.venv/lib/python3.14/site-packages/sympy/polys/rootisolation.py

"""Real and complex root isolation and refinement algorithms. """


from sympy.polys.densearith import (
    dup_neg, dup_rshift, dup_rem,
    dup_l2_norm_squared)
from sympy.polys.densebasic import (
    dup_LC, dup_TC, dup_degree,
    dup_strip, dup_reverse,
    dup_convert,
    dup_terms_gcd)
from sympy.polys.densetools import (
    dup_clear_denoms,
    dup_mirror, dup_scale, dup_shift,
    dup_transform,
    dup_diff,
    dup_eval, dmp_eval_in,
    dup_sign_variations,
    dup_real_imag)
from sympy.polys.euclidtools import (
    dup_discriminant)
from sympy.polys.factortools import (
    dup_factor_list)
from sympy.polys.polyerrors import (
    RefinementFailed,
    DomainError,
    PolynomialError)
from sympy.polys.sqfreetools import (
    dup_sqf_part, dup_sqf_list)


def dup_sturm(f, K):
    """
    Computes the Sturm sequence of ``f`` in ``F[x]``.

    Given a univariate, square-free polynomial ``f(x)`` returns the
    associated Sturm sequence ``f_0(x), ..., f_n(x)`` defined by::

       f_0(x), f_1(x) = f(x), f'(x)
       f_n = -rem(f_{n-2}(x), f_{n-1}(x))

    Examples
    ========

    >>> from sympy.polys import ring, QQ
    >>> R, x = ring("x", QQ)

    >>> R.dup_sturm(x**3 - 2*x**2 + x - 3)
    [x**3 - 2*x**2 + x - 3, 3*x**2 - 4*x + 1, 2/9*x + 25/9, -2079/4]

    References
    ==========

    .. [1] [Davenport88]_

    """
    if not K.is_Field:
        raise DomainError("Cannot compute Sturm sequence over %s" % K)

    f = dup_sqf_part(f, K)

    sturm = [f, dup_diff(f, 1, K)]

    while sturm[-1]:
        s = dup_rem(sturm[-2], sturm[-1], K)
        sturm.append(dup_neg(s, K))

    return sturm[:-1]

def dup_root_upper_bound(f, K):
    """Compute the LMQ upper bound for the positive roots of `f`;
       LMQ (Local Max Quadratic) was developed by Akritas-Strzebonski-Vigklas.

    References
    ==========
    .. [1] Alkiviadis G. Akritas: "Linear and Quadratic Complexity Bounds on the
        Values of the Positive Roots of Polynomials"
        Journal of Universal Computer Science, Vol. 15, No. 3, 523-537, 2009.
    """
    n, P = len(f), []
    t = n * [K.one]
    if dup_LC(f, K) < 0:
        f = dup_neg(f, K)
    f = list(reversed(f))

    for i in range(0, n):
        if f[i] >= 0:
            continue

        a, QL = K.log(-f[i], 2), []

        for j in range(i + 1, n):

            if f[j] <= 0:
                continue

            q = t[j] + a - K.log(f[j], 2)
            QL.append([q // (j - i), j])

        if not QL:
            continue

        q = min(QL)

        t[q[1]] = t[q[1]] + 1

        P.append(q[0])

    if not P:
        return None
    else:
        return K.get_field()(2)**(max(P) + 1)

def dup_root_lower_bound(f, K):
    """Compute the LMQ lower bound for the positive roots of `f`;
       LMQ (Local Max Quadratic) was developed by Akritas-Strzebonski-Vigklas.

       References
       ==========
       .. [1] Alkiviadis G. Akritas: "Linear and Quadratic Complexity Bounds on the
              Values of the Positive Roots of Polynomials"
              Journal of Universal Computer Science, Vol. 15, No. 3, 523-537, 2009.
    """
    bound = dup_root_upper_bound(dup_reverse(f), K)

    if bound is not None:
        return 1/bound
    else:
        return None

def dup_cauchy_upper_bound(f, K):
    """
    Compute the Cauchy upper bound on the absolute value of all roots of f,
    real or complex.

    References
    ==========
    .. [1] https://en.wikipedia.org/wiki/Geometrical_properties_of_polynomial_roots#Lagrange's_and_Cauchy's_bounds
    """
    n = dup_degree(f)
    if n < 1:
        raise PolynomialError('Polynomial has no roots.')

    if K.is_ZZ:
        L = K.get_field()
        f, K = dup_convert(f, K, L), L
    elif not K.is_QQ or K.is_RR or K.is_CC:
        # We need to compute absolute value, and we are not supporting cases
        # where this would take us outside the domain (or its quotient field).
        raise DomainError('Cauchy bound not supported over %s' % K)
    else:
        f = f[:]

    while K.is_zero(f[-1]):
        f.pop()
    if len(f) == 1:
        # Monomial. All roots are zero.
        return K.zero

    lc = f[0]
    return K.one + max(abs(n / lc) for n in f[1:])

def dup_cauchy_lower_bound(f, K):
    """Compute the Cauchy lower bound on the absolute value of all non-zero
       roots of f, real or complex."""
    g = dup_reverse(f)
    if len(g) < 2:
        raise PolynomialError('Polynomial has no non-zero roots.')
    if K.is_ZZ:
        K = K.get_field()
    b = dup_cauchy_upper_bound(g, K)
    return K.one / b

def dup_mignotte_sep_bound_squared(f, K):
    """
    Return the square of the Mignotte lower bound on separation between
    distinct roots of f. The square is returned so that the bound lies in
    K or its quotient field.

    References
    ==========

    .. [1] Mignotte, Maurice. "Some useful bounds." Computer algebra.
        Springer, Vienna, 1982. 259-263.
        https://people.dm.unipi.it/gianni/AC-EAG/Mignotte.pdf
    """
    n = dup_degree(f)
    if n < 2:
        raise PolynomialError('Polynomials of degree < 2 have no distinct roots.')

    if K.is_ZZ:
        L = K.get_field()
        f, K = dup_convert(f, K, L), L
    elif not K.is_QQ or K.is_RR or K.is_CC:
        # We need to compute absolute value, and we are not supporting cases
        # where this would take us outside the domain (or its quotient field).
        raise DomainError('Mignotte bound not supported over %s' % K)

    D = dup_discriminant(f, K)
    l2sq = dup_l2_norm_squared(f, K)
    return K(3)*K.abs(D) / ( K(n)**(n+1) * l2sq**(n-1) )

def _mobius_from_interval(I, field):
    """Convert an open interval to a Mobius transform. """
    s, t = I

    a, c = field.numer(s), field.denom(s)
    b, d = field.numer(t), field.denom(t)

    return a, b, c, d

def _mobius_to_interval(M, field):
    """Convert a Mobius transform to an open interval. """
    a, b, c, d = M

    s, t = field(a, c), field(b, d)

    if s <= t:
        return (s, t)
    else:
        return (t, s)

def dup_step_refine_real_root(f, M, K, fast=False):
    """One step of positive real root refinement algorithm. """
    a, b, c, d = M

    if a == b and c == d:
        return f, (a, b, c, d)

    A = dup_root_lower_bound(f, K)

    if A is not None:
        A = K(int(A))
    else:
        A = K.zero

    if fast and A > 16:
        f = dup_scale(f, A, K)
        a, c, A = A*a, A*c, K.one

    if A >= K.one:
        f = dup_shift(f, A, K)
        b, d = A*a + b, A*c + d

        if not dup_eval(f, K.zero, K):
            return f, (b, b, d, d)

    f, g = dup_shift(f, K.one, K), f

    a1, b1, c1, d1 = a, a + b, c, c + d

    if not dup_eval(f, K.zero, K):
        return f, (b1, b1, d1, d1)

    k = dup_sign_variations(f, K)

    if k == 1:
        a, b, c, d = a1, b1, c1, d1
    else:
        f = dup_shift(dup_reverse(g), K.one, K)

        if not dup_eval(f, K.zero, K):
            f = dup_rshift(f, 1, K)

        a, b, c, d = b, a + b, d, c + d

    return f, (a, b, c, d)

def dup_inner_refine_real_root(f, M, K, eps=None, steps=None, disjoint=None, fast=False, mobius=False):
    """Refine a positive root of `f` given a Mobius transform or an interval. """
    F = K.get_field()

    if len(M) == 2:
        a, b, c, d = _mobius_from_interval(M, F)
    else:
        a, b, c, d = M

    while not c:
        f, (a, b, c, d) = dup_step_refine_real_root(f, (a, b, c,
            d), K, fast=fast)

    if eps is not None and steps is not None:
        for i in range(0, steps):
            if abs(F(a, c) - F(b, d)) >= eps:
                f, (a, b, c, d) = dup_step_refine_real_root(f, (a, b, c, d), K, fast=fast)
            else:
                break
    else:
        if eps is not None:
            while abs(F(a, c) - F(b, d)) >= eps:
                f, (a, b, c, d) = dup_step_refine_real_root(f, (a, b, c, d), K, fast=fast)

        if steps is not None:
            for i in range(0, steps):
                f, (a, b, c, d) = dup_step_refine_real_root(f, (a, b, c, d), K, fast=fast)

    if disjoint is not None:
        while True:
            u, v = _mobius_to_interval((a, b, c, d), F)

