Esempio n. 1
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def dup_sqf_part(f, K):
    """
    Returns square-free part of a polynomial in ``K[x]``.

    Examples
    ========

    >>> from diofant.polys import ring, ZZ
    >>> R, x = ring("x", ZZ)

    >>> R.dup_sqf_part(x**3 - 3*x - 2)
    x**2 - x - 2

    """
    if K.is_FiniteField:
        return dup_gf_sqf_part(f, K)

    if not f:
        return f

    if K.is_negative(dup_LC(f, K)):
        f = dup_neg(f, K)

    gcd = dup_gcd(f, dup_diff(f, 1, K), K)
    sqf = dup_quo(f, gcd, K)

    if K.has_Field:
        return dup_monic(sqf, K)
    else:
        return dup_primitive(sqf, K)[1]
Esempio n. 2
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def dup_zz_factor_sqf(f, K):
    """Factor square-free (non-primitive) polyomials in `Z[x]`. """
    cont, g = dup_primitive(f, K)

    n = dup_degree(g)

    if dup_LC(g, K) < 0:
        cont, g = -cont, dup_neg(g, K)

    if n <= 0:
        return cont, []
    elif n == 1:
        return cont, [g]

    if query('USE_IRREDUCIBLE_IN_FACTOR'):
        if dup_zz_irreducible_p(g, K):
            return cont, [g]

    factors = None

    if query('USE_CYCLOTOMIC_FACTOR'):
        factors = dup_zz_cyclotomic_factor(g, K)

    if factors is None:
        factors = dup_zz_zassenhaus(g, K)

    return cont, _sort_factors(factors, multiple=False)
Esempio n. 3
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def dup_factor_list(f, K0):
    """Factor polynomials into irreducibles in `K[x]`. """
    j, f = dup_terms_gcd(f, K0)
    cont, f = dup_primitive(f, K0)

    if K0.is_FiniteField:
        coeff, factors = dup_gf_factor(f, K0)
    elif K0.is_Algebraic:
        coeff, factors = dup_ext_factor(f, K0)
    else:
        if not K0.is_Exact:
            K0_inexact, K0 = K0, K0.get_exact()
            f = dup_convert(f, K0_inexact, K0)
        else:
            K0_inexact = None

        if K0.has_Field:
            K = K0.get_ring()

            denom, f = dup_clear_denoms(f, K0, K)
            f = dup_convert(f, K0, K)
        else:
            K = K0

        if K.is_ZZ:
            coeff, factors = dup_zz_factor(f, K)
        elif K.is_Poly:
            f, u = dmp_inject(f, 0, K)

            coeff, factors = dmp_factor_list(f, u, K.domain)

            for i, (f, k) in enumerate(factors):
                factors[i] = (dmp_eject(f, u, K), k)

            coeff = K.convert(coeff, K.domain)
        else:  # pragma: no cover
            raise DomainError('factorization not supported over %s' % K0)

        if K0.has_Field:
            for i, (f, k) in enumerate(factors):
                factors[i] = (dup_convert(f, K, K0), k)

            coeff = K0.convert(coeff, K)

            if K0_inexact is None:
                coeff = coeff / denom
            else:
                for i, (f, k) in enumerate(factors):
                    f = dup_quo_ground(f, denom, K0)
                    f = dup_convert(f, K0, K0_inexact)
                    factors[i] = (f, k)

                coeff = K0_inexact.convert(coeff * denom**i, K0)
                K0 = K0_inexact

    if j:
        factors.insert(0, ([K0.one, K0.zero], j))

    return coeff * cont, _sort_factors(factors)
Esempio n. 4
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def dup_sqf_list(f, K, all=False):
    """
    Return square-free decomposition of a polynomial in ``K[x]``.

    Examples
    ========

    >>> from diofant.polys import ring, ZZ

    >>> R, x = ring("x", ZZ)

    >>> f = 2*x**5 + 16*x**4 + 50*x**3 + 76*x**2 + 56*x + 16

    >>> R.dup_sqf_list(f)
    (2, [(x + 1, 2), (x + 2, 3)])
    >>> R.dup_sqf_list(f, all=True)
    (2, [(1, 1), (x + 1, 2), (x + 2, 3)])
    """
    if K.is_FiniteField:
        return dup_gf_sqf_list(f, K, all=all)

    if K.has_Field:
        coeff = dup_LC(f, K)
        f = dup_monic(f, K)
    else:
        coeff, f = dup_primitive(f, K)

        if K.is_negative(dup_LC(f, K)):
            f = dup_neg(f, K)
            coeff = -coeff

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

    result, i = [], 1

    h = dup_diff(f, 1, K)
    g, p, q = dup_inner_gcd(f, h, K)

    while True:
        d = dup_diff(p, 1, K)
        h = dup_sub(q, d, K)

        if not h:
            result.append((p, i))
            break

        g, p, q = dup_inner_gcd(p, h, K)

        if all or dup_degree(g) > 0:
            result.append((g, i))

        i += 1

    return coeff, result
Esempio n. 5
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def dup_rr_prs_gcd(f, g, K):
    """
    Computes polynomial GCD using subresultants over a ring.

