Esempio n. 1
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def _solve_radical(f, symbol, solveset_solver):
    """ Helper function to solve equations with radicals """
    eq, cov = unrad(f)
    if not cov:
        result = solveset_solver(eq, symbol) - \
            Union(*[solveset_solver(g, symbol) for g in denoms(f, [symbol])])
    else:
        y, yeq = cov
        if not solveset_solver(y - I, y):
            yreal = Dummy('yreal', real=True)
            yeq = yeq.xreplace({y: yreal})
            eq = eq.xreplace({y: yreal})
            y = yreal
        g_y_s = solveset_solver(yeq, symbol)
        f_y_sols = solveset_solver(eq, y)
        result = Union(*[imageset(Lambda(y, g_y), f_y_sols) for g_y in g_y_s])

    return FiniteSet(*[s for s in result if checksol(f, symbol, s) is True])
Esempio n. 2
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def _solve_radical(f, symbol, solveset_solver):
    """ Helper function to solve equations with radicals """
    eq, cov = unrad(f)
    if not cov:
        result = solveset_solver(eq, symbol) - \
            Union(*[solveset_solver(g, symbol) for g in denoms(f, [symbol])])
    else:
        y, yeq = cov
        if not solveset_solver(y - I, y):
            yreal = Dummy('yreal', real=True)
            yeq = yeq.xreplace({y: yreal})
            eq = eq.xreplace({y: yreal})
            y = yreal
        g_y_s = solveset_solver(yeq, symbol)
        f_y_sols = solveset_solver(eq, y)
        result = Union(*[imageset(Lambda(y, g_y), f_y_sols)
                         for g_y in g_y_s])

    return FiniteSet(*[s for s in result if checksol(f, symbol, s) is True])
Esempio n. 3
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def _solve_inequality(ie, s, linear=False):
    """Return the inequality with s isolated on the left, if possible.
    If the relationship is non-linear, a solution involving And or Or
    may be returned. False or True are returned if the relationship
    is never True or always True, respectively.

    If `linear` is True (default is False) an `s`-dependent expression
    will be isoloated on the left, if possible
    but it will not be solved for `s` unless the expression is linear
    in `s`. Furthermore, only "safe" operations which don't change the
    sense of the relationship are applied: no division by an unsigned
    value is attempted unless the relationship involves Eq or Ne and
    no division by a value not known to be nonzero is ever attempted.

    Examples
    ========

    >>> from sympy import Eq, Symbol
    >>> from sympy.solvers.inequalities import _solve_inequality as f
    >>> from sympy.abc import x, y

    For linear expressions, the symbol can be isolated:

    >>> f(x - 2 < 0, x)
    x < 2
    >>> f(-x - 6 < x, x)
    x > -3

    Sometimes nonlinear relationships will be False

    >>> f(x**2 + 4 < 0, x)
    False

    Or they may involve more than one region of values:

    >>> f(x**2 - 4 < 0, x)
    (-2 < x) & (x < 2)

    To restrict the solution to a relational, set linear=True
    and only the x-dependent portion will be isolated on the left:

    >>> f(x**2 - 4 < 0, x, linear=True)
    x**2 < 4

    Division of only nonzero quantities is allowed, so x cannot
    be isolated by dividing by y:

    >>> y.is_nonzero is None  # it is unknown whether it is 0 or not
    True
    >>> f(x*y < 1, x)
    x*y < 1

    And while an equality (or unequality) still holds after dividing by a
    non-zero quantity

    >>> nz = Symbol('nz', nonzero=True)
    >>> f(Eq(x*nz, 1), x)
    Eq(x, 1/nz)

    the sign must be known for other inequalities involving > or <:

    >>> f(x*nz <= 1, x)
    nz*x <= 1
    >>> p = Symbol('p', positive=True)
    >>> f(x*p <= 1, x)
    x <= 1/p

    When there are denominators in the original expression that
    are removed by expansion, conditions for them will be returned
    as part of the result:

    >>> f(x < x*(2/x - 1), x)
    (x < 1) & Ne(x, 0)
    """
    from sympy.solvers.solvers import denoms
    if s not in ie.free_symbols:
        return ie
    if ie.rhs == s:
        ie = ie.reversed
    if ie.lhs == s and s not in ie.rhs.free_symbols:
        return ie

    def classify(ie, s, i):
        # return True or False if ie evaluates when substituting s with
        # i else None (if unevaluated) or NaN (when there is an error
        # in evaluating)
        try:
            v = ie.subs(s, i)
            if v is S.NaN:
                return v
            elif v not in (True, False):
                return
            return v
        except TypeError:
            return S.NaN

    rv = None
    oo = S.Infinity
    expr = ie.lhs - ie.rhs
    try:
        p = Poly(expr, s)
        if p.degree() == 0:
            rv = ie.func(p.as_expr(), 0)
        elif not linear and p.degree() > 1:
            # handle in except clause
            raise NotImplementedError
    except (PolynomialError, NotImplementedError):
        if not linear:
            try:
                rv = reduce_rational_inequalities([[ie]], s)
            except PolynomialError:
                rv = solve_univariate_inequality(ie, s)
            # remove restrictions wrt +/-oo that may have been
            # applied when using sets to simplify the relationship
            okoo = classify(ie, s, oo)
            if okoo is S.true and classify(rv, s, oo) is S.false:
                rv = rv.subs(s < oo, True)
            oknoo = classify(ie, s, -oo)
            if (oknoo is S.true and classify(rv, s, -oo) is S.false):
                rv = rv.subs(-oo < s, True)
                rv = rv.subs(s > -oo, True)
            if rv is S.true:
                rv = (s <= oo) if okoo is S.true else (s < oo)
                if oknoo is not S.true:
                    rv = And(-oo < s, rv)
        else:
            p = Poly(expr)

    conds = []
    if rv is None:
        e = p.as_expr()  # this is in expanded form
        # Do a safe inversion of e, moving non-s terms
        # to the rhs and dividing by a nonzero factor if
        # the relational is Eq/Ne; for other relationals
        # the sign must also be positive or negative
        rhs = 0
        b, ax = e.as_independent(s, as_Add=True)
        e -= b
        rhs -= b
        ef = factor_terms(e)
        a, e = ef.as_independent(s, as_Add=False)
        if (a.is_zero != False or  # don't divide by potential 0
                a.is_negative == a.is_positive == None
                and  # if sign is not known then
                ie.rel_op not in ('!=', '==')):  # reject if not Eq/Ne
            e = ef
            a = S.One
        rhs /= a
        if a.is_positive:
            rv = ie.func(e, rhs)
        else:
            rv = ie.reversed.func(e, rhs)

        # return conditions under which the value is
        # valid, too.
        beginning_denoms = denoms(ie.lhs) | denoms(ie.rhs)
        current_denoms = denoms(rv)
        for d in beginning_denoms - current_denoms:
            c = _solve_inequality(Eq(d, 0), s, linear=linear)
            if isinstance(c, Eq) and c.lhs == s:
                if classify(rv, s, c.rhs) is S.true:
                    # rv is permitting this value but it shouldn't
                    conds.append(~c)
        for i in (-oo, oo):
            if (classify(rv, s, i) is S.true
                    and classify(ie, s, i) is not S.true):
                conds.append(s < i if i is oo else i < s)

    conds.append(rv)
    return And(*conds)
Esempio n. 4
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def solve_univariate_inequality(expr,
                                gen,
                                relational=True,
                                domain=S.Reals,
                                continuous=False):
    """Solves a real univariate inequality.