            if v <= disjoint or disjoint <= u:
                break
            else:
                f, (a, b, c, d) = dup_step_refine_real_root(f, (a, b, c, d), K, fast=fast)

    if not mobius:
        return _mobius_to_interval((a, b, c, d), F)
    else:
        return f, (a, b, c, d)

def dup_outer_refine_real_root(f, s, t, K, eps=None, steps=None, disjoint=None, fast=False):
    """Refine a positive root of `f` given an interval `(s, t)`. """
    a, b, c, d = _mobius_from_interval((s, t), K.get_field())

    f = dup_transform(f, dup_strip([a, b]),
                         dup_strip([c, d]), K)

    if dup_sign_variations(f, K) != 1:
        raise RefinementFailed("there should be exactly one root in (%s, %s) interval" % (s, t))

    return dup_inner_refine_real_root(f, (a, b, c, d), K, eps=eps, steps=steps, disjoint=disjoint, fast=fast)

def dup_refine_real_root(f, s, t, K, eps=None, steps=None, disjoint=None, fast=False):
    """Refine real root's approximating interval to the given precision. """
    if K.is_QQ:
        (_, f), K = dup_clear_denoms(f, K, convert=True), K.get_ring()
    elif not K.is_ZZ:
        raise DomainError("real root refinement not supported over %s" % K)

    if s == t:
        return (s, t)

    if s > t:
        s, t = t, s

    negative = False

    if s < 0:
        if t <= 0:
            f, s, t, negative = dup_mirror(f, K), -t, -s, True
        else:
            raise ValueError("Cannot refine a real root in (%s, %s)" % (s, t))

    if negative and disjoint is not None:
        if disjoint < 0:
            disjoint = -disjoint
        else:
            disjoint = None

    s, t = dup_outer_refine_real_root(
        f, s, t, K, eps=eps, steps=steps, disjoint=disjoint, fast=fast)

    if negative:
        return (-t, -s)
    else:
        return ( s, t)

def dup_inner_isolate_real_roots(f, K, eps=None, fast=False):
    """Internal function for isolation positive roots up to given precision.

       References
       ==========
           1. Alkiviadis G. Akritas and Adam W. Strzebonski: A Comparative Study of Two Real Root
           Isolation Methods . Nonlinear Analysis: Modelling and Control, Vol. 10, No. 4, 297-304, 2005.
           2. Alkiviadis G. Akritas, Adam W. Strzebonski and Panagiotis S. Vigklas: Improving the
           Performance of the Continued Fractions Method Using new Bounds of Positive Roots. Nonlinear
           Analysis: Modelling and Control, Vol. 13, No. 3, 265-279, 2008.
    """
    a, b, c, d = K.one, K.zero, K.zero, K.one

    k = dup_sign_variations(f, K)

    if k == 0:
        return []
    if k == 1:
        roots = [dup_inner_refine_real_root(
            f, (a, b, c, d), K, eps=eps, fast=fast, mobius=True)]
    else:
        roots, stack = [], [(a, b, c, d, f, k)]

        while stack:
            a, b, c, d, f, k = stack.pop()

            A = dup_root_lower_bound(f, K)

            if A is not None:
                A = K(int(A))
            else:
                A = K.zero

            if fast and A > 16:
                f = dup_scale(f, A, K)
                a, c, A = A*a, A*c, K.one

            if A >= K.one:
                f = dup_shift(f, A, K)
                b, d = A*a + b, A*c + d

                if not dup_TC(f, K):
                    roots.append((f, (b, b, d, d)))
                    f = dup_rshift(f, 1, K)

                k = dup_sign_variations(f, K)

                if k == 0:
                    continue
                if k == 1:
                    roots.append(dup_inner_refine_real_root(
                        f, (a, b, c, d), K, eps=eps, fast=fast, mobius=True))
                    continue

            f1 = dup_shift(f, K.one, K)

            a1, b1, c1, d1, r = a, a + b, c, c + d, 0

            if not dup_TC(f1, K):
                roots.append((f1, (b1, b1, d1, d1)))
                f1, r = dup_rshift(f1, 1, K), 1

            k1 = dup_sign_variations(f1, K)
            k2 = k - k1 - r

            a2, b2, c2, d2 = b, a + b, d, c + d

            if k2 > 1:
                f2 = dup_shift(dup_reverse(f), K.one, K)

                if not dup_TC(f2, K):
                    f2 = dup_rshift(f2, 1, K)

                k2 = dup_sign_variations(f2, K)
            else:
                f2 = None

            if k1 < k2:
                a1, a2, b1, b2 = a2, a1, b2, b1
                c1, c2, d1, d2 = c2, c1, d2, d1
                f1, f2, k1, k2 = f2, f1, k2, k1

            if not k1:
                continue

            if f1 is None:
                f1 = dup_shift(dup_reverse(f), K.one, K)

                if not dup_TC(f1, K):
                    f1 = dup_rshift(f1, 1, K)

            if k1 == 1:
                roots.append(dup_inner_refine_real_root(
                    f1, (a1, b1, c1, d1), K, eps=eps, fast=fast, mobius=True))
            else:
                stack.append((a1, b1, c1, d1, f1, k1))

            if not k2:
                continue

            if f2 is None:
                f2 = dup_shift(dup_reverse(f), K.one, K)

                if not dup_TC(f2, K):
                    f2 = dup_rshift(f2, 1, K)

            if k2 == 1:
                roots.append(dup_inner_refine_real_root(
                    f2, (a2, b2, c2, d2), K, eps=eps, fast=fast, mobius=True))
            else:
                stack.append((a2, b2, c2, d2, f2, k2))

    return roots

def _discard_if_outside_interval(f, M, inf, sup, K, negative, fast, mobius):
    """Discard an isolating interval if outside ``(inf, sup)``. """
    F = K.get_field()

    while True:
        u, v = _mobius_to_interval(M, F)

        if negative:
            u, v = -v, -u

        if (inf is None or u >= inf) and (sup is None or v <= sup):
            if not mobius:
                return u, v
            else:
                return f, M
        elif (sup is not None and u > sup) or (inf is not None and v < inf):
            return None
        else:
            f, M = dup_step_refine_real_root(f, M, K, fast=fast)

def dup_inner_isolate_positive_roots(f, K, eps=None, inf=None, sup=None, fast=False, mobius=False):
    """Iteratively compute disjoint positive root isolation intervals. """
    if sup is not None and sup < 0:
        return []

    roots = dup_inner_isolate_real_roots(f, K, eps=eps, fast=fast)

    F, results = K.get_field(), []

    if inf is not None or sup is not None:
        for f, M in roots:
            result = _discard_if_outside_interval(f, M, inf, sup, K, False, fast, mobius)

            if result is not None:
                results.append(result)
    elif not mobius:
        results.extend(_mobius_to_interval(M, F) for _, M in roots)
    else:
        results = roots

    return results

def dup_inner_isolate_negative_roots(f, K, inf=None, sup=None, eps=None, fast=False, mobius=False):
    """Iteratively compute disjoint negative root isolation intervals. """
    if inf is not None and inf >= 0:
        return []

    roots = dup_inner_isolate_real_roots(dup_mirror(f, K), K, eps=eps, fast=fast)

    F, results = K.get_field(), []

    if inf is not None or sup is not None:
        for f, M in roots:
            result = _discard_if_outside_interval(f, M, inf, sup, K, True, fast, mobius)

            if result is not None:
                results.append(result)
    elif not mobius:
        for f, M in roots:
            u, v = _mobius_to_interval(M, F)
            results.append((-v, -u))
    else:
        results = roots

    return results

def _isolate_zero(f, K, inf, sup, basis=False, sqf=False):
    """Handle special case of CF algorithm when ``f`` is homogeneous. """
    j, f = dup_terms_gcd(f, K)

    if j > 0:
        F = K.get_field()

        if (inf is None or inf <= 0) and (sup is None or 0 <= sup):
            if not sqf:
                if not basis:
                    return [((F.zero, F.zero), j)], f
                else:
                    return [((F.zero, F.zero), j, [K.one, K.zero])], f
            else:
                return [(F.zero, F.zero)], f

    return [], f

def dup_isolate_real_roots_sqf(f, K, eps=None, inf=None, sup=None, fast=False, blackbox=False):
    """Isolate real roots of a square-free polynomial using the Vincent-Akritas-Strzebonski (VAS) CF approach.

       References
       ==========
       .. [1] Alkiviadis G. Akritas and Adam W. Strzebonski: A Comparative
              Study of Two Real Root Isolation Methods. Nonlinear Analysis:
              Modelling and Control, Vol. 10, No. 4, 297-304, 2005.
       .. [2] Alkiviadis G. Akritas, Adam W. Strzebonski and Panagiotis S.
              Vigklas: Improving the Performance of the Continued Fractions
              Method Using New Bounds of Positive Roots. Nonlinear Analysis:
              Modelling and Control, Vol. 13, No. 3, 265-279, 2008.