    Returns ``(h, cff, cfg)`` such that ``a = gcd(f, g)``, ``cff = quo(f, h)``,
    and ``cfg = quo(g, h)``.

    Examples
    ========

    >>> from diofant.polys import ring, ZZ
    >>> R, x = ring("x", ZZ)

    >>> R.dup_rr_prs_gcd(x**2 - 1, x**2 - 3*x + 2)
    (x - 1, x + 1, x - 2)

    """
    result = _dup_rr_trivial_gcd(f, g, K)

    if result is not None:
        return result

    fc, F = dup_primitive(f, K)
    gc, G = dup_primitive(g, K)

    c = K.gcd(fc, gc)

    h = dup_subresultants(F, G, K)[-1]
    _, h = dup_primitive(h, K)

    if K.is_negative(dup_LC(h, K)):
        c = -c

    h = dup_mul_ground(h, c, K)

    cff = dup_quo(f, h, K)
    cfg = dup_quo(g, h, K)

    return h, cff, cfg
Esempio n. 6
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def dup_rr_lcm(f, g, K):
    """
    Computes polynomial LCM over a ring in `K[x]`.

    Examples
    ========

    >>> from diofant.polys import ring, ZZ
    >>> R, x = ring("x", ZZ)

    >>> R.dup_rr_lcm(x**2 - 1, x**2 - 3*x + 2)
    x**3 - 2*x**2 - x + 2

    """
    fc, f = dup_primitive(f, K)
    gc, g = dup_primitive(g, K)

    c = K.lcm(fc, gc)

    h = dup_quo(dup_mul(f, g, K), dup_gcd(f, g, K), K)

    return dup_mul_ground(h, c, K)
Esempio n. 7
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def dup_primitive_prs(f, g, K):
    """
    Primitive polynomial remainder sequence (PRS) in `K[x]`.

    Examples
    ========

    >>> from diofant.polys import ring, ZZ
    >>> R, x = ring("x", ZZ)

    >>> f = x**8 + x**6 - 3*x**4 - 3*x**3 + 8*x**2 + 2*x - 5
    >>> g = 3*x**6 + 5*x**4 - 4*x**2 - 9*x + 21

    >>> prs = R.dup_primitive_prs(f, g)

    >>> prs[0]
    x**8 + x**6 - 3*x**4 - 3*x**3 + 8*x**2 + 2*x - 5
    >>> prs[1]
    3*x**6 + 5*x**4 - 4*x**2 - 9*x + 21
    >>> prs[2]
    -5*x**4 + x**2 - 3
    >>> prs[3]
    13*x**2 + 25*x - 49
    >>> prs[4]
    4663*x - 6150
    >>> prs[5]
    1

    """
    prs = [f, g]
    _, h = dup_primitive(dup_prem(f, g, K), K)

    while h:
        prs.append(h)
        f, g = g, h
        _, h = dup_primitive(dup_prem(f, g, K), K)

    return prs
Esempio n. 8
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def dmp_zz_wang_test_points(f, T, ct, A, u, K):
    """Wang/EEZ: Test evaluation points for suitability. """
    if not dmp_eval_tail(dmp_LC(f, K), A, u - 1, K):
        raise EvaluationFailed('no luck')

    g = dmp_eval_tail(f, A, u, K)

    if not dup_sqf_p(g, K):
        raise EvaluationFailed('no luck')

    c, h = dup_primitive(g, K)

    if K.is_negative(dup_LC(h, K)):
        c, h = -c, dup_neg(h, K)

    v = u - 1

    E = [dmp_eval_tail(t, A, v, K) for t, _ in T]
    D = dmp_zz_wang_non_divisors(E, c, ct, K)

    if D is not None:
        return c, h, E
    else:
        raise EvaluationFailed('no luck')
Esempio n. 9
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def test_dup_primitive():
    assert dup_primitive([], ZZ) == (ZZ(0), [])
    assert dup_primitive([ZZ(1)], ZZ) == (ZZ(1), [ZZ(1)])
    assert dup_primitive([ZZ(1), ZZ(1)], ZZ) == (ZZ(1), [ZZ(1), ZZ(1)])
    assert dup_primitive([ZZ(2), ZZ(2)], ZZ) == (ZZ(2), [ZZ(1), ZZ(1)])
    assert dup_primitive([ZZ(1), ZZ(2), ZZ(1)],
                         ZZ) == (ZZ(1), [ZZ(1), ZZ(2), ZZ(1)])
    assert dup_primitive([ZZ(2), ZZ(4), ZZ(2)],
                         ZZ) == (ZZ(2), [ZZ(1), ZZ(2), ZZ(1)])

