectively. `solveset` ignores the assumptions on the
    variable being solved for and instead, uses the `domain` parameter to
    decide which solver to use.


    See Also
    ========

    solveset_real: solver for real domain
    solveset_complex: solver for complex domain

    Examples
    ========

    >>> from sympy import exp, Symbol, Eq, pprint, S, solveset
    >>> from sympy.abc import x

    * The default domain is complex. Not specifying a domain will lead to the
      solving of the equation in the complex domain.

    >>> pprint(solveset(exp(x) - 1, x), use_unicode=False)
    {2*n*I*pi | n in Integers()}

    * If you want to solve equation in real domain by the `solveset`
      interface, then specify that the domain is real. Alternatively use
      `solveset\_real`.

    >>> x = Symbol('x')
    >>> solveset(exp(x) - 1, x, S.Reals)
    {0}
    >>> solveset(Eq(exp(x), 1), x, S.Reals)
    {0}

    * Inequalities can be solved over the real domain only. Use of a complex
      domain leads to a NotImplementedError.

    >>> solveset(exp(x) > 1, x, S.Reals)
    (0, oo)

    """

    from sympy.solvers.inequalities import solve_univariate_inequality

    if symbol is None:
        free_symbols = f.free_symbols
        if len(free_symbols) == 1:
            symbol = free_symbols.pop()
        else:
            raise ValueError(filldedent('''
                The independent variable must be specified for a
                multivariate equation.'''))
    elif not symbol.is_Symbol:
        raise ValueError('A Symbol must be given, not type %s: %s' % (type(symbol), symbol))

    f = sympify(f)

    if f is S.false:
        return EmptySet()

    if f is S.true:
        return domain

#.........这里部分代码省略.........

def is_convex(f, *syms, **kwargs):
    """Determines the  convexity of the function passed in the argument.

    Parameters
    ==========

    f : Expr
        The concerned function.
    syms : Tuple of symbols
        The variables with respect to which the convexity is to be determined.
    domain : Interval, optional
        The domain over which the convexity of the function has to be checked.
        If unspecified, S.Reals will be the default domain.

    Returns
    =======

    Boolean
        The method returns `True` if the function is convex otherwise it
        returns `False`.

    Raises
    ======

    NotImplementedError
        The check for the convexity of multivariate functions is not implemented yet.

    Notes
    =====

    To determine concavity of a function pass `-f` as the concerned function.
    To determine logarithmic convexity of a function pass log(f) as
    concerned function.
    To determine logartihmic concavity of a function pass -log(f) as
    concerned function.

    Currently, convexity check of multivariate functions is not handled.

    Examples
    ========

    >>> from sympy import symbols, exp, oo, Interval
    >>> from sympy.calculus.util import is_convex
    >>> x = symbols('x')
    >>> is_convex(exp(x), x)
    True
    >>> is_convex(x**3, x, domain = Interval(-1, oo))
    False

    References
    ==========

    .. [1] https://en.wikipedia.org/wiki/Convex_function
    .. [2] http://www.ifp.illinois.edu/~angelia/L3_convfunc.pdf
    .. [3] https://en.wikipedia.org/wiki/Logarithmically_convex_function
    .. [4] https://en.wikipedia.org/wiki/Logarithmically_concave_function
    .. [5] https://en.wikipedia.org/wiki/Concave_function

    """

    if len(syms) > 1:
        raise NotImplementedError(
            "The check for the convexity of multivariate functions is not implemented yet.")

    f = _sympify(f)
    domain = kwargs.get('domain', S.Reals)
    var = syms[0]
    condition = f.diff(var, 2) < 0
    if solve_univariate_inequality(condition, var, False, domain):
        return False
    return True

def continuous_domain(f, symbol, domain):
    """
    Returns the intervals in the given domain for which the function
    is continuous.
    This method is limited by the ability to determine the various
    singularities and discontinuities of the given function.

    Parameters
    ==========

    f : Expr
        The concerned function.
    symbol : Symbol
        The variable for which the intervals are to be determined.
    domain : Interval
        The domain over which the continuity of the symbol has to be checked.

    Examples
    ========

    >>> from sympy import Symbol, S, tan, log, pi, sqrt
    >>> from sympy.sets import Interval
    >>> from sympy.calculus.util import continuous_domain
    >>> x = Symbol('x')
    >>> continuous_domain(1/x, x, S.Reals)
    Union(Interval.open(-oo, 0), Interval.open(0, oo))
    >>> continuous_domain(tan(x), x, Interval(0, pi))
    Union(Interval.Ropen(0, pi/2), Interval.Lopen(pi/2, pi))
    >>> continuous_domain(sqrt(x - 2), x, Interval(-5, 5))
    Interval(2, 5)
    >>> continuous_domain(log(2*x - 1), x, S.Reals)
    Interval.open(1/2, oo)

    Returns
    =======

    Interval
        Union of all intervals where the function is continuous.

    Raises
    ======
    NotImplementedError
        If the method to determine continuity of such a function
        has not yet been developed.

    """
    from sympy.solvers.inequalities import solve_univariate_inequality
    from sympy.solvers.solveset import solveset, _has_rational_power

    if domain.is_subset(S.Reals):
        constrained_interval = domain
        for atom in f.atoms(Pow):
            predicate, denomin = _has_rational_power(atom, symbol)
            constraint = S.EmptySet
            if predicate and denomin == 2:
                constraint = solve_univariate_inequality(atom.base >= 0,
                                                         symbol).as_set()
                constrained_interval = Intersection(constraint,
                                                    constrained_interval)

        for atom in f.atoms(log):
            constraint = solve_univariate_inequality(atom.args[0] > 0,
                                                     symbol).as_set()
            constrained_interval = Intersection(constraint,
                                                constrained_interval)

        domain = constrained_interval

    try:
        sings = S.EmptySet
        if f.has(Abs):
            sings = solveset(1/f, symbol, domain) + \
                solveset(denom(together(f)), symbol, domain)
        else:
            for atom in f.atoms(Pow):
                predicate, denomin = _has_rational_power(atom, symbol)
                if predicate and denomin == 2:
                    sings = solveset(1/f, symbol, domain) +\
                        solveset(denom(together(f)), symbol, domain)
                    break
            else:
                sings = Intersection(solveset(1/f, symbol), domain) + \
                    solveset(denom(together(f)), symbol, domain)

    except NotImplementedError:
        import sys
        raise (NotImplementedError("Methods for determining the continuous domains"
                                   " of this function have not been developed."),
               None,
               sys.exc_info()[2])

    return domain - sings