    """
    if K.is_QQ:
        (_, f), K = dup_clear_denoms(f, K, convert=True), K.get_ring()
    elif not K.is_ZZ:
        raise DomainError("isolation of real roots not supported over %s" % K)

    if dup_degree(f) <= 0:
        return []

    I_zero, f = _isolate_zero(f, K, inf, sup, basis=False, sqf=True)

    I_neg = dup_inner_isolate_negative_roots(f, K, eps=eps, inf=inf, sup=sup, fast=fast)
    I_pos = dup_inner_isolate_positive_roots(f, K, eps=eps, inf=inf, sup=sup, fast=fast)

    roots = sorted(I_neg + I_zero + I_pos)

    if not blackbox:
        return roots
    else:
        return [ RealInterval((a, b), f, K) for (a, b) in roots ]

def dup_isolate_real_roots(f, K, eps=None, inf=None, sup=None, basis=False, fast=False):
    """Isolate real roots using Vincent-Akritas-Strzebonski (VAS) continued fractions approach.

       References
       ==========

       .. [1] Alkiviadis G. Akritas and Adam W. Strzebonski: A Comparative
              Study of Two Real Root Isolation Methods. Nonlinear Analysis:
              Modelling and Control, Vol. 10, No. 4, 297-304, 2005.
       .. [2] Alkiviadis G. Akritas, Adam W. Strzebonski and Panagiotis S.
              Vigklas: Improving the Performance of the Continued Fractions
              Method Using New Bounds of Positive Roots.
              Nonlinear Analysis: Modelling and Control, Vol. 13, No. 3, 265-279, 2008.

    """
    if K.is_QQ:
        (_, f), K = dup_clear_denoms(f, K, convert=True), K.get_ring()
    elif not K.is_ZZ:
        raise DomainError("isolation of real roots not supported over %s" % K)

    if dup_degree(f) <= 0:
        return []

    I_zero, f = _isolate_zero(f, K, inf, sup, basis=basis, sqf=False)

    _, factors = dup_sqf_list(f, K)

    if len(factors) == 1:
        ((f, k),) = factors

        I_neg = dup_inner_isolate_negative_roots(f, K, eps=eps, inf=inf, sup=sup, fast=fast)
        I_pos = dup_inner_isolate_positive_roots(f, K, eps=eps, inf=inf, sup=sup, fast=fast)

        I_neg = [ ((u, v), k) for u, v in I_neg ]
        I_pos = [ ((u, v), k) for u, v in I_pos ]
    else:
        I_neg, I_pos = _real_isolate_and_disjoin(factors, K,
            eps=eps, inf=inf, sup=sup, basis=basis, fast=fast)

    return sorted(I_neg + I_zero + I_pos)

def dup_isolate_real_roots_list(polys, K, eps=None, inf=None, sup=None, strict=False, basis=False, fast=False):
    """Isolate real roots of a list of polynomial using Vincent-Akritas-Strzebonski (VAS) CF approach.

       References
       ==========

       .. [1] Alkiviadis G. Akritas and Adam W. Strzebonski: A Comparative
              Study of Two Real Root Isolation Methods. Nonlinear Analysis:
              Modelling and Control, Vol. 10, No. 4, 297-304, 2005.
       .. [2] Alkiviadis G. Akritas, Adam W. Strzebonski and Panagiotis S.
              Vigklas: Improving the Performance of the Continued Fractions
              Method Using New Bounds of Positive Roots.
              Nonlinear Analysis: Modelling and Control, Vol. 13, No. 3, 265-279, 2008.

    """
    if K.is_QQ:
        K, F, polys = K.get_ring(), K, polys[:]

        for i, p in enumerate(polys):
            polys[i] = dup_clear_denoms(p, F, K, convert=True)[1]
    elif not K.is_ZZ:
        raise DomainError("isolation of real roots not supported over %s" % K)

    zeros, factors_dict = False, {}

    if (inf is None or inf <= 0) and (sup is None or 0 <= sup):
        zeros, zero_indices = True, {}

    for i, p in enumerate(polys):
        j, p = dup_terms_gcd(p, K)

        if zeros and j > 0:
            zero_indices[i] = j

        for f, k in dup_factor_list(p, K)[1]:
            f = tuple(f)

            if f not in factors_dict:
                factors_dict[f] = {i: k}
            else:
                factors_dict[f][i] = k

    factors_list = [(list(f), indices) for f, indices in factors_dict.items()]
    I_neg, I_pos = _real_isolate_and_disjoin(factors_list, K, eps=eps,
        inf=inf, sup=sup, strict=strict, basis=basis, fast=fast)

    F = K.get_field()

    if not zeros or not zero_indices:
        I_zero = []
    else:
        if not basis:
            I_zero = [((F.zero, F.zero), zero_indices)]
        else:
            I_zero = [((F.zero, F.zero), zero_indices, [K.one, K.zero])]

    return sorted(I_neg + I_zero + I_pos)

def _disjoint_p(M, N, strict=False):
    """Check if Mobius transforms define disjoint intervals. """
    a1, b1, c1, d1 = M
    a2, b2, c2, d2 = N

    a1d1, b1c1 = a1*d1, b1*c1
    a2d2, b2c2 = a2*d2, b2*c2

    if a1d1 == b1c1 and a2d2 == b2c2:
        return True

    if a1d1 > b1c1:
        a1, c1, b1, d1 = b1, d1, a1, c1

    if a2d2 > b2c2:
        a2, c2, b2, d2 = b2, d2, a2, c2

    if not strict:
        return a2*d1 >= c2*b1 or b2*c1 <= d2*a1
    else:
        return a2*d1 > c2*b1 or b2*c1 < d2*a1

def _real_isolate_and_disjoin(factors, K, eps=None, inf=None, sup=None, strict=False, basis=False, fast=False):
    """Isolate real roots of a list of polynomials and disjoin intervals. """
    I_pos, I_neg = [], []

    for i, (f, k) in enumerate(factors):
        for F, M in dup_inner_isolate_positive_roots(f, K, eps=eps, inf=inf, sup=sup, fast=fast, mobius=True):
            I_pos.append((F, M, k, f))

        for G, N in dup_inner_isolate_negative_roots(f, K, eps=eps, inf=inf, sup=sup, fast=fast, mobius=True):
            I_neg.append((G, N, k, f))

    for i, (f, M, k, F) in enumerate(I_pos):
        for j, (g, N, m, G) in enumerate(I_pos[i + 1:]):
            while not _disjoint_p(M, N, strict=strict):
                f, M = dup_inner_refine_real_root(f, M, K, steps=1, fast=fast, mobius=True)
                g, N = dup_inner_refine_real_root(g, N, K, steps=1, fast=fast, mobius=True)

            I_pos[i + j + 1] = (g, N, m, G)

        I_pos[i] = (f, M, k, F)

    for i, (f, M, k, F) in enumerate(I_neg):
        for j, (g, N, m, G) in enumerate(I_neg[i + 1:]):
            while not _disjoint_p(M, N, strict=strict):
                f, M = dup_inner_refine_real_root(f, M, K, steps=1, fast=fast, mobius=True)
                g, N = dup_inner_refine_real_root(g, N, K, steps=1, fast=fast, mobius=True)

            I_neg[i + j + 1] = (g, N, m, G)

        I_neg[i] = (f, M, k, F)

    if strict:
        for i, (f, M, k, F) in enumerate(I_neg):
            if not M[0]:
                while not M[0]:
                    f, M = dup_inner_refine_real_root(f, M, K, steps=1, fast=fast, mobius=True)

                I_neg[i] = (f, M, k, F)
                break

        for j, (g, N, m, G) in enumerate(I_pos):
            if not N[0]:
                while not N[0]:
                    g, N = dup_inner_refine_real_root(g, N, K, steps=1, fast=fast, mobius=True)

                I_pos[j] = (g, N, m, G)
                break

    field = K.get_field()

    I_neg = [ (_mobius_to_interval(M, field), k, f) for (_, M, k, f) in I_neg ]
    I_pos = [ (_mobius_to_interval(M, field), k, f) for (_, M, k, f) in I_pos ]

    I_neg = [((-v, -u), k, f) for ((u, v), k, f) in I_neg]

    if not basis:
        I_neg = [((u, v), k) for ((u, v), k, _) in I_neg]
        I_pos = [((u, v), k) for ((u, v), k, _) in I_pos]

    return I_neg, I_pos

def dup_count_real_roots(f, K, inf=None, sup=None):
    """Returns the number of distinct real roots of ``f`` in ``[inf, sup]``. """
    if dup_degree(f) <= 0:
        return 0

    if not K.is_Field:
        R, K = K, K.get_field()
        f = dup_convert(f, R, K)

    sturm = dup_sturm(f, K)

    if inf is None:
        signs_inf = dup_sign_variations([ dup_LC(s, K)*(-1)**dup_degree(s) for s in sturm ], K)
    else:
        signs_inf = dup_sign_variations([ dup_eval(s, inf, K) for s in sturm ], K)

    if sup is None:
        signs_sup = dup_sign_variations([ dup_LC(s, K) for s in sturm ], K)
    else:
        signs_sup = dup_sign_variations([ dup_eval(s, sup, K) for s in sturm ], K)

    count = abs(signs_inf - signs_sup)

    if inf is not None and not dup_eval(f, inf, K):
        count += 1

    return count

OO = 'OO'  # Origin of (re, im) coordinate system

Q1 = 'Q1'  # Quadrant #1 (++): re > 0 and im > 0
Q2 = 'Q2'  # Quadrant #2 (-+): re < 0 and im > 0
Q3 = 'Q3'  # Quadrant #3 (--): re < 0 and im < 0
Q4 = 'Q4'  # Quadrant #4 (+-): re > 0 and im < 0