    Parameters
    ==========

    expr : Relational
        The target inequality
    gen : Symbol
        The variable for which the inequality is solved
    relational : bool
        A Relational type output is expected or not
    domain : Set
        The domain over which the equation is solved
    continuous: bool
        True if expr is known to be continuous over the given domain
        (and so continuous_domain() doesn't need to be called on it)

    Raises
    ======

    NotImplementedError
        The solution of the inequality cannot be determined due to limitation
        in `solvify`.

    Notes
    =====

    Currently, we cannot solve all the inequalities due to limitations in
    `solvify`. Also, the solution returned for trigonometric inequalities
    are restricted in its periodic interval.

    See Also
    ========

    solvify: solver returning solveset solutions with solve's output API

    Examples
    ========

    >>> from sympy.solvers.inequalities import solve_univariate_inequality
    >>> from sympy import Symbol, sin, Interval, S
    >>> x = Symbol('x')

    >>> solve_univariate_inequality(x**2 >= 4, x)
    ((2 <= x) & (x < oo)) | ((x <= -2) & (-oo < x))

    >>> solve_univariate_inequality(x**2 >= 4, x, relational=False)
    Union(Interval(-oo, -2), Interval(2, oo))

    >>> domain = Interval(0, S.Infinity)
    >>> solve_univariate_inequality(x**2 >= 4, x, False, domain)
    Interval(2, oo)

    >>> solve_univariate_inequality(sin(x) > 0, x, relational=False)
    Interval.open(0, pi)

    """
    from sympy import im
    from sympy.calculus.util import (continuous_domain, periodicity,
                                     function_range)
    from sympy.solvers.solvers import denoms
    from sympy.solvers.solveset import solveset_real, solvify, solveset
    from sympy.solvers.solvers import solve

    # This keeps the function independent of the assumptions about `gen`.
    # `solveset` makes sure this function is called only when the domain is
    # real.
    _gen = gen
    _domain = domain
    if gen.is_real is False:
        rv = S.EmptySet
        return rv if not relational else rv.as_relational(_gen)
    elif gen.is_real is None:
        gen = Dummy('gen', real=True)
        try:
            expr = expr.xreplace({_gen: gen})
        except TypeError:
            raise TypeError(
                filldedent('''
                When gen is real, the relational has a complex part
                which leads to an invalid comparison like I < 0.
                '''))

    rv = None

    if expr is S.true:
        rv = domain

    elif expr is S.false:
        rv = S.EmptySet

    else:
        e = expr.lhs - expr.rhs
        period = periodicity(e, gen)
        if period is S.Zero:
            e = expand_mul(e)
            const = expr.func(e, 0)
            if const is S.true:
                rv = domain
            elif const is S.false:
                rv = S.EmptySet
        elif period is not None:
            frange = function_range(e, gen, domain)

            rel = expr.rel_op
            if rel == '<' or rel == '<=':
                if expr.func(frange.sup, 0):
                    rv = domain
                elif not expr.func(frange.inf, 0):
                    rv = S.EmptySet

            elif rel == '>' or rel == '>=':
                if expr.func(frange.inf, 0):
                    rv = domain
                elif not expr.func(frange.sup, 0):
                    rv = S.EmptySet

            inf, sup = domain.inf, domain.sup
            if sup - inf is S.Infinity:
                domain = Interval(0, period, False, True)

        if rv is None:
            n, d = e.as_numer_denom()
            try:
                if gen not in n.free_symbols and len(e.free_symbols) > 1:
                    raise ValueError
                # this might raise ValueError on its own
                # or it might give None...
                solns = solvify(e, gen, domain)
                if solns is None:
                    # in which case we raise ValueError
                    raise ValueError
            except (ValueError, NotImplementedError):
                # replace gen with generic x since it's
                # univariate anyway
                raise NotImplementedError(
                    filldedent('''
                    The inequality, %s, cannot be solved using
                    solve_univariate_inequality.
                    ''' % expr.subs(gen, Symbol('x'))))

            expanded_e = expand_mul(e)

            def valid(x):
                # this is used to see if gen=x satisfies the
                # relational by substituting it into the
                # expanded form and testing against 0, e.g.
                # if expr = x*(x + 1) < 2 then e = x*(x + 1) - 2
                # and expanded_e = x**2 + x - 2; the test is
                # whether a given value of x satisfies
                # x**2 + x - 2 < 0
                #
                # expanded_e, expr and gen used from enclosing scope
                v = expanded_e.subs(gen, expand_mul(x))
                try:
                    r = expr.func(v, 0)
                except TypeError:
                    r = S.false
                if r in (S.true, S.false):
                    return r
                if v.is_real is False:
                    return S.false
                else:
                    v = v.n(2)
                    if v.is_comparable:
                        return expr.func(v, 0)
                    # not comparable or couldn't be evaluated
                    raise NotImplementedError(
                        'relationship did not evaluate: %s' % r)

            singularities = []
            for d in denoms(expr, gen):
                singularities.extend(solvify(d, gen, domain))
            if not continuous:
                domain = continuous_domain(expanded_e, gen, domain)

            include_x = '=' in expr.rel_op and expr.rel_op != '!='

            try:
                discontinuities = set(domain.boundary -
                                      FiniteSet(domain.inf, domain.sup))
                # remove points that are not between inf and sup of domain
                critical_points = FiniteSet(
                    *(solns + singularities +
                      list(discontinuities))).intersection(
                          Interval(domain.inf, domain.sup, domain.inf
                                   not in domain, domain.sup not in domain))
                if all(r.is_number for r in critical_points):
                    reals = _nsort(critical_points, separated=True)[0]
                else:
                    from sympy.utilities.iterables import sift
                    sifted = sift(critical_points, lambda x: x.is_real)
                    if sifted[None]:
                        # there were some roots that weren't known
                        # to be real
                        raise NotImplementedError
                    try:
                        reals = sifted[True]
                        if len(reals) > 1:
                            reals = list(sorted(reals))
                    except TypeError:
                        raise NotImplementedError
            except NotImplementedError:
                raise NotImplementedError(
                    'sorting of these roots is not supported')