    assert dup_primitive([], QQ) == (QQ(0), [])
    assert dup_primitive([QQ(1)], QQ) == (QQ(1), [QQ(1)])
    assert dup_primitive([QQ(1), QQ(1)], QQ) == (QQ(1), [QQ(1), QQ(1)])
    assert dup_primitive([QQ(2), QQ(2)], QQ) == (QQ(2), [QQ(1), QQ(1)])
    assert dup_primitive([QQ(1), QQ(2), QQ(1)],
                         QQ) == (QQ(1), [QQ(1), QQ(2), QQ(1)])
    assert dup_primitive([QQ(2), QQ(4), QQ(2)],
                         QQ) == (QQ(2), [QQ(1), QQ(2), QQ(1)])

    assert dup_primitive([QQ(2, 3), QQ(4, 9)], QQ) == (QQ(2,
                                                          9), [QQ(3),
                                                               QQ(2)])
    assert dup_primitive([QQ(2, 3), QQ(4, 5)], QQ) == (QQ(2,
                                                          15), [QQ(5),
                                                                QQ(6)])
Esempio n. 10
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def dup_zz_heu_gcd(f, g, K):
    """
    Heuristic polynomial GCD in `Z[x]`.

    Given univariate polynomials `f` and `g` in `Z[x]`, returns
    their GCD and cofactors, i.e. polynomials ``h``, ``cff`` and ``cfg``
    such that::

          h = gcd(f, g), cff = quo(f, h) and cfg = quo(g, h)

    The algorithm is purely heuristic which means it may fail to compute
    the GCD. This will be signaled by raising an exception. In this case
    you will need to switch to another GCD method.

    The algorithm computes the polynomial GCD by evaluating polynomials
    f and g at certain points and computing (fast) integer GCD of those
    evaluations. The polynomial GCD is recovered from the integer image
    by interpolation.  The final step is to verify if the result is the
    correct GCD. This gives cofactors as a side effect.

    Examples
    ========

    >>> from diofant.polys import ring, ZZ
    >>> R, x = ring("x", ZZ)

    >>> R.dup_zz_heu_gcd(x**2 - 1, x**2 - 3*x + 2)
    (x - 1, x + 1, x - 2)

    References
    ==========

    .. [1] [Liao95]_
    """
    result = _dup_rr_trivial_gcd(f, g, K)

    if result is not None:
        return result

    df = dup_degree(f)
    dg = dup_degree(g)

    gcd, f, g = dup_extract(f, g, K)

    if df == 0 or dg == 0:
        return [gcd], f, g

    f_norm = dup_max_norm(f, K)
    g_norm = dup_max_norm(g, K)

    B = K(2 * min(f_norm, g_norm) + 29)

    x = max(
        min(B, 99 * K.sqrt(B)),
        2 * min(f_norm // abs(dup_LC(f, K)), g_norm // abs(dup_LC(g, K))) + 2)

    for i in range(0, HEU_GCD_MAX):
        ff = dup_eval(f, x, K)
        gg = dup_eval(g, x, K)

        if ff and gg:
            h = K.gcd(ff, gg)

            cff = ff // h
            cfg = gg // h

            h = _dup_zz_gcd_interpolate(h, x, K)
            h = dup_primitive(h, K)[1]

            cff_, r = dup_div(f, h, K)

            if not r:
                cfg_, r = dup_div(g, h, K)

                if not r:
                    h = dup_mul_ground(h, gcd, K)
                    return h, cff_, cfg_

            cff = _dup_zz_gcd_interpolate(cff, x, K)

            h, r = dup_div(f, cff, K)

            if not r:
                cfg_, r = dup_div(g, h, K)

                if not r:
                    h = dup_mul_ground(h, gcd, K)
                    return h, cff, cfg_

            cfg = _dup_zz_gcd_interpolate(cfg, x, K)

            h, r = dup_div(g, cfg, K)

            if not r:
                cff_, r = dup_div(f, h, K)

                if not r:
                    h = dup_mul_ground(h, gcd, K)
                    return h, cff_, cfg

        x = 73794 * x * K.sqrt(K.sqrt(x)) // 27011

    raise HeuristicGCDFailed('no luck')
Esempio n. 11
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def dup_zz_factor(f, K):
    """
    Factor (non square-free) polynomials in `Z[x]`.