A1 = 'A1'  # Axis #1 (+0): re > 0 and im = 0
A2 = 'A2'  # Axis #2 (0+): re = 0 and im > 0
A3 = 'A3'  # Axis #3 (-0): re < 0 and im = 0
A4 = 'A4'  # Axis #4 (0-): re = 0 and im < 0

_rules_simple = {
    # Q --> Q (same) => no change
    (Q1, Q1): 0,
    (Q2, Q2): 0,
    (Q3, Q3): 0,
    (Q4, Q4): 0,

    # A -- CCW --> Q => +1/4 (CCW)
    (A1, Q1): 1,
    (A2, Q2): 1,
    (A3, Q3): 1,
    (A4, Q4): 1,

    # A --  CW --> Q => -1/4 (CCW)
    (A1, Q4): 2,
    (A2, Q1): 2,
    (A3, Q2): 2,
    (A4, Q3): 2,

    # Q -- CCW --> A => +1/4 (CCW)
    (Q1, A2): 3,
    (Q2, A3): 3,
    (Q3, A4): 3,
    (Q4, A1): 3,

    # Q --  CW --> A => -1/4 (CCW)
    (Q1, A1): 4,
    (Q2, A2): 4,
    (Q3, A3): 4,
    (Q4, A4): 4,

    # Q -- CCW --> Q => +1/2 (CCW)
    (Q1, Q2): +5,
    (Q2, Q3): +5,
    (Q3, Q4): +5,
    (Q4, Q1): +5,

    # Q --  CW --> Q => -1/2 (CW)
    (Q1, Q4): -5,
    (Q2, Q1): -5,
    (Q3, Q2): -5,
    (Q4, Q3): -5,
}

_rules_ambiguous = {
    # A -- CCW --> Q => { +1/4 (CCW), -9/4 (CW) }
    (A1, OO, Q1): -1,
    (A2, OO, Q2): -1,
    (A3, OO, Q3): -1,
    (A4, OO, Q4): -1,

    # A --  CW --> Q => { -1/4 (CCW), +7/4 (CW) }
    (A1, OO, Q4): -2,
    (A2, OO, Q1): -2,
    (A3, OO, Q2): -2,
    (A4, OO, Q3): -2,

    # Q -- CCW --> A => { +1/4 (CCW), -9/4 (CW) }
    (Q1, OO, A2): -3,
    (Q2, OO, A3): -3,
    (Q3, OO, A4): -3,
    (Q4, OO, A1): -3,

    # Q --  CW --> A => { -1/4 (CCW), +7/4 (CW) }
    (Q1, OO, A1): -4,
    (Q2, OO, A2): -4,
    (Q3, OO, A3): -4,
    (Q4, OO, A4): -4,

    # A --  OO --> A => { +1 (CCW), -1 (CW) }
    (A1, A3): 7,
    (A2, A4): 7,
    (A3, A1): 7,
    (A4, A2): 7,

    (A1, OO, A3): 7,
    (A2, OO, A4): 7,
    (A3, OO, A1): 7,
    (A4, OO, A2): 7,

    # Q -- DIA --> Q => { +1 (CCW), -1 (CW) }
    (Q1, Q3): 8,
    (Q2, Q4): 8,
    (Q3, Q1): 8,
    (Q4, Q2): 8,

    (Q1, OO, Q3): 8,
    (Q2, OO, Q4): 8,
    (Q3, OO, Q1): 8,
    (Q4, OO, Q2): 8,

    # A --- R ---> A => { +1/2 (CCW), -3/2 (CW) }
    (A1, A2): 9,
    (A2, A3): 9,
    (A3, A4): 9,
    (A4, A1): 9,

    (A1, OO, A2): 9,
    (A2, OO, A3): 9,
    (A3, OO, A4): 9,
    (A4, OO, A1): 9,

    # A --- L ---> A => { +3/2 (CCW), -1/2 (CW) }
    (A1, A4): 10,
    (A2, A1): 10,
    (A3, A2): 10,
    (A4, A3): 10,

    (A1, OO, A4): 10,
    (A2, OO, A1): 10,
    (A3, OO, A2): 10,
    (A4, OO, A3): 10,

    # Q --- 1 ---> A => { +3/4 (CCW), -5/4 (CW) }
    (Q1, A3): 11,
    (Q2, A4): 11,
    (Q3, A1): 11,
    (Q4, A2): 11,

    (Q1, OO, A3): 11,
    (Q2, OO, A4): 11,
    (Q3, OO, A1): 11,
    (Q4, OO, A2): 11,

    # Q --- 2 ---> A => { +5/4 (CCW), -3/4 (CW) }
    (Q1, A4): 12,
    (Q2, A1): 12,
    (Q3, A2): 12,
    (Q4, A3): 12,

    (Q1, OO, A4): 12,
    (Q2, OO, A1): 12,
    (Q3, OO, A2): 12,
    (Q4, OO, A3): 12,

    # A --- 1 ---> Q => { +5/4 (CCW), -3/4 (CW) }
    (A1, Q3): 13,
    (A2, Q4): 13,
    (A3, Q1): 13,
    (A4, Q2): 13,

    (A1, OO, Q3): 13,
    (A2, OO, Q4): 13,
    (A3, OO, Q1): 13,
    (A4, OO, Q2): 13,

    # A --- 2 ---> Q => { +3/4 (CCW), -5/4 (CW) }
    (A1, Q2): 14,
    (A2, Q3): 14,
    (A3, Q4): 14,
    (A4, Q1): 14,

    (A1, OO, Q2): 14,
    (A2, OO, Q3): 14,
    (A3, OO, Q4): 14,
    (A4, OO, Q1): 14,

    # Q --> OO --> Q => { +1/2 (CCW), -3/2 (CW) }
    (Q1, OO, Q2): 15,
    (Q2, OO, Q3): 15,
    (Q3, OO, Q4): 15,
    (Q4, OO, Q1): 15,

    # Q --> OO --> Q => { +3/2 (CCW), -1/2 (CW) }
    (Q1, OO, Q4): 16,
    (Q2, OO, Q1): 16,
    (Q3, OO, Q2): 16,
    (Q4, OO, Q3): 16,

    # A --> OO --> A => { +2 (CCW), 0 (CW) }
    (A1, OO, A1): 17,
    (A2, OO, A2): 17,
    (A3, OO, A3): 17,
    (A4, OO, A4): 17,

    # Q --> OO --> Q => { +2 (CCW), 0 (CW) }
    (Q1, OO, Q1): 18,
    (Q2, OO, Q2): 18,
    (Q3, OO, Q3): 18,
    (Q4, OO, Q4): 18,
}

_values = {
    0: [( 0, 1)],
    1: [(+1, 4)],
    2: [(-1, 4)],
    3: [(+1, 4)],
    4: [(-1, 4)],
    -1: [(+9, 4), (+1, 4)],
    -2: [(+7, 4), (-1, 4)],
    -3: [(+9, 4), (+1, 4)],
    -4: [(+7, 4), (-1, 4)],
    +5: [(+1, 2)],
    -5: [(-1, 2)],
    7: [(+1, 1), (-1, 1)],
    8: [(+1, 1), (-1, 1)],
    9: [(+1, 2), (-3, 2)],
    10: [(+3, 2), (-1, 2)],
    11: [(+3, 4), (-5, 4)],
    12: [(+5, 4), (-3, 4)],
    13: [(+5, 4), (-3, 4)],
    14: [(+3, 4), (-5, 4)],
    15: [(+1, 2), (-3, 2)],
    16: [(+3, 2), (-1, 2)],
    17: [(+2, 1), ( 0, 1)],
    18: [(+2, 1), ( 0, 1)],
}

def _classify_point(re, im):
    """Return the half-axis (or origin) on which (re, im) point is located. """
    if not re and not im:
        return OO

    if not re:
        if im > 0:
            return A2
        else:
            return A4
    elif not im:
        if re > 0:
            return A1
        else:
            return A3

def _intervals_to_quadrants(intervals, f1, f2, s, t, F):
    """Generate a sequence of extended quadrants from a list of critical points. """
    if not intervals:
        return []