            # If expr contains imaginary coefficients, only take real
            # values of x for which the imaginary part is 0
            make_real = S.Reals
            if im(expanded_e) != S.Zero:
                check = True
                im_sol = FiniteSet()
                try:
                    a = solveset(im(expanded_e), gen, domain)
                    if not isinstance(a, Interval):
                        for z in a:
                            if z not in singularities and valid(
                                    z) and z.is_real:
                                im_sol += FiniteSet(z)
                    else:
                        start, end = a.inf, a.sup
                        for z in _nsort(critical_points + FiniteSet(end)):
                            valid_start = valid(start)
                            if start != end:
                                valid_z = valid(z)
                                pt = _pt(start, z)
                                if pt not in singularities and pt.is_real and valid(
                                        pt):
                                    if valid_start and valid_z:
                                        im_sol += Interval(start, z)
                                    elif valid_start:
                                        im_sol += Interval.Ropen(start, z)
                                    elif valid_z:
                                        im_sol += Interval.Lopen(start, z)
                                    else:
                                        im_sol += Interval.open(start, z)
                            start = z
                        for s in singularities:
                            im_sol -= FiniteSet(s)
                except (TypeError):
                    im_sol = S.Reals
                    check = False

                if isinstance(im_sol, EmptySet):
                    raise ValueError(
                        filldedent('''
                        %s contains imaginary parts which cannot be
                        made 0 for any value of %s satisfying the
                        inequality, leading to relations like I < 0.
                        ''' % (expr.subs(gen, _gen), _gen)))

                make_real = make_real.intersect(im_sol)

            empty = sol_sets = [S.EmptySet]

            start = domain.inf
            if valid(start) and start.is_finite:
                sol_sets.append(FiniteSet(start))

            for x in reals:
                end = x

                if valid(_pt(start, end)):
                    sol_sets.append(Interval(start, end, True, True))

                if x in singularities:
                    singularities.remove(x)
                else:
                    if x in discontinuities:
                        discontinuities.remove(x)
                        _valid = valid(x)
                    else:  # it's a solution
                        _valid = include_x
                    if _valid:
                        sol_sets.append(FiniteSet(x))

                start = end

            end = domain.sup
            if valid(end) and end.is_finite:
                sol_sets.append(FiniteSet(end))

            if valid(_pt(start, end)):
                sol_sets.append(Interval.open(start, end))

            if im(expanded_e) != S.Zero and check:
                rv = (make_real).intersect(_domain)
            else:
                rv = Intersection((Union(*sol_sets)), make_real,
                                  _domain).subs(gen, _gen)

    return rv if not relational else rv.as_relational(_gen)
Esempio n. 5
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def heurisch_wrapper(f, x, rewrite=False, hints=None, mappings=None, retries=3,
                     degree_offset=0, unnecessary_permutations=None,
                     _try_heurisch=None):
    """
    A wrapper around the heurisch integration algorithm.

    Explanation
    ===========

    This method takes the result from heurisch and checks for poles in the
    denominator. For each of these poles, the integral is reevaluated, and
    the final integration result is given in terms of a Piecewise.

    Examples
    ========

    >>> from sympy import cos, symbols
    >>> from sympy.integrals.heurisch import heurisch, heurisch_wrapper
    >>> n, x = symbols('n x')
    >>> heurisch(cos(n*x), x)
    sin(n*x)/n
    >>> heurisch_wrapper(cos(n*x), x)
    Piecewise((sin(n*x)/n, Ne(n, 0)), (x, True))

    See Also
    ========

    heurisch
    """
    from sympy.solvers.solvers import solve, denoms
    f = sympify(f)
    if not f.has_free(x):
        return f*x

    res = heurisch(f, x, rewrite, hints, mappings, retries, degree_offset,
                   unnecessary_permutations, _try_heurisch)
    if not isinstance(res, Basic):
        return res
    # We consider each denominator in the expression, and try to find
    # cases where one or more symbolic denominator might be zero. The
    # conditions for these cases are stored in the list slns.
    slns = []
    for d in denoms(res):
        try:
            slns += solve(d, dict=True, exclude=(x,))
        except NotImplementedError:
            pass
    if not slns:
        return res
    slns = list(uniq(slns))
    # Remove the solutions corresponding to poles in the original expression.
    slns0 = []
    for d in denoms(f):
        try:
            slns0 += solve(d, dict=True, exclude=(x,))
        except NotImplementedError:
            pass
    slns = [s for s in slns if s not in slns0]
    if not slns:
        return res
    if len(slns) > 1:
        eqs = []
        for sub_dict in slns:
            eqs.extend([Eq(key, value) for key, value in sub_dict.items()])
        slns = solve(eqs, dict=True, exclude=(x,)) + slns
    # For each case listed in the list slns, we reevaluate the integral.
    pairs = []
    for sub_dict in slns:
        expr = heurisch(f.subs(sub_dict), x, rewrite, hints, mappings, retries,
                        degree_offset, unnecessary_permutations,
                        _try_heurisch)
        cond = And(*[Eq(key, value) for key, value in sub_dict.items()])
        generic = Or(*[Ne(key, value) for key, value in sub_dict.items()])
        if expr is None:
            expr = integrate(f.subs(sub_dict),x)
        pairs.append((expr, cond))
    # If there is one condition, put the generic case first. Otherwise,
    # doing so may lead to longer Piecewise formulas
    if len(pairs) == 1:
        pairs = [(heurisch(f, x, rewrite, hints, mappings, retries,
                              degree_offset, unnecessary_permutations,
                              _try_heurisch),
                              generic),
                 (pairs[0][0], True)]
    else:
        pairs.append((heurisch(f, x, rewrite, hints, mappings, retries,
                              degree_offset, unnecessary_permutations,
                              _try_heurisch),
                              True))
    return Piecewise(*pairs)
Esempio n. 6
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def heurisch_wrapper(f, x, rewrite=False, hints=None, mappings=None, retries=3,
                     degree_offset=0, unnecessary_permutations=None):
    """
    A wrapper around the heurisch integration algorithm.

    This method takes the result from heurisch and checks for poles in the
    denominator. For each of these poles, the integral is reevaluated, and
    the final integration result is given in terms of a Piecewise.

    Examples
    ========

    >>> from sympy.core import symbols
    >>> from sympy.functions import cos
    >>> from sympy.integrals.heurisch import heurisch, heurisch_wrapper
    >>> n, x = symbols('n x')
    >>> heurisch(cos(n*x), x)
    sin(n*x)/n
    >>> heurisch_wrapper(cos(n*x), x)
    Piecewise((x, n == 0), (sin(n*x)/n, True))

    See Also
    ========

    heurisch
    """
    f = sympify(f)
    if x not in f.free_symbols:
        return f*x

    res = heurisch(f, x, rewrite, hints, mappings, retries, degree_offset,
                   unnecessary_permutations)
    if not isinstance(res, Basic):
        return res
    # We consider each denominator in the expression, and try to find
    # cases where one or more symbolic denominator might be zero. The
    # conditions for these cases are stored in the list slns.
    slns = []
    for d in denoms(res):
        try:
            slns += solve(d, dict=True, exclude=(x,))
        except NotImplementedError:
            pass
    if not slns:
        return res
    slns = list(uniq(slns))
    # Remove the solutions corresponding to poles in the original expression.
    slns0 = []
    for d in denoms(f):
        try:
            slns0 += solve(d, dict=True, exclude=(x,))
        except NotImplementedError:
            pass
    slns = [s for s in slns if s not in slns0]
    if not slns:
        return res
    if len(slns) > 1:
        eqs = []
        for sub_dict in slns:
            eqs.extend([Eq(key, value) for key, value in sub_dict.items()])
        slns = solve(eqs, dict=True, exclude=(x,)) + slns
    # For each case listed in the list slns, we reevaluate the integral.
    pairs = []
    for sub_dict in slns:
        expr = heurisch(f.subs(sub_dict), x, rewrite, hints, mappings, retries,
                        degree_offset, unnecessary_permutations)
        cond = And(*[Eq(key, value) for key, value in sub_dict.items()])
        pairs.append((expr, cond))
    pairs.append((heurisch(f, x, rewrite, hints, mappings, retries,
                           degree_offset, unnecessary_permutations), True))
    return Piecewise(*pairs)
Esempio n. 7
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def _invalid_solutions(f, symbol, domain):
    bad = S.EmptySet
    for d in denoms(f):
        bad += _solveset(d, symbol, domain, _check=False)
    return bad
Esempio n. 8
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def solve_univariate_inequality(expr, gen, relational=True):
    """Solves a real univariate inequality.