    Given a univariate polynomial `f` in `Z[x]` computes its complete
    factorization `f_1, ..., f_n` into irreducibles over integers::

                f = content(f) f_1**k_1 ... f_n**k_n

    The factorization is computed by reducing the input polynomial
    into a primitive square-free polynomial and factoring it using
    Zassenhaus algorithm. Trial division is used to recover the
    multiplicities of factors.

    The result is returned as a tuple consisting of::

              (content(f), [(f_1, k_1), ..., (f_n, k_n))

    Consider polynomial `f = 2*x**4 - 2`::

        >>> from diofant.polys import ring, ZZ

        >>> R, x = ring("x", ZZ)

        >>> R.dup_zz_factor(2*x**4 - 2)
        (2, [(x - 1, 1), (x + 1, 1), (x**2 + 1, 1)])

    In result we got the following factorization::

                 f = 2 (x - 1) (x + 1) (x**2 + 1)

    Note that this is a complete factorization over integers,
    however over Gaussian integers we can factor the last term.

    By default, polynomials `x**n - 1` and `x**n + 1` are factored
    using cyclotomic decomposition to speedup computations. To
    disable this behaviour set cyclotomic=False.

    References
    ==========

    .. [1] [Gathen99]_
    """
    cont, g = dup_primitive(f, K)

    n = dup_degree(g)

    if dup_LC(g, K) < 0:
        cont, g = -cont, dup_neg(g, K)

    if n <= 0:
        return cont, []
    elif n == 1:
        return cont, [(g, 1)]

    if query('USE_IRREDUCIBLE_IN_FACTOR'):
        if dup_zz_irreducible_p(g, K):
            return cont, [(g, 1)]

    g = dup_sqf_part(g, K)
    H = None

    if query('USE_CYCLOTOMIC_FACTOR'):
        H = dup_zz_cyclotomic_factor(g, K)

    if H is None:
        H = dup_zz_zassenhaus(g, K)

    factors = dup_trial_division(f, H, K)
    return cont, factors
Esempio n. 12
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def dup_zz_zassenhaus(f, K):
    """Factor primitive square-free polynomials in `Z[x]`. """
    n = dup_degree(f)

    if n == 1:
        return [f]

    fc = f[-1]
    A = dup_max_norm(f, K)
    b = dup_LC(f, K)
    B = int(abs(K.sqrt(K(n + 1)) * 2**n * A * b))
    C = int((n + 1)**(2 * n) * A**(2 * n - 1))
    gamma = int(_ceil(2 * _log(C, 2)))
    bound = int(2 * gamma * _log(gamma))
    a = []
    # choose a prime number `p` such that `f` be square free in Z_p
    # if there are many factors in Z_p, choose among a few different `p`
    # the one with fewer factors
    for px in range(3, bound + 1):
        if not isprime(px) or b % px == 0:
            continue

        px = K.convert(px)

        F = gf_from_int_poly(f, px)

        if not gf_sqf_p(F, px, K):
            continue
        fsqfx = gf_factor_sqf(F, px, K)[1]
        a.append((px, fsqfx))
        if len(fsqfx) < 15 or len(a) > 4:
            break
    p, fsqf = min(a, key=lambda x: len(x[1]))

    l = int(_ceil(_log(2 * B + 1, p)))

    modular = [gf_to_int_poly(ff, p) for ff in fsqf]

    g = dup_zz_hensel_lift(p, f, modular, l, K)

    sorted_T = range(len(g))
    T = set(sorted_T)
    factors, s = [], 1
    pl = p**l

    while 2 * s <= len(T):
        for S in subsets(sorted_T, s):
            # lift the constant coefficient of the product `G` of the factors
            # in the subset `S`; if it is does not divide `fc`, `G` does
            # not divide the input polynomial

            if b == 1:
                q = 1
                for i in S:
                    q = q * g[i][-1]
                q = q % pl
                if not _test_pl(fc, q, pl):
                    continue
            else:
                G = [b]
                for i in S:
                    G = dup_mul(G, g[i], K)
                G = dup_trunc(G, pl, K)
                G = dup_primitive(G, K)[1]
                q = G[-1]
                if q and fc % q != 0:
                    continue

            H = [b]
            S = set(S)
            T_S = T - S

            if b == 1:
                G = [b]
                for i in S:
                    G = dup_mul(G, g[i], K)
                G = dup_trunc(G, pl, K)

            for i in T_S:
                H = dup_mul(H, g[i], K)

            H = dup_trunc(H, pl, K)

            G_norm = dup_l1_norm(G, K)
            H_norm = dup_l1_norm(H, K)

            if G_norm * H_norm <= B:
                T = T_S
                sorted_T = [i for i in sorted_T if i not in S]

                G = dup_primitive(G, K)[1]
                f = dup_primitive(H, K)[1]

                factors.append(G)
                b = dup_LC(f, K)

                break
        else:
            s += 1

    return factors + [f]