    Q = []

    if not f1:
        (a, b), _, _ = intervals[0]

        if a == b == s:
            if len(intervals) == 1:
                if dup_eval(f2, t, F) > 0:
                    return [OO, A2]
                else:
                    return [OO, A4]
            else:
                (a, _), _, _ = intervals[1]

                if dup_eval(f2, (s + a)/2, F) > 0:
                    Q.extend([OO, A2])
                    f2_sgn = +1
                else:
                    Q.extend([OO, A4])
                    f2_sgn = -1

                intervals = intervals[1:]
        else:
            if dup_eval(f2, s, F) > 0:
                Q.append(A2)
                f2_sgn = +1
            else:
                Q.append(A4)
                f2_sgn = -1

        for (a, _), indices, _ in intervals:
            Q.append(OO)

            if indices[1] % 2 == 1:
                f2_sgn = -f2_sgn

            if a != t:
                if f2_sgn > 0:
                    Q.append(A2)
                else:
                    Q.append(A4)

        return Q

    if not f2:
        (a, b), _, _ = intervals[0]

        if a == b == s:
            if len(intervals) == 1:
                if dup_eval(f1, t, F) > 0:
                    return [OO, A1]
                else:
                    return [OO, A3]
            else:
                (a, _), _, _ = intervals[1]

                if dup_eval(f1, (s + a)/2, F) > 0:
                    Q.extend([OO, A1])
                    f1_sgn = +1
                else:
                    Q.extend([OO, A3])
                    f1_sgn = -1

                intervals = intervals[1:]
        else:
            if dup_eval(f1, s, F) > 0:
                Q.append(A1)
                f1_sgn = +1
            else:
                Q.append(A3)
                f1_sgn = -1

        for (a, _), indices, _ in intervals:
            Q.append(OO)

            if indices[0] % 2 == 1:
                f1_sgn = -f1_sgn

            if a != t:
                if f1_sgn > 0:
                    Q.append(A1)
                else:
                    Q.append(A3)

        return Q

    re = dup_eval(f1, s, F)
    im = dup_eval(f2, s, F)

    if not re or not im:
        Q.append(_classify_point(re, im))

        if len(intervals) == 1:
            re = dup_eval(f1, t, F)
            im = dup_eval(f2, t, F)
        else:
            (a, _), _, _ = intervals[1]

            re = dup_eval(f1, (s + a)/2, F)
            im = dup_eval(f2, (s + a)/2, F)

        intervals = intervals[1:]

    if re > 0:
        f1_sgn = +1
    else:
        f1_sgn = -1

    if im > 0:
        f2_sgn = +1
    else:
        f2_sgn = -1

    sgn = {
        (+1, +1): Q1,
        (-1, +1): Q2,
        (-1, -1): Q3,
        (+1, -1): Q4,
    }

    Q.append(sgn[(f1_sgn, f2_sgn)])

    for (a, b), indices, _ in intervals:
        if a == b:
            re = dup_eval(f1, a, F)
            im = dup_eval(f2, a, F)

            cls = _classify_point(re, im)

            if cls is not None:
                Q.append(cls)

        if 0 in indices:
            if indices[0] % 2 == 1:
                f1_sgn = -f1_sgn

        if 1 in indices:
            if indices[1] % 2 == 1:
                f2_sgn = -f2_sgn

        if not (a == b and b == t):
            Q.append(sgn[(f1_sgn, f2_sgn)])

    return Q

def _traverse_quadrants(Q_L1, Q_L2, Q_L3, Q_L4, exclude=None):
    """Transform sequences of quadrants to a sequence of rules. """
    if exclude is True:
        edges = [1, 1, 0, 0]

        corners = {
            (0, 1): 1,
            (1, 2): 1,
            (2, 3): 0,
            (3, 0): 1,
        }
    else:
        edges = [0, 0, 0, 0]

        corners = {
            (0, 1): 0,
            (1, 2): 0,
            (2, 3): 0,
            (3, 0): 0,
        }

    if exclude is not None and exclude is not True:
        exclude = set(exclude)

        for i, edge in enumerate(['S', 'E', 'N', 'W']):
            if edge in exclude:
                edges[i] = 1

        for i, corner in enumerate(['SW', 'SE', 'NE', 'NW']):
            if corner in exclude:
                corners[((i - 1) % 4, i)] = 1

    QQ, rules = [Q_L1, Q_L2, Q_L3, Q_L4], []

    for i, Q in enumerate(QQ):
        if not Q:
            continue

        if Q[-1] == OO:
            Q = Q[:-1]

        if Q[0] == OO:
            j, Q = (i - 1) % 4, Q[1:]
            qq = (QQ[j][-2], OO, Q[0])

            if qq in _rules_ambiguous:
                rules.append((_rules_ambiguous[qq], corners[(j, i)]))
            else:
                raise NotImplementedError("3 element rule (corner): " + str(qq))

        q1, k = Q[0], 1

        while k < len(Q):
            q2, k = Q[k], k + 1

            if q2 != OO:
                qq = (q1, q2)

                if qq in _rules_simple:
                    rules.append((_rules_simple[qq], 0))
                elif qq in _rules_ambiguous:
                    rules.append((_rules_ambiguous[qq], edges[i]))
                else:
                    raise NotImplementedError("2 element rule (inside): " + str(qq))
            else:
                qq, k = (q1, q2, Q[k]), k + 1

                if qq in _rules_ambiguous:
                    rules.append((_rules_ambiguous[qq], edges[i]))
                else:
                    raise NotImplementedError("3 element rule (edge): " + str(qq))

            q1 = qq[-1]

    return rules

def _reverse_intervals(intervals):
    """Reverse intervals for traversal from right to left and from top to bottom. """
    return [ ((b, a), indices, f) for (a, b), indices, f in reversed(intervals) ]

def _winding_number(T, field):
    """Compute the winding number of the input polynomial, i.e. the number of roots. """
    return int(sum(field(*_values[t][i]) for t, i in T) / field(2))

def dup_count_complex_roots(f, K, inf=None, sup=None, exclude=None):
    """Count all roots in [u + v*I, s + t*I] rectangle using Collins-Krandick algorithm. """
    if not K.is_ZZ and not K.is_QQ:
        raise DomainError("complex root counting is not supported over %s" % K)

    if K.is_ZZ:
        R, F = K, K.get_field()
    else:
        R, F = K.get_ring(), K

    f = dup_convert(f, K, F)

    if inf is None or sup is None:
        _, lc = dup_degree(f), abs(dup_LC(f, F))
        B = 2*max(F.quo(abs(c), lc) for c in f)

    if inf is None:
        (u, v) = (-B, -B)
    else:
        (u, v) = inf

    if sup is None:
        (s, t) = (+B, +B)
    else:
        (s, t) = sup

    f1, f2 = dup_real_imag(f, F)

    f1L1F = dmp_eval_in(f1, v, 1, 1, F)
    f2L1F = dmp_eval_in(f2, v, 1, 1, F)

    _, f1L1R = dup_clear_denoms(f1L1F, F, R, convert=True)
    _, f2L1R = dup_clear_denoms(f2L1F, F, R, convert=True)

    f1L2F = dmp_eval_in(f1, s, 0, 1, F)
    f2L2F = dmp_eval_in(f2, s, 0, 1, F)

    _, f1L2R = dup_clear_denoms(f1L2F, F, R, convert=True)
    _, f2L2R = dup_clear_denoms(f2L2F, F, R, convert=True)

    f1L3F = dmp_eval_in(f1, t, 1, 1, F)
    f2L3F = dmp_eval_in(f2, t, 1, 1, F)

    _, f1L3R = dup_clear_denoms(f1L3F, F, R, convert=True)
    _, f2L3R = dup_clear_denoms(f2L3F, F, R, convert=True)

    f1L4F = dmp_eval_in(f1, u, 0, 1, F)
    f2L4F = dmp_eval_in(f2, u, 0, 1, F)

    _, f1L4R = dup_clear_denoms(f1L4F, F, R, convert=True)
    _, f2L4R = dup_clear_denoms(f2L4F, F, R, convert=True)

    S_L1 = [f1L1R, f2L1R]
    S_L2 = [f1L2R, f2L2R]
    S_L3 = [f1L3R, f2L3R]
    S_L4 = [f1L4R, f2L4R]

    I_L1 = dup_isolate_real_roots_list(S_L1, R, inf=u, sup=s, fast=True, basis=True, strict=True)
    I_L2 = dup_isolate_real_roots_list(S_L2, R, inf=v, sup=t, fast=True, basis=True, strict=True)
    I_L3 = dup_isolate_real_roots_list(S_L3, R, inf=u, sup=s, fast=True, basis=True, strict=True)
    I_L4 = dup_isolate_real_roots_list(S_L4, R, inf=v, sup=t, fast=True, basis=True, strict=True)

    I_L3 = _reverse_intervals(I_L3)
    I_L4 = _reverse_intervals(I_L4)

    Q_L1 = _intervals_to_quadrants(I_L1, f1L1F, f2L1F, u, s, F)
    Q_L2 = _intervals_to_quadrants(I_L2, f1L2F, f2L2F, v, t, F)
    Q_L3 = _intervals_to_quadrants(I_L3, f1L3F, f2L3F, s, u, F)
    Q_L4 = _intervals_to_quadrants(I_L4, f1L4F, f2L4F, t, v, F)

    T = _traverse_quadrants(Q_L1, Q_L2, Q_L3, Q_L4, exclude=exclude)

    return _winding_number(T, F)

def _vertical_bisection(N, a, b, I, Q, F1, F2, f1, f2, F):
    """Vertical bisection step in Collins-Krandick root isolation algorithm. """
    (u, v), (s, t) = a, b

    I_L1, I_L2, I_L3, I_L4 = I
    Q_L1, Q_L2, Q_L3, Q_L4 = Q

    f1L1F, f1L2F, f1L3F, f1L4F = F1
    f2L1F, f2L2F, f2L3F, f2L4F = F2

    x = (u + s) / 2

    f1V = dmp_eval_in(f1, x, 0, 1, F)
    f2V = dmp_eval_in(f2, x, 0, 1, F)

    I_V = dup_isolate_real_roots_list([f1V, f2V], F, inf=v, sup=t, fast=True, strict=True, basis=True)