    Examples
    ========

    >>> from sympy.solvers.inequalities import solve_univariate_inequality
    >>> from sympy.core.symbol import Symbol
    >>> x = Symbol('x')

    >>> solve_univariate_inequality(x**2 >= 4, x)
    Or(And(-oo < x, x <= -2), And(2 <= x, x < oo))

    >>> solve_univariate_inequality(x**2 >= 4, x, relational=False)
    (-oo, -2] U [2, oo)

    """

    from sympy.solvers.solvers import solve, denoms

    # This keeps the function independent of the assumptions about `gen`.
    # `solveset` makes sure this function is called only when the domain is
    # real.
    d = Dummy(real=True)
    expr = expr.subs(gen, d)
    _gen = gen
    gen = d

    if expr is S.true:
        rv = S.Reals
    elif expr is S.false:
        rv = S.EmptySet
    else:
        e = expr.lhs - expr.rhs
        parts = n, d = e.as_numer_denom()
        if all(i.is_polynomial(gen) for i in parts):
            solns = solve(n, gen, check=False)
            singularities = solve(d, gen, check=False)
        else:
            solns = solve(e, gen, check=False)
            singularities = []
            for d in denoms(e):
                singularities.extend(solve(d, gen))

        include_x = expr.func(0, 0)

        def valid(x):
            v = e.subs(gen, x)
            try:
                r = expr.func(v, 0)
            except TypeError:
                r = S.false
            if r in (S.true, S.false):
                return r
            if v.is_real is False:
                return S.false
            else:
                v = v.n(2)
                if v.is_comparable:
                    return expr.func(v, 0)
                return S.false

        start = S.NegativeInfinity
        sol_sets = [S.EmptySet]
        try:
            reals = _nsort(set(solns + singularities), separated=True)[0]
        except NotImplementedError:
            raise NotImplementedError('sorting of these roots is not supported')
        for x in reals:
            end = x

            if end in [S.NegativeInfinity, S.Infinity]:
                if valid(S(0)):
                    sol_sets.append(Interval(start, S.Infinity, True, True))
                    break

            pt = ((start + end)/2 if start is not S.NegativeInfinity else
                (end/2 if end.is_positive else
                (2*end if end.is_negative else
                end - 1)))
            if valid(pt):
                sol_sets.append(Interval(start, end, True, True))

            if x in singularities:
                singularities.remove(x)
            elif include_x:
                sol_sets.append(FiniteSet(x))

            start = end

        end = S.Infinity

        # in case start == -oo then there were no solutions so we just
        # check a point between -oo and oo (e.g. 0) else pick a point
        # past the last solution (which is start after the end of the
        # for-loop above
        pt = (0 if start is S.NegativeInfinity else
            (start/2 if start.is_negative else
            (2*start if start.is_positive else
            start + 1)))
        if valid(pt):
            sol_sets.append(Interval(start, end, True, True))

        rv = Union(*sol_sets).subs(gen, _gen)

    return rv if not relational else rv.as_relational(_gen)
Esempio n. 9
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def solve_univariate_inequality(expr, gen, relational=True):
    """Solves a real univariate inequality.

    Examples
    ========

    >>> from sympy.solvers.inequalities import solve_univariate_inequality
    >>> from sympy.core.symbol import Symbol
    >>> x = Symbol('x', real=True)

    >>> solve_univariate_inequality(x**2 >= 4, x)
    Or(And(-oo < x, x <= -2), And(2 <= x, x < oo))

    >>> solve_univariate_inequality(x**2 >= 4, x, relational=False)
    (-oo, -2] U [2, oo)

    """

    from sympy.solvers.solvers import solve, denoms

    e = expr.lhs - expr.rhs
    parts = n, d = e.as_numer_denom()
    if all(i.is_polynomial(gen) for i in parts):
        solns = solve(n, gen, check=False)
        singularities = solve(d, gen, check=False)
    else:
        solns = solve(e, gen, check=False)
        singularities = []
        for d in denoms(e):
            singularities.extend(solve(d, gen))

    include_x = expr.func(0, 0)

    def valid(x):
        v = e.subs(gen, x)
        try:
            r = expr.func(v, 0)
        except TypeError:
            r = S.false
        if r in (S.true, S.false):
            return r
        if v.is_real is False:
            return S.false
        else:
            v = v.n(2)
            if v.is_comparable:
                return expr.func(v, 0)
            return S.false

    start = S.NegativeInfinity
    sol_sets = [S.EmptySet]
    try:
        reals = _nsort(set(solns + singularities), separated=True)[0]
    except NotImplementedError:
        raise NotImplementedError('sorting of these roots is not supported')
    for x in reals:
        end = x

        if end in [S.NegativeInfinity, S.Infinity]:
            if valid(S(0)):
                sol_sets.append(Interval(start, S.Infinity, True, True))
                break

        if valid((start + end)/2 if start != S.NegativeInfinity else end - 1):
            sol_sets.append(Interval(start, end, True, True))

        if x in singularities:
            singularities.remove(x)
        elif include_x:
            sol_sets.append(FiniteSet(x))

        start = end

    end = S.Infinity

    if valid(start + 1):
        sol_sets.append(Interval(start, end, True, True))

    rv = Union(*sol_sets)
    return rv if not relational else rv.as_relational(gen)
Esempio n. 10
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def heurisch_wrapper(f, x, rewrite=False, hints=None, mappings=None, retries=3,
                     degree_offset=0, unnecessary_permutations=None):
    """
    A wrapper around the heurisch integration algorithm.

    This method takes the result from heurisch and checks for poles in the
    denominator. For each of these poles, the integral is reevaluated, and
    the final integration result is given in terms of a Piecewise.

    Examples
    ========

    >>> from sympy.core import symbols
    >>> from sympy.functions import cos
    >>> from sympy.integrals.heurisch import heurisch, heurisch_wrapper
    >>> n, x = symbols('n x')
    >>> heurisch(cos(n*x), x)
    sin(n*x)/n
    >>> heurisch_wrapper(cos(n*x), x)
    Piecewise((x, n == 0), (sin(n*x)/n, True))

    See Also
    ========

    heurisch
    """
    f = sympify(f)
    if x not in f.free_symbols:
        return f*x

    res = heurisch(f, x, rewrite, hints, mappings, retries, degree_offset,
                   unnecessary_permutations)
    if not isinstance(res, Basic):
        return res
    # We consider each denominator in the expression, and try to find
    # cases where one or more symbolic denominator might be zero. The
    # conditions for these cases are stored in the list slns.
    slns = []
    for d in denoms(res):
        try:
            slns += solve(d, dict=True, exclude=(x,))
        except NotImplementedError:
            pass
    if not slns:
        return res
    slns = list(uniq(slns))
    # Remove the solutions corresponding to poles in the original expression.
    slns0 = []
    for d in denoms(f):
        try:
            slns0 += solve(d, dict=True, exclude=(x,))
        except NotImplementedError:
            pass
    slns = [s for s in slns if s not in slns0]
    if not slns:
        return res
    if len(slns) > 1:
        eqs = []
        for sub_dict in slns:
            eqs.extend([Eq(key, value) for key, value in sub_dict.items()])
        slns = solve(eqs, dict=True, exclude=(x,)) + slns
    # For each case listed in the list slns, we reevaluate the integral.
    pairs = []
    for sub_dict in slns:
        expr = heurisch(f.subs(sub_dict), x, rewrite, hints, mappings, retries,
                        degree_offset, unnecessary_permutations)
        cond = And(*[Eq(key, value) for key, value in sub_dict.items()])
        pairs.append((expr, cond))
    pairs.append((heurisch(f, x, rewrite, hints, mappings, retries,
                           degree_offset, unnecessary_permutations), True))
    return Piecewise(*pairs)
Esempio n. 11
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def intersection_sets(self, other):  # noqa:F811
    from sympy.solvers.diophantine import diophantine