    I_L1_L, I_L1_R = [], []
    I_L2_L, I_L2_R = I_V, I_L2
    I_L3_L, I_L3_R = [], []
    I_L4_L, I_L4_R = I_L4, _reverse_intervals(I_V)

    for I in I_L1:
        (a, b), indices, h = I

        if a == b:
            if a == x:
                I_L1_L.append(I)
                I_L1_R.append(I)
            elif a < x:
                I_L1_L.append(I)
            else:
                I_L1_R.append(I)
        else:
            if b <= x:
                I_L1_L.append(I)
            elif a >= x:
                I_L1_R.append(I)
            else:
                a, b = dup_refine_real_root(h, a, b, F.get_ring(), disjoint=x, fast=True)

                if b <= x:
                    I_L1_L.append(((a, b), indices, h))
                if a >= x:
                    I_L1_R.append(((a, b), indices, h))

    for I in I_L3:
        (b, a), indices, h = I

        if a == b:
            if a == x:
                I_L3_L.append(I)
                I_L3_R.append(I)
            elif a < x:
                I_L3_L.append(I)
            else:
                I_L3_R.append(I)
        else:
            if b <= x:
                I_L3_L.append(I)
            elif a >= x:
                I_L3_R.append(I)
            else:
                a, b = dup_refine_real_root(h, a, b, F.get_ring(), disjoint=x, fast=True)

                if b <= x:
                    I_L3_L.append(((b, a), indices, h))
                if a >= x:
                    I_L3_R.append(((b, a), indices, h))

    Q_L1_L = _intervals_to_quadrants(I_L1_L, f1L1F, f2L1F, u, x, F)
    Q_L2_L = _intervals_to_quadrants(I_L2_L, f1V, f2V, v, t, F)
    Q_L3_L = _intervals_to_quadrants(I_L3_L, f1L3F, f2L3F, x, u, F)
    Q_L4_L = Q_L4

    Q_L1_R = _intervals_to_quadrants(I_L1_R, f1L1F, f2L1F, x, s, F)
    Q_L2_R = Q_L2
    Q_L3_R = _intervals_to_quadrants(I_L3_R, f1L3F, f2L3F, s, x, F)
    Q_L4_R = _intervals_to_quadrants(I_L4_R, f1V, f2V, t, v, F)

    T_L = _traverse_quadrants(Q_L1_L, Q_L2_L, Q_L3_L, Q_L4_L, exclude=True)
    T_R = _traverse_quadrants(Q_L1_R, Q_L2_R, Q_L3_R, Q_L4_R, exclude=True)

    N_L = _winding_number(T_L, F)
    N_R = _winding_number(T_R, F)

    I_L = (I_L1_L, I_L2_L, I_L3_L, I_L4_L)
    Q_L = (Q_L1_L, Q_L2_L, Q_L3_L, Q_L4_L)

    I_R = (I_L1_R, I_L2_R, I_L3_R, I_L4_R)
    Q_R = (Q_L1_R, Q_L2_R, Q_L3_R, Q_L4_R)

    F1_L = (f1L1F, f1V, f1L3F, f1L4F)
    F2_L = (f2L1F, f2V, f2L3F, f2L4F)

    F1_R = (f1L1F, f1L2F, f1L3F, f1V)
    F2_R = (f2L1F, f2L2F, f2L3F, f2V)

    a, b = (u, v), (x, t)
    c, d = (x, v), (s, t)

    D_L = (N_L, a, b, I_L, Q_L, F1_L, F2_L)
    D_R = (N_R, c, d, I_R, Q_R, F1_R, F2_R)

    return D_L, D_R

def _horizontal_bisection(N, a, b, I, Q, F1, F2, f1, f2, F):
    """Horizontal bisection step in Collins-Krandick root isolation algorithm. """
    (u, v), (s, t) = a, b

    I_L1, I_L2, I_L3, I_L4 = I
    Q_L1, Q_L2, Q_L3, Q_L4 = Q

    f1L1F, f1L2F, f1L3F, f1L4F = F1
    f2L1F, f2L2F, f2L3F, f2L4F = F2

    y = (v + t) / 2

    f1H = dmp_eval_in(f1, y, 1, 1, F)
    f2H = dmp_eval_in(f2, y, 1, 1, F)

    I_H = dup_isolate_real_roots_list([f1H, f2H], F, inf=u, sup=s, fast=True, strict=True, basis=True)

    I_L1_B, I_L1_U = I_L1, I_H
    I_L2_B, I_L2_U = [], []
    I_L3_B, I_L3_U = _reverse_intervals(I_H), I_L3
    I_L4_B, I_L4_U = [], []

    for I in I_L2:
        (a, b), indices, h = I

        if a == b:
            if a == y:
                I_L2_B.append(I)
                I_L2_U.append(I)
            elif a < y:
                I_L2_B.append(I)
            else:
                I_L2_U.append(I)
        else:
            if b <= y:
                I_L2_B.append(I)
            elif a >= y:
                I_L2_U.append(I)
            else:
                a, b = dup_refine_real_root(h, a, b, F.get_ring(), disjoint=y, fast=True)

                if b <= y:
                    I_L2_B.append(((a, b), indices, h))
                if a >= y:
                    I_L2_U.append(((a, b), indices, h))

    for I in I_L4:
        (b, a), indices, h = I

        if a == b:
            if a == y:
                I_L4_B.append(I)
                I_L4_U.append(I)
            elif a < y:
                I_L4_B.append(I)
            else:
                I_L4_U.append(I)
        else:
            if b <= y:
                I_L4_B.append(I)
            elif a >= y:
                I_L4_U.append(I)
            else:
                a, b = dup_refine_real_root(h, a, b, F.get_ring(), disjoint=y, fast=True)

                if b <= y:
                    I_L4_B.append(((b, a), indices, h))
                if a >= y:
                    I_L4_U.append(((b, a), indices, h))

    Q_L1_B = Q_L1
    Q_L2_B = _intervals_to_quadrants(I_L2_B, f1L2F, f2L2F, v, y, F)
    Q_L3_B = _intervals_to_quadrants(I_L3_B, f1H, f2H, s, u, F)
    Q_L4_B = _intervals_to_quadrants(I_L4_B, f1L4F, f2L4F, y, v, F)

    Q_L1_U = _intervals_to_quadrants(I_L1_U, f1H, f2H, u, s, F)
    Q_L2_U = _intervals_to_quadrants(I_L2_U, f1L2F, f2L2F, y, t, F)
    Q_L3_U = Q_L3
    Q_L4_U = _intervals_to_quadrants(I_L4_U, f1L4F, f2L4F, t, y, F)

    T_B = _traverse_quadrants(Q_L1_B, Q_L2_B, Q_L3_B, Q_L4_B, exclude=True)
    T_U = _traverse_quadrants(Q_L1_U, Q_L2_U, Q_L3_U, Q_L4_U, exclude=True)

    N_B = _winding_number(T_B, F)
    N_U = _winding_number(T_U, F)

    I_B = (I_L1_B, I_L2_B, I_L3_B, I_L4_B)
    Q_B = (Q_L1_B, Q_L2_B, Q_L3_B, Q_L4_B)

    I_U = (I_L1_U, I_L2_U, I_L3_U, I_L4_U)
    Q_U = (Q_L1_U, Q_L2_U, Q_L3_U, Q_L4_U)

    F1_B = (f1L1F, f1L2F, f1H, f1L4F)
    F2_B = (f2L1F, f2L2F, f2H, f2L4F)

    F1_U = (f1H, f1L2F, f1L3F, f1L4F)
    F2_U = (f2H, f2L2F, f2L3F, f2L4F)

    a, b = (u, v), (s, y)
    c, d = (u, y), (s, t)

    D_B = (N_B, a, b, I_B, Q_B, F1_B, F2_B)
    D_U = (N_U, c, d, I_U, Q_U, F1_U, F2_U)

    return D_B, D_U

def _depth_first_select(rectangles):
    """Find a rectangle of minimum area for bisection. """
    min_area, j = None, None

    for i, (_, (u, v), (s, t), _, _, _, _) in enumerate(rectangles):
        area = (s - u)*(t - v)

        if min_area is None or area < min_area:
            min_area, j = area, i

    return rectangles.pop(j)

def _rectangle_small_p(a, b, eps):
    """Return ``True`` if the given rectangle is small enough. """
    (u, v), (s, t) = a, b

    if eps is not None:
        return s - u < eps and t - v < eps
    else:
        return True

def dup_isolate_complex_roots_sqf(f, K, eps=None, inf=None, sup=None, blackbox=False):
    """Isolate complex roots of a square-free polynomial using Collins-Krandick algorithm. """
    if not K.is_ZZ and not K.is_QQ:
        raise DomainError("isolation of complex roots is not supported over %s" % K)

    if dup_degree(f) <= 0:
        return []

    if K.is_ZZ:
        F = K.get_field()
    else:
        F = K

    f = dup_convert(f, K, F)

    lc = abs(dup_LC(f, F))
    B = 2*max(F.quo(abs(c), lc) for c in f)