    # Only handle the straight-forward univariate case
    if (len(self.lamda.variables) > 1
            or self.lamda.signature != self.lamda.variables):
        return None
    base_set = self.base_sets[0]

    # Intersection between ImageSets with Integers as base set
    # For {f(n) : n in Integers} & {g(m) : m in Integers} we solve the
    # diophantine equations f(n)=g(m).
    # If the solutions for n are {h(t) : t in Integers} then we return
    # {f(h(t)) : t in integers}.
    # If the solutions for n are {n_1, n_2, ..., n_k} then we return
    # {f(n_i) : 1 <= i <= k}.
    if base_set is S.Integers:
        gm = None
        if isinstance(other, ImageSet) and other.base_sets == (S.Integers, ):
            gm = other.lamda.expr
            var = other.lamda.variables[0]
            # Symbol of second ImageSet lambda must be distinct from first
            m = Dummy('m')
            gm = gm.subs(var, m)
        elif other is S.Integers:
            m = gm = Dummy('m')
        if gm is not None:
            fn = self.lamda.expr
            n = self.lamda.variables[0]
            try:
                solns = list(diophantine(fn - gm, syms=(n, m), permute=True))
            except (TypeError, NotImplementedError):
                # TypeError if equation not polynomial with rational coeff.
                # NotImplementedError if correct format but no solver.
                return
            # 3 cases are possible for solns:
            # - empty set,
            # - one or more parametric (infinite) solutions,
            # - a finite number of (non-parametric) solution couples.
            # Among those, there is one type of solution set that is
            # not helpful here: multiple parametric solutions.
            if len(solns) == 0:
                return EmptySet
            elif any(not isinstance(s, int) and s.free_symbols
                     for tupl in solns for s in tupl):
                if len(solns) == 1:
                    soln, solm = solns[0]
                    (t, ) = soln.free_symbols
                    expr = fn.subs(n, soln.subs(t, n)).expand()
                    return imageset(Lambda(n, expr), S.Integers)
                else:
                    return
            else:
                return FiniteSet(*(fn.subs(n, s[0]) for s in solns))

    if other == S.Reals:
        from sympy.core.function import expand_complex
        from sympy.solvers.solvers import denoms, solve_linear
        from sympy.core.relational import Eq

        def _solution_union(exprs, sym):
            # return a union of linear solutions to i in expr;
            # if i cannot be solved, use a ConditionSet for solution
            sols = []
            for i in exprs:
                x, xis = solve_linear(i, 0, [sym])
                if x == sym:
                    sols.append(FiniteSet(xis))
                else:
                    sols.append(ConditionSet(sym, Eq(i, 0)))
            return Union(*sols)

        f = self.lamda.expr
        n = self.lamda.variables[0]

        n_ = Dummy(n.name, real=True)
        f_ = f.subs(n, n_)

        re, im = f_.as_real_imag()
        im = expand_complex(im)

        re = re.subs(n_, n)
        im = im.subs(n_, n)
        ifree = im.free_symbols
        lam = Lambda(n, re)
        if im.is_zero:
            # allow re-evaluation
            # of self in this case to make
            # the result canonical
            pass
        elif im.is_zero is False:
            return S.EmptySet
        elif ifree != {n}:
            return None
        else:
            # univarite imaginary part in same variable;
            # use numer instead of as_numer_denom to keep
            # this as fast as possible while still handling
            # simple cases
            base_set &= _solution_union(Mul.make_args(numer(im)), n)
        # exclude values that make denominators 0
        base_set -= _solution_union(denoms(f), n)
        return imageset(lam, base_set)

    elif isinstance(other, Interval):
        from sympy.solvers.solveset import (invert_real, invert_complex,
                                            solveset)

        f = self.lamda.expr
        n = self.lamda.variables[0]
        new_inf, new_sup = None, None
        new_lopen, new_ropen = other.left_open, other.right_open

        if f.is_real:
            inverter = invert_real
        else:
            inverter = invert_complex

        g1, h1 = inverter(f, other.inf, n)
        g2, h2 = inverter(f, other.sup, n)

        if all(isinstance(i, FiniteSet) for i in (h1, h2)):
            if g1 == n:
                if len(h1) == 1:
                    new_inf = h1.args[0]
            if g2 == n:
                if len(h2) == 1:
                    new_sup = h2.args[0]
            # TODO: Design a technique to handle multiple-inverse
            # functions

            # Any of the new boundary values cannot be determined
            if any(i is None for i in (new_sup, new_inf)):
                return

            range_set = S.EmptySet

            if all(i.is_real for i in (new_sup, new_inf)):
                # this assumes continuity of underlying function
                # however fixes the case when it is decreasing
                if new_inf > new_sup:
                    new_inf, new_sup = new_sup, new_inf
                new_interval = Interval(new_inf, new_sup, new_lopen, new_ropen)
                range_set = base_set.intersect(new_interval)
            else:
                if other.is_subset(S.Reals):
                    solutions = solveset(f, n, S.Reals)
                    if not isinstance(range_set, (ImageSet, ConditionSet)):
                        range_set = solutions.intersect(other)
                    else:
                        return

            if range_set is S.EmptySet:
                return S.EmptySet
            elif isinstance(range_set,
                            Range) and range_set.size is not S.Infinity:
                range_set = FiniteSet(*list(range_set))

            if range_set is not None:
                return imageset(Lambda(n, f), range_set)
            return
        else:
            return
Esempio n. 12
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def _solve_inequality(ie, s, linear=False):
    """Return the inequality with s isolated on the left, if possible.
    If the relationship is non-linear, a solution involving And or Or
    may be returned. False or True are returned if the relationship
    is never True or always True, respectively.

    If `linear` is True (default is False) an `s`-dependent expression
    will be isoloated on the left, if possible
    but it will not be solved for `s` unless the expression is linear
    in `s`. Furthermore, only "safe" operations which don't change the
    sense of the relationship are applied: no division by an unsigned
    value is attempted unless the relationship involves Eq or Ne and
    no division by a value not known to be nonzero is ever attempted.