    (u, v), (s, t) = (-B, F.zero), (B, B)

    if inf is not None:
        u = inf

    if sup is not None:
        s = sup

    if v < 0 or t <= v or s <= u:
        raise ValueError("not a valid complex isolation rectangle")

    f1, f2 = dup_real_imag(f, F)

    f1L1 = dmp_eval_in(f1, v, 1, 1, F)
    f2L1 = dmp_eval_in(f2, v, 1, 1, F)

    f1L2 = dmp_eval_in(f1, s, 0, 1, F)
    f2L2 = dmp_eval_in(f2, s, 0, 1, F)

    f1L3 = dmp_eval_in(f1, t, 1, 1, F)
    f2L3 = dmp_eval_in(f2, t, 1, 1, F)

    f1L4 = dmp_eval_in(f1, u, 0, 1, F)
    f2L4 = dmp_eval_in(f2, u, 0, 1, F)

    S_L1 = [f1L1, f2L1]
    S_L2 = [f1L2, f2L2]
    S_L3 = [f1L3, f2L3]
    S_L4 = [f1L4, f2L4]

    I_L1 = dup_isolate_real_roots_list(S_L1, F, inf=u, sup=s, fast=True, strict=True, basis=True)
    I_L2 = dup_isolate_real_roots_list(S_L2, F, inf=v, sup=t, fast=True, strict=True, basis=True)
    I_L3 = dup_isolate_real_roots_list(S_L3, F, inf=u, sup=s, fast=True, strict=True, basis=True)
    I_L4 = dup_isolate_real_roots_list(S_L4, F, inf=v, sup=t, fast=True, strict=True, basis=True)

    I_L3 = _reverse_intervals(I_L3)
    I_L4 = _reverse_intervals(I_L4)

    Q_L1 = _intervals_to_quadrants(I_L1, f1L1, f2L1, u, s, F)
    Q_L2 = _intervals_to_quadrants(I_L2, f1L2, f2L2, v, t, F)
    Q_L3 = _intervals_to_quadrants(I_L3, f1L3, f2L3, s, u, F)
    Q_L4 = _intervals_to_quadrants(I_L4, f1L4, f2L4, t, v, F)

    T = _traverse_quadrants(Q_L1, Q_L2, Q_L3, Q_L4)
    N = _winding_number(T, F)

    if not N:
        return []

    I = (I_L1, I_L2, I_L3, I_L4)
    Q = (Q_L1, Q_L2, Q_L3, Q_L4)

    F1 = (f1L1, f1L2, f1L3, f1L4)
    F2 = (f2L1, f2L2, f2L3, f2L4)

    rectangles, roots = [(N, (u, v), (s, t), I, Q, F1, F2)], []

    while rectangles:
        N, (u, v), (s, t), I, Q, F1, F2 = _depth_first_select(rectangles)

        if s - u > t - v:
            D_L, D_R = _vertical_bisection(N, (u, v), (s, t), I, Q, F1, F2, f1, f2, F)

            N_L, a, b, I_L, Q_L, F1_L, F2_L = D_L
            N_R, c, d, I_R, Q_R, F1_R, F2_R = D_R

            if N_L >= 1:
                if N_L == 1 and _rectangle_small_p(a, b, eps):
                    roots.append(ComplexInterval(a, b, I_L, Q_L, F1_L, F2_L, f1, f2, F))
                else:
                    rectangles.append(D_L)

            if N_R >= 1:
                if N_R == 1 and _rectangle_small_p(c, d, eps):
                    roots.append(ComplexInterval(c, d, I_R, Q_R, F1_R, F2_R, f1, f2, F))
                else:
                    rectangles.append(D_R)
        else:
            D_B, D_U = _horizontal_bisection(N, (u, v), (s, t), I, Q, F1, F2, f1, f2, F)

            N_B, a, b, I_B, Q_B, F1_B, F2_B = D_B
            N_U, c, d, I_U, Q_U, F1_U, F2_U = D_U

            if N_B >= 1:
                if N_B == 1 and _rectangle_small_p(a, b, eps):
                    roots.append(ComplexInterval(
                        a, b, I_B, Q_B, F1_B, F2_B, f1, f2, F))
                else:
                    rectangles.append(D_B)

            if N_U >= 1:
                if N_U == 1 and _rectangle_small_p(c, d, eps):
                    roots.append(ComplexInterval(
                        c, d, I_U, Q_U, F1_U, F2_U, f1, f2, F))
                else:
                    rectangles.append(D_U)

    _roots, roots = sorted(roots, key=lambda r: (r.ax, r.ay)), []

    for root in _roots:
        roots.extend([root.conjugate(), root])

    if blackbox:
        return roots
    else:
        return [ r.as_tuple() for r in roots ]

def dup_isolate_all_roots_sqf(f, K, eps=None, inf=None, sup=None, fast=False, blackbox=False):
    """Isolate real and complex roots of a square-free polynomial ``f``. """
    return (
        dup_isolate_real_roots_sqf( f, K, eps=eps, inf=inf, sup=sup, fast=fast, blackbox=blackbox),
        dup_isolate_complex_roots_sqf(f, K, eps=eps, inf=inf, sup=sup, blackbox=blackbox))

def dup_isolate_all_roots(f, K, eps=None, inf=None, sup=None, fast=False):
    """Isolate real and complex roots of a non-square-free polynomial ``f``. """
    if not K.is_ZZ and not K.is_QQ:
        raise DomainError("isolation of real and complex roots is not supported over %s" % K)

    _, factors = dup_sqf_list(f, K)

    if len(factors) == 1:
        ((f, k),) = factors

        real_part, complex_part = dup_isolate_all_roots_sqf(
            f, K, eps=eps, inf=inf, sup=sup, fast=fast)

        real_part = [ ((a, b), k) for (a, b) in real_part ]
        complex_part = [ ((a, b), k) for (a, b) in complex_part ]

        return real_part, complex_part
    else:
        raise NotImplementedError( "only trivial square-free polynomials are supported")

class RealInterval:
    """A fully qualified representation of a real isolation interval. """

    def __init__(self, data, f, dom):
        """Initialize new real interval with complete information. """
        if len(data) == 2:
            s, t = data

            self.neg = False

            if s < 0:
                if t <= 0:
                    f, s, t, self.neg = dup_mirror(f, dom), -t, -s, True
                else:
                    raise ValueError("Cannot refine a real root in (%s, %s)" % (s, t))

            a, b, c, d = _mobius_from_interval((s, t), dom.get_field())

            f = dup_transform(f, dup_strip([a, b]),
                                 dup_strip([c, d]), dom)

            self.mobius = a, b, c, d
        else:
            self.mobius = data[:-1]
            self.neg = data[-1]

        self.f, self.dom = f, dom

    @property
    def func(self):
        return RealInterval

    @property
    def args(self):
        i = self
        return (i.mobius + (i.neg,), i.f, i.dom)

    def __eq__(self, other):
        if type(other) is not type(self):
            return False
        return self.args == other.args

    @property
    def a(self):
        """Return the position of the left end. """
        field = self.dom.get_field()
        a, b, c, d = self.mobius

        if not self.neg:
            if a*d < b*c:
                return field(a, c)
            return field(b, d)
        else:
            if a*d > b*c:
                return -field(a, c)
            return -field(b, d)

    @property
    def b(self):
        """Return the position of the right end. """
        was = self.neg
        self.neg = not was
        rv = -self.a
        self.neg = was
        return rv

    @property
    def dx(self):
        """Return width of the real isolating interval. """
        return self.b - self.a

    @property
    def center(self):
        """Return the center of the real isolating interval. """
        return (self.a + self.b)/2

    @property
    def max_denom(self):
        """Return the largest denominator occurring in either endpoint. """
        return max(self.a.denominator, self.b.denominator)

    def as_tuple(self):
        """Return tuple representation of real isolating interval. """
        return (self.a, self.b)

    def __repr__(self):
        return "(%s, %s)" % (self.a, self.b)

    def __contains__(self, item):
        """
        Say whether a complex number belongs to this real interval.

        Parameters
        ==========

        item : pair (re, im) or number re
            Either a pair giving the real and imaginary parts of the number,
            or else a real number.

        """
        if isinstance(item, tuple):
            re, im = item
        else:
            re, im = item, 0
        return im == 0 and self.a <= re <= self.b

    def is_disjoint(self, other):
        """Return ``True`` if two isolation intervals are disjoint. """
        if isinstance(other, RealInterval):
            return (self.b < other.a or other.b < self.a)
        assert isinstance(other, ComplexInterval)
        return (self.b < other.ax or other.bx < self.a
            or other.ay*other.by > 0)

    def _inner_refine(self):
        """Internal one step real root refinement procedure. """
        if self.mobius is None:
            return self

        f, mobius = dup_inner_refine_real_root(
            self.f, self.mobius, self.dom, steps=1, mobius=True)

        return RealInterval(mobius + (self.neg,), f, self.dom)

    def refine_disjoint(self, other):
        """Refine an isolating interval until it is disjoint with another one. """
        expr = self
        while not expr.is_disjoint(other):
            expr, other = expr._inner_refine(), other._inner_refine()

        return expr, other

    def refine_size(self, dx):
        """Refine an isolating interval until it is of sufficiently small size. """
        expr = self
        while not (expr.dx < dx):
            expr = expr._inner_refine()

        return expr

    def refine_step(self, steps=1):
        """Perform several steps of real root refinement algorithm. """
        expr = self
        for _ in range(steps):
            expr = expr._inner_refine()

        return expr

    def refine(self):
        """Perform one step of real root refinement algorithm. """
        return self._inner_refine()


class ComplexInterval:
    """A fully qualified representation of a complex isolation interval.
    The printed form is shown as (ax, bx) x (ay, by) where (ax, ay)
    and (bx, by) are the coordinates of the southwest and northeast
    corners of the interval's rectangle, respectively.

    Examples
    ========

    >>> from sympy import CRootOf, S
    >>> from sympy.abc import x
    >>> CRootOf.clear_cache()  # for doctest reproducibility
    >>> root = CRootOf(x**10 - 2*x + 3, 9)
    >>> i = root._get_interval(); i
    (3/64, 3/32) x (9/8, 75/64)

    The real part of the root lies within the range [0, 3/4] while
    the imaginary part lies within the range [9/8, 3/2]:

    >>> root.n(3)
    0.0766 + 1.14*I

    The width of the ranges in the x and y directions on the complex
    plane are:

    >>> i.dx, i.dy
    (3/64, 3/64)

    The center of the range is

    >>> i.center
    (9/128, 147/128)

    The northeast coordinate of the rectangle bounding the root in the
    complex plane is given by attribute b and the x and y components
    are accessed by bx and by:

    >>> i.b, i.bx, i.by
    ((3/32, 75/64), 3/32, 75/64)

    The southwest coordinate is similarly given by i.a

    >>> i.a, i.ax, i.ay
    ((3/64, 9/8), 3/64, 9/8)

    Although the interval prints to show only the real and imaginary
    range of the root, all the information of the underlying root
    is contained as properties of the interval.

    For example, an interval with a nonpositive imaginary range is
    considered to be the conjugate. Since the y values of y are in the
    range [0, 1/4] it is not the conjugate:

    >>> i.conj
    False

    The conjugate's interval is

    >>> ic = i.conjugate(); ic
    (3/64, 3/32) x (-75/64, -9/8)

        NOTE: the values printed still represent the x and y range
        in which the root -- conjugate, in this case -- is located,
        but the underlying a and b values of a root and its conjugate
        are the same:

        >>> assert i.a == ic.a and i.b == ic.b

        What changes are the reported coordinates of the bounding rectangle:

        >>> (i.ax, i.ay), (i.bx, i.by)
        ((3/64, 9/8), (3/32, 75/64))
        >>> (ic.ax, ic.ay), (ic.bx, ic.by)
        ((3/64, -75/64), (3/32, -9/8))

    The interval can be refined once:

    >>> i  # for reference, this is the current interval
    (3/64, 3/32) x (9/8, 75/64)

    >>> i.refine()
    (3/64, 3/32) x (9/8, 147/128)

    Several refinement steps can be taken:

    >>> i.refine_step(2)  # 2 steps
    (9/128, 3/32) x (9/8, 147/128)

    It is also possible to refine to a given tolerance:

    >>> tol = min(i.dx, i.dy)/2
    >>> i.refine_size(tol)
    (9/128, 21/256) x (9/8, 291/256)

    A disjoint interval is one whose bounding rectangle does not
    overlap with another. An interval, necessarily, is not disjoint with
    itself, but any interval is disjoint with a conjugate since the
    conjugate rectangle will always be in the lower half of the complex
    plane and the non-conjugate in the upper half:

    >>> i.is_disjoint(i), i.is_disjoint(i.conjugate())
    (False, True)

    The following interval j is not disjoint from i:

    >>> close = CRootOf(x**10 - 2*x + 300/S(101), 9)
    >>> j = close._get_interval(); j
    (75/1616, 75/808) x (225/202, 1875/1616)
    >>> i.is_disjoint(j)
    False

    The two can be made disjoint, however:

    >>> newi, newj = i.refine_disjoint(j)
    >>> newi
    (39/512, 159/2048) x (2325/2048, 4653/4096)
    >>> newj
    (3975/51712, 2025/25856) x (29325/25856, 117375/103424)

    Even though the real ranges overlap, the imaginary do not, so
    the roots have been resolved as distinct. Intervals are disjoint
    when either the real or imaginary component of the intervals is
    distinct. In the case above, the real components have not been
    resolved (so we do not know, yet, which root has the smaller real
    part) but the imaginary part of ``close`` is larger than ``root``:

    >>> close.n(3)
    0.0771 + 1.13*I
    >>> root.n(3)
    0.0766 + 1.14*I
    """

    def __init__(self, a, b, I, Q, F1, F2, f1, f2, dom, conj=False):
        """Initialize new complex interval with complete information. """
        # a and b are the SW and NE corner of the bounding interval,
        # (ax, ay) and (bx, by), respectively, for the NON-CONJUGATE
        # root (the one with the positive imaginary part); when working
        # with the conjugate, the a and b value are still non-negative
        # but the ay, by are reversed and have oppositite sign
        self.a, self.b = a, b
        self.I, self.Q = I, Q

        self.f1, self.F1 = f1, F1
        self.f2, self.F2 = f2, F2

        self.dom = dom
        self.conj = conj

    @property
    def func(self):
        return ComplexInterval

    @property
    def args(self):
        i = self
        return (i.a, i.b, i.I, i.Q, i.F1, i.F2, i.f1, i.f2, i.dom, i.conj)

    def __eq__(self, other):
        if type(other) is not type(self):
            return False
        return self.args == other.args

    @property
    def ax(self):
        """Return ``x`` coordinate of south-western corner. """
        return self.a[0]

    @property
    def ay(self):
        """Return ``y`` coordinate of south-western corner. """
        if not self.conj:
            return self.a[1]
        else:
            return -self.b[1]

    @property
    def bx(self):
        """Return ``x`` coordinate of north-eastern corner. """
        return self.b[0]

    @property
    def by(self):
        """Return ``y`` coordinate of north-eastern corner. """
        if not self.conj:
            return self.b[1]
        else:
            return -self.a[1]

    @property
    def dx(self):
        """Return width of the complex isolating interval. """
        return self.b[0] - self.a[0]

    @property
    def dy(self):
        """Return height of the complex isolating interval. """
        return self.b[1] - self.a[1]

    @property
    def center(self):
        """Return the center of the complex isolating interval. """
        return ((self.ax + self.bx)/2, (self.ay + self.by)/2)

    @property
    def max_denom(self):
        """Return the largest denominator occurring in either endpoint. """
        return max(self.ax.denominator, self.bx.denominator,
                   self.ay.denominator, self.by.denominator)

    def as_tuple(self):
        """Return tuple representation of the complex isolating
        interval's SW and NE corners, respectively. """
        return ((self.ax, self.ay), (self.bx, self.by))

    def __repr__(self):
        return "(%s, %s) x (%s, %s)" % (self.ax, self.bx, self.ay, self.by)

    def conjugate(self):
        """This complex interval really is located in lower half-plane. """
        return ComplexInterval(self.a, self.b, self.I, self.Q,
            self.F1, self.F2, self.f1, self.f2, self.dom, conj=True)

    def __contains__(self, item):
        """
        Say whether a complex number belongs to this complex rectangular
        region.

        Parameters
        ==========

        item : pair (re, im) or number re
            Either a pair giving the real and imaginary parts of the number,
            or else a real number.

        """
        if isinstance(item, tuple):
            re, im = item
        else:
            re, im = item, 0
        return self.ax <= re <= self.bx and self.ay <= im <= self.by

    def is_disjoint(self, other):
        """Return ``True`` if two isolation intervals are disjoint. """
        if isinstance(other, RealInterval):
            return other.is_disjoint(self)
        if self.conj != other.conj:  # above and below real axis
            return True
        re_distinct = (self.bx < other.ax or other.bx < self.ax)
        if re_distinct:
            return True
        im_distinct = (self.by < other.ay or other.by < self.ay)
        return im_distinct

    def _inner_refine(self):
        """Internal one step complex root refinement procedure. """
        (u, v), (s, t) = self.a, self.b

        I, Q = self.I, self.Q

        f1, F1 = self.f1, self.F1
        f2, F2 = self.f2, self.F2

        dom = self.dom

        if s - u > t - v:
            D_L, D_R = _vertical_bisection(1, (u, v), (s, t), I, Q, F1, F2, f1, f2, dom)

            if D_L[0] == 1:
                _, a, b, I, Q, F1, F2 = D_L
            else:
                _, a, b, I, Q, F1, F2 = D_R
        else:
            D_B, D_U = _horizontal_bisection(1, (u, v), (s, t), I, Q, F1, F2, f1, f2, dom)

            if D_B[0] == 1:
                _, a, b, I, Q, F1, F2 = D_B
            else:
                _, a, b, I, Q, F1, F2 = D_U

        return ComplexInterval(a, b, I, Q, F1, F2, f1, f2, dom, self.conj)

    def refine_disjoint(self, other):
        """Refine an isolating interval until it is disjoint with another one. """
        expr = self
        while not expr.is_disjoint(other):
            expr, other = expr._inner_refine(), other._inner_refine()

        return expr, other

    def refine_size(self, dx, dy=None):
        """Refine an isolating interval until it is of sufficiently small size. """
        if dy is None:
            dy = dx
        expr = self
        while not (expr.dx < dx and expr.dy < dy):
            expr = expr._inner_refine()

        return expr

    def refine_step(self, steps=1):
        """Perform several steps of complex root refinement algorithm. """
        expr = self
        for _ in range(steps):
            expr = expr._inner_refine()

        return expr

    def refine(self):
        """Perform one step of complex root refinement algorithm. """
        return self._inner_refine()