    Examples
    ========

    >>> from sympy import Eq, Symbol
    >>> from sympy.solvers.inequalities import _solve_inequality as f
    >>> from sympy.abc import x, y

    For linear expressions, the symbol can be isolated:

    >>> f(x - 2 < 0, x)
    x < 2
    >>> f(-x - 6 < x, x)
    x > -3

    Sometimes nonlinear relationships will be False

    >>> f(x**2 + 4 < 0, x)
    False

    Or they may involve more than one region of values:

    >>> f(x**2 - 4 < 0, x)
    (-2 < x) & (x < 2)

    To restrict the solution to a relational, set linear=True
    and only the x-dependent portion will be isolated on the left:

    >>> f(x**2 - 4 < 0, x, linear=True)
    x**2 < 4

    Division of only nonzero quantities is allowed, so x cannot
    be isolated by dividing by y:

    >>> y.is_nonzero is None  # it is unknown whether it is 0 or not
    True
    >>> f(x*y < 1, x)
    x*y < 1

    And while an equality (or unequality) still holds after dividing by a
    non-zero quantity

    >>> nz = Symbol('nz', nonzero=True)
    >>> f(Eq(x*nz, 1), x)
    Eq(x, 1/nz)

    the sign must be known for other inequalities involving > or <:

    >>> f(x*nz <= 1, x)
    nz*x <= 1
    >>> p = Symbol('p', positive=True)
    >>> f(x*p <= 1, x)
    x <= 1/p

    When there are denominators in the original expression that
    are removed by expansion, conditions for them will be returned
    as part of the result:

    >>> f(x < x*(2/x - 1), x)
    (x < 1) & Ne(x, 0)
    """
    from sympy.solvers.solvers import denoms
    if s not in ie.free_symbols:
        return ie
    if ie.rhs == s:
        ie = ie.reversed
    if ie.lhs == s and s not in ie.rhs.free_symbols:
        return ie
    expr = ie.lhs - ie.rhs
    rv = None
    try:
        p = Poly(expr, s)
        if p.degree() == 0:
            rv = ie.func(p.as_expr(), 0)
        elif not linear and p.degree() > 1:
            # handle in except clause
            raise NotImplementedError
    except (PolynomialError, NotImplementedError):
        if not linear:
            try:
                return reduce_rational_inequalities([[ie]], s)
            except PolynomialError:
                return solve_univariate_inequality(ie, s)
        else:
            p = Poly(expr)

    e = expanded = p.as_expr()  # this is in exanded form
    if rv is None:
        # Do a safe inversion of e, moving non-s terms
        # to the rhs and dividing by a nonzero factor if
        # the relational is Eq/Ne; for other relationals
        # the sign must also be positive or negative
        rhs = 0
        b, ax = e.as_independent(s, as_Add=True)
        e -= b
        rhs -= b
        ef = factor_terms(e)
        a, e = ef.as_independent(s, as_Add=False)
        if (a.is_zero != False or  # don't divide by potential 0
                a.is_negative ==
                a.is_positive == None and  # if sign is not known then
                ie.rel_op not in ('!=', '==')): # reject if not Eq/Ne
            e = ef
            a = S.One
        rhs /= a
        if a.is_positive:
            rv = ie.func(e, rhs)
        else:
            rv = ie.reversed.func(e, rhs)
    # return conditions under which the value is
    # valid, too.
    conds = [rv]
    beginning_denoms = denoms(ie.lhs) | denoms(ie.rhs)
    current_denoms = denoms(expanded)
    for d in beginning_denoms - current_denoms:
        conds.append(_solve_inequality(Ne(d, 0), s, linear=linear))
    return And(*conds)
Esempio n. 13
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def solve_univariate_inequality(expr, gen, relational=True, domain=S.Reals, continuous=False):
    """Solves a real univariate inequality.

    Parameters
    ==========

    expr : Relational
        The target inequality
    gen : Symbol
        The variable for which the inequality is solved
    relational : bool
        A Relational type output is expected or not
    domain : Set
        The domain over which the equation is solved
    continuous: bool
        True if expr is known to be continuous over the given domain
        (and so continuous_domain() doesn't need to be called on it)

    Raises
    ======

    NotImplementedError
        The solution of the inequality cannot be determined due to limitation
        in `solvify`.

    Notes
    =====

    Currently, we cannot solve all the inequalities due to limitations in
    `solvify`. Also, the solution returned for trigonometric inequalities
    are restricted in its periodic interval.

    See Also
    ========

    solvify: solver returning solveset solutions with solve's output API

    Examples
    ========

    >>> from sympy.solvers.inequalities import solve_univariate_inequality
    >>> from sympy import Symbol, sin, Interval, S
    >>> x = Symbol('x')

    >>> solve_univariate_inequality(x**2 >= 4, x)
    ((2 <= x) & (x < oo)) | ((x <= -2) & (-oo < x))

    >>> solve_univariate_inequality(x**2 >= 4, x, relational=False)
    Union(Interval(-oo, -2), Interval(2, oo))

    >>> domain = Interval(0, S.Infinity)
    >>> solve_univariate_inequality(x**2 >= 4, x, False, domain)
    Interval(2, oo)

    >>> solve_univariate_inequality(sin(x) > 0, x, relational=False)
    Interval.open(0, pi)

    """
    from sympy import im
    from sympy.calculus.util import (continuous_domain, periodicity,
        function_range)
    from sympy.solvers.solvers import denoms
    from sympy.solvers.solveset import solveset_real, solvify, solveset
    from sympy.solvers.solvers import solve

    # This keeps the function independent of the assumptions about `gen`.
    # `solveset` makes sure this function is called only when the domain is
    # real.
    _gen = gen
    _domain = domain
    if gen.is_real is False:
        rv = S.EmptySet
        return rv if not relational else rv.as_relational(_gen)
    elif gen.is_real is None:
        gen = Dummy('gen', real=True)
        try:
            expr = expr.xreplace({_gen: gen})
        except TypeError:
            raise TypeError(filldedent('''
When gen is real, the relational has a complex part
which leads to an invalid comparison like I < 0.
            '''))

    rv = None

    if expr is S.true:
        rv = domain

    elif expr is S.false:
        rv = S.EmptySet

    else:
        e = expr.lhs - expr.rhs
        period = periodicity(e, gen)
        if period is not None:
            frange = function_range(e, gen, domain)

            rel = expr.rel_op
            if rel == '<' or rel == '<=':
                if expr.func(frange.sup, 0):
                    rv = domain
                elif not expr.func(frange.inf, 0):
                    rv = S.EmptySet

            elif rel == '>' or rel == '>=':
                if expr.func(frange.inf, 0):
                    rv = domain
                elif not expr.func(frange.sup, 0):
                    rv = S.EmptySet

            inf, sup = domain.inf, domain.sup
            if sup - inf is S.Infinity:
                domain = Interval(0, period, False, True)

        if rv is None:
            n, d = e.as_numer_denom()
            try:
                if gen not in n.free_symbols and len(e.free_symbols) > 1:
                    raise ValueError
                # this might raise ValueError on its own
                # or it might give None...
                solns = solvify(e, gen, domain)
                if solns is None:
                    # in which case we raise ValueError
                    raise ValueError
            except (ValueError, NotImplementedError):
                raise NotImplementedError(filldedent('''
The inequality cannot be solved using solve_univariate_inequality.
                        '''))

            expanded_e = expand_mul(e)
            def valid(x):
                # this is used to see if gen=x satisfies the
                # relational by substituting it into the
                # expanded form and testing against 0, e.g.
                # if expr = x*(x + 1) < 2 then e = x*(x + 1) - 2
                # and expanded_e = x**2 + x - 2; the test is
                # whether a given value of x satisfies
                # x**2 + x - 2 < 0
                #
                # expanded_e, expr and gen used from enclosing scope
                v = expanded_e.subs(gen, x)
                try:
                    r = expr.func(v, 0)
                except TypeError:
                    r = S.false
                if r in (S.true, S.false):
                    return r
                if v.is_real is False:
                    return S.false
                else:
                    v = v.n(2)
                    if v.is_comparable:
                        return expr.func(v, 0)
                    # not comparable or couldn't be evaluated
                    raise NotImplementedError(
                        'relationship did not evaluate: %s' % r)

            singularities = []
            for d in denoms(expr, gen):
                singularities.extend(solvify(d, gen, domain))
            if not continuous:
                domain = continuous_domain(e, gen, domain)

            include_x = '=' in expr.rel_op and expr.rel_op != '!='

            try:
                discontinuities = set(domain.boundary -
                    FiniteSet(domain.inf, domain.sup))
                # remove points that are not between inf and sup of domain
                critical_points = FiniteSet(*(solns + singularities + list(
                    discontinuities))).intersection(
                    Interval(domain.inf, domain.sup,
                    domain.inf not in domain, domain.sup not in domain))
                if all(r.is_number for r in critical_points):
                    reals = _nsort(critical_points, separated=True)[0]
                else:
                    from sympy.utilities.iterables import sift
                    sifted = sift(critical_points, lambda x: x.is_real)
                    if sifted[None]:
                        # there were some roots that weren't known
                        # to be real
                        raise NotImplementedError
                    try:
                        reals = sifted[True]
                        if len(reals) > 1:
                            reals = list(sorted(reals))
                    except TypeError:
                        raise NotImplementedError
            except NotImplementedError:
                raise NotImplementedError('sorting of these roots is not supported')

            #If expr contains imaginary coefficients
            #Only real values of x for which the imaginary part is 0 are taken
            make_real = S.Reals
            if im(expanded_e) != S.Zero:
                check = True
                im_sol = FiniteSet()
                try:
                    a = solveset(im(expanded_e), gen, domain)
                    if not isinstance(a, Interval):
                        for z in a:
                            if z not in singularities and valid(z) and z.is_real:
                                im_sol += FiniteSet(z)
                    else:
                        start, end = a.inf, a.sup
                        for z in _nsort(critical_points + FiniteSet(end)):
                            valid_start = valid(start)
                            if start != end:
                                valid_z = valid(z)
                                pt = _pt(start, z)
                                if pt not in singularities and pt.is_real and valid(pt):
                                    if valid_start and valid_z:
                                        im_sol += Interval(start, z)
                                    elif valid_start:
                                        im_sol += Interval.Ropen(start, z)
                                    elif valid_z:
                                        im_sol += Interval.Lopen(start, z)
                                    else:
                                        im_sol += Interval.open(start, z)
                            start = z
                        for s in singularities:
                            im_sol -= FiniteSet(s)
                except (TypeError):
                    im_sol = S.Reals
                    check = False

                if isinstance(im_sol, EmptySet):
                    raise ValueError(filldedent('''
%s contains imaginary parts which cannot be made 0 for any value of %s
satisfying the inequality, leading to relations like I < 0.
                '''  % (expr.subs(gen, _gen), _gen)))

                make_real = make_real.intersect(im_sol)

            empty = sol_sets = [S.EmptySet]

            start = domain.inf
            if valid(start) and start.is_finite:
                sol_sets.append(FiniteSet(start))

            for x in reals:
                end = x

                if valid(_pt(start, end)):
                    sol_sets.append(Interval(start, end, True, True))

                if x in singularities:
                    singularities.remove(x)
                else:
                    if x in discontinuities:
                        discontinuities.remove(x)
                        _valid = valid(x)
                    else:  # it's a solution
                        _valid = include_x
                    if _valid:
                        sol_sets.append(FiniteSet(x))

                start = end

            end = domain.sup
            if valid(end) and end.is_finite:
                sol_sets.append(FiniteSet(end))

            if valid(_pt(start, end)):
                sol_sets.append(Interval.open(start, end))

            if im(expanded_e) != S.Zero and check:
                rv = (make_real).intersect(_domain)
            else:
                rv = Intersection(
                    (Union(*sol_sets)), make_real, _domain).subs(gen, _gen)

    return rv if not relational else rv.as_relational(_gen)
Esempio n. 14
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def solve_univariate_inequality(expr,
                                gen,
                                relational=True,
                                domain=S.Reals,
                                continuous=False):
    """Solves a real univariate inequality.

    Parameters
    ==========

    expr : Relational
        The target inequality
    gen : Symbol
        The variable for which the inequality is solved
    relational : bool
        A Relational type output is expected or not
    domain : Set
        The domain over which the equation is solved
    continuous: bool
        True if expr is known to be continuous over the given domain
        (and so continuous_domain() doesn't need to be called on it)

    Raises
    ======

    NotImplementedError
        The solution of the inequality cannot be determined due to limitation
        in `solvify`.

    Notes
    =====

    Currently, we cannot solve all the inequalities due to limitations in
    `solvify`. Also, the solution returned for trigonometric inequalities
    are restricted in its periodic interval.

    See Also
    ========

    solvify: solver returning solveset solutions with solve's output API

    Examples
    ========

    >>> from sympy.solvers.inequalities import solve_univariate_inequality
    >>> from sympy import Symbol, sin, Interval, S
    >>> x = Symbol('x')

    >>> solve_univariate_inequality(x**2 >= 4, x)
    ((2 <= x) & (x < oo)) | ((x <= -2) & (-oo < x))

    >>> solve_univariate_inequality(x**2 >= 4, x, relational=False)
    (-oo, -2] U [2, oo)

    >>> domain = Interval(0, S.Infinity)
    >>> solve_univariate_inequality(x**2 >= 4, x, False, domain)
    [2, oo)

    >>> solve_univariate_inequality(sin(x) > 0, x, relational=False)
    (0, pi)

    """
    from sympy.calculus.util import (continuous_domain, periodicity,
                                     function_range)
    from sympy.solvers.solvers import denoms
    from sympy.solvers.solveset import solveset_real, solvify

    # This keeps the function independent of the assumptions about `gen`.
    # `solveset` makes sure this function is called only when the domain is
    # real.
    d = Dummy(real=True)
    expr = expr.subs(gen, d)
    _gen = gen
    gen = d
    rv = None

    if expr is S.true:
        rv = domain

    elif expr is S.false:
        rv = S.EmptySet

    else:
        e = expr.lhs - expr.rhs
        period = periodicity(e, gen)
        if period is not None:
            frange = function_range(e, gen, domain)

            rel = expr.rel_op
            if rel == '<' or rel == '<=':
                if expr.func(frange.sup, 0):
                    rv = domain
                elif not expr.func(frange.inf, 0):
                    rv = S.EmptySet

            elif rel == '>' or rel == '>=':
                if expr.func(frange.inf, 0):
                    rv = domain
                elif not expr.func(frange.sup, 0):
                    rv = S.EmptySet

            inf, sup = domain.inf, domain.sup
            if sup - inf is S.Infinity:
                domain = Interval(0, period, False, True)

        if rv is None:
            singularities = []
            for d in denoms(e):
                singularities.extend(solvify(d, gen, domain))
            if not continuous:
                domain = continuous_domain(e, gen, domain)
            solns = solvify(e, gen, domain)

            if solns is None:
                raise NotImplementedError(
                    filldedent('''The inequality cannot be
                    solved using solve_univariate_inequality.'''))

            include_x = expr.func(0, 0)

            def valid(x):
                v = e.subs(gen, x)
                try:
                    r = expr.func(v, 0)
                except TypeError:
                    r = S.false
                if r in (S.true, S.false):
                    return r
                if v.is_real is False:
                    return S.false
                else:
                    v = v.n(2)
                    if v.is_comparable:
                        return expr.func(v, 0)
                    return S.false

            start = domain.inf
            sol_sets = [S.EmptySet]
            try:
                discontinuities = domain.boundary - FiniteSet(
                    domain.inf, domain.sup)
                critical_points = set(solns + singularities +
                                      list(discontinuities))
                reals = _nsort(critical_points, separated=True)[0]

            except NotImplementedError:
                raise NotImplementedError(
                    'sorting of these roots is not supported')

            if valid(start) and start.is_finite:
                sol_sets.append(FiniteSet(start))

            for x in reals:
                end = x

                if end in [S.NegativeInfinity, S.Infinity]:
                    if valid(S(0)):
                        sol_sets.append(Interval(start, S.Infinity, True,
                                                 True))
                        break

                pt = ((start + end) / 2 if start is not S.NegativeInfinity else
                      (end / 2 if end.is_positive else
                       (2 * end if end.is_negative else end - 1)))
                if valid(pt):
                    sol_sets.append(Interval(start, end, True, True))

                if x in singularities:
                    singularities.remove(x)
                elif include_x:
                    sol_sets.append(FiniteSet(x))

                start = end

            end = domain.sup

            # in case start == -oo then there were no solutions so we just
            # check a point between -oo and oo (e.g. 0) else pick a point
            # past the last solution (which is start after the end of the
            # for-loop above
            pt = (0 if start is S.NegativeInfinity else
                  (start / 2 if start.is_negative else
                   (2 * start if start.is_positive else start + 1)))

            if pt >= end:
                pt = (start + end) / 2

            if valid(pt):
                sol_sets.append(Interval(start, end, True, True))

            rv = Union(*sol_sets).subs(gen, _gen)

    return rv if not relational else rv.as_relational(_gen)
Esempio n. 15
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def solve_univariate_inequality(expr, gen, relational=True, domain=S.Reals, continuous=False):
    """Solves a real univariate inequality.

    Parameters
    ==========

    expr : Relational
        The target inequality
    gen : Symbol
        The variable for which the inequality is solved
    relational : bool
        A Relational type output is expected or not
    domain : Set
        The domain over which the equation is solved
    continuous: bool
        True if expr is known to be continuous over the given domain
        (and so continuous_domain() doesn't need to be called on it)

    Raises
    ======

    NotImplementedError
        The solution of the inequality cannot be determined due to limitation
        in `solvify`.

    Notes
    =====

    Currently, we cannot solve all the inequalities due to limitations in
    `solvify`. Also, the solution returned for trigonometric inequalities
    are restricted in its periodic interval.

    See Also
    ========

    solvify: solver returning solveset solutions with solve's output API

    Examples
    ========

    >>> from sympy.solvers.inequalities import solve_univariate_inequality
    >>> from sympy import Symbol, sin, Interval, S
    >>> x = Symbol('x')

    >>> solve_univariate_inequality(x**2 >= 4, x)
    ((2 <= x) & (x < oo)) | ((x <= -2) & (-oo < x))

    >>> solve_univariate_inequality(x**2 >= 4, x, relational=False)
    (-oo, -2] U [2, oo)

    >>> domain = Interval(0, S.Infinity)
    >>> solve_univariate_inequality(x**2 >= 4, x, False, domain)
    [2, oo)

    >>> solve_univariate_inequality(sin(x) > 0, x, relational=False)
    (0, pi)

    """
    from sympy.calculus.util import (continuous_domain, periodicity,
        function_range)
    from sympy.solvers.solvers import denoms
    from sympy.solvers.solveset import solveset_real, solvify

    # This keeps the function independent of the assumptions about `gen`.
    # `solveset` makes sure this function is called only when the domain is
    # real.
    d = Dummy(real=True)
    expr = expr.subs(gen, d)
    _gen = gen
    gen = d
    rv = None

    if expr is S.true:
        rv = domain

    elif expr is S.false:
        rv = S.EmptySet

    else:
        e = expr.lhs - expr.rhs
        period = periodicity(e, gen)
        if period is not None:
            frange = function_range(e, gen, domain)

            rel = expr.rel_op
            if rel == '<' or rel == '<=':
                if expr.func(frange.sup, 0):
                    rv = domain
                elif not expr.func(frange.inf, 0):
                    rv = S.EmptySet

            elif rel == '>' or rel == '>=':
                if expr.func(frange.inf, 0):
                    rv = domain
                elif not expr.func(frange.sup, 0):
                    rv = S.EmptySet

            inf, sup = domain.inf, domain.sup
            if sup - inf is S.Infinity:
                domain = Interval(0, period, False, True)

        if rv is None:
            singularities = []
            for d in denoms(e):
                singularities.extend(solvify(d, gen, domain))
            if not continuous:
                domain = continuous_domain(e, gen, domain)
            solns = solvify(e, gen, domain)

            if solns is None:
                raise NotImplementedError(filldedent('''The inequality cannot be
                    solved using solve_univariate_inequality.'''))

            include_x = expr.func(0, 0)

            def valid(x):
                v = e.subs(gen, x)
                try:
                    r = expr.func(v, 0)
                except TypeError:
                    r = S.false
                if r in (S.true, S.false):
                    return r
                if v.is_real is False:
                    return S.false
                else:
                    v = v.n(2)
                    if v.is_comparable:
                        return expr.func(v, 0)
                    return S.false

            start = domain.inf
            sol_sets = [S.EmptySet]
            try:
                discontinuities = domain.boundary - FiniteSet(domain.inf, domain.sup)
                critical_points = set(solns + singularities + list(discontinuities))
                reals = _nsort(critical_points, separated=True)[0]

            except NotImplementedError:
                raise NotImplementedError('sorting of these roots is not supported')

            if valid(start) and start.is_finite:
                sol_sets.append(FiniteSet(start))

            for x in reals:
                end = x

                if end in [S.NegativeInfinity, S.Infinity]:
                    if valid(S(0)):
                        sol_sets.append(Interval(start, S.Infinity, True, True))
                        break

                pt = ((start + end)/2 if start is not S.NegativeInfinity else
                    (end/2 if end.is_positive else
                    (2*end if end.is_negative else
                    end - 1)))
                if valid(pt):
                    sol_sets.append(Interval(start, end, True, True))

                if x in singularities:
                    singularities.remove(x)
                elif include_x:
                    sol_sets.append(FiniteSet(x))

                start = end

            end = domain.sup

            # in case start == -oo then there were no solutions so we just
            # check a point between -oo and oo (e.g. 0) else pick a point
            # past the last solution (which is start after the end of the
            # for-loop above
            pt = (0 if start is S.NegativeInfinity else
                (start/2 if start.is_negative else
                (2*start if start.is_positive else
                start + 1)))

            if pt >= end:
                pt = (start + end)/2

            if valid(pt):
                sol_sets.append(Interval(start, end, True, True))

            rv = Union(*sol_sets).subs(gen, _gen)

    return rv if not relational else rv.as_relational(_gen)