Python densearith.dmp_neg函数代码示例

def dmp_cancel(f, g, u, K, multout=True):
    """
    Cancel common factors in a rational function ``f/g``.

    **Examples**

    >>> from sympy.polys.domains import ZZ
    >>> from sympy.polys.euclidtools import dmp_cancel

    >>> f = ZZ.map([[2], [0], [-2]])
    >>> g = ZZ.map([[1], [-2], [1]])

    >>> dmp_cancel(f, g, 1, ZZ)
    ([[2], [2]], [[1], [-1]])

    """
    if dmp_zero_p(f, u) or dmp_zero_p(g, u):
        if multout:
            return f, g
        else:
            return K.one, K.one, f, g

    K0 = None

    if K.has_Field and K.has_assoc_Ring:
        K0, K = K, K.get_ring()

        cq, f = dmp_clear_denoms(f, u, K0, K, convert=True)
        cp, g = dmp_clear_denoms(g, u, K0, K, convert=True)
    else:
        cp, cq = K.one, K.one

    _, p, q = dmp_inner_gcd(f, g, u, K)

    if K0 is not None:
        p = dmp_convert(p, u, K, K0)
        q = dmp_convert(q, u, K, K0)

        K = K0

    p_neg = K.is_negative(dmp_ground_LC(p, u, K))
    q_neg = K.is_negative(dmp_ground_LC(q, u, K))

    if p_neg and q_neg:
        p, q = dmp_neg(p, u, K), dmp_neg(q, u, K)
    elif p_neg:
        cp, p = -cp, dmp_neg(p, u, K)
    elif q_neg:
        cp, q = -cp, dmp_neg(q, u, K)

    if not multout:
        return cp, cq, p, q

    p = dmp_mul_ground(p, cp, u, K)
    q = dmp_mul_ground(q, cq, u, K)

    return p, q

def dmp_cancel(f, g, u, K, include=True):
    """
    Cancel common factors in a rational function `f/g`.

    Examples
    ========

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

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

    """
    K0 = None

    if K.has_Field and K.has_assoc_Ring:
        K0, K = K, K.get_ring()

        cq, f = dmp_clear_denoms(f, u, K0, K, convert=True)
        cp, g = dmp_clear_denoms(g, u, K0, K, convert=True)
    else:
        cp, cq = K.one, K.one

    _, p, q = dmp_inner_gcd(f, g, u, K)

    if K0 is not None:
        _, cp, cq = K.cofactors(cp, cq)

        p = dmp_convert(p, u, K, K0)
        q = dmp_convert(q, u, K, K0)

        K = K0

    p_neg = K.is_negative(dmp_ground_LC(p, u, K))
    q_neg = K.is_negative(dmp_ground_LC(q, u, K))

    if p_neg and q_neg:
        p, q = dmp_neg(p, u, K), dmp_neg(q, u, K)
    elif p_neg:
        cp, p = -cp, dmp_neg(p, u, K)
    elif q_neg:
        cp, q = -cp, dmp_neg(q, u, K)

    if not include:
        return cp, cq, p, q

    p = dmp_mul_ground(p, cp, u, K)
    q = dmp_mul_ground(q, cq, u, K)

    return p, q

def dmp_sqf_part(f, u, K):
    """
    Returns square-free part of a polynomial in ``K[X]``.

    Examples
    ========

    >>> from sympy.polys.domains import ZZ
    >>> from sympy.polys.sqfreetools import dmp_sqf_part

    >>> f = ZZ.map([[1], [2, 0], [1, 0, 0], []])

    >>> dmp_sqf_part(f, 1, ZZ)
    [[1], [1, 0], []]

    """
    if not u:
        return dup_sqf_part(f, K)

    if not K.has_CharacteristicZero:
        return dmp_gf_sqf_part(f, u, K)

    if dmp_zero_p(f, u):
        return f

    if K.is_negative(dmp_ground_LC(f, u, K)):
        f = dmp_neg(f, u, K)

    gcd = dmp_gcd(f, dmp_diff(f, 1, u, K), u, K)
    sqf = dmp_quo(f, gcd, u, K)

    if K.has_Field or not K.is_Exact:
        return dmp_ground_monic(sqf, u, K)
    else:
        return dmp_ground_primitive(sqf, u, K)[1]

def dmp_content(f, u, K):
    """
    Returns GCD of multivariate coefficients.

    **Examples**

    >>> from sympy.polys.domains import ZZ
    >>> from sympy.polys.euclidtools import dmp_content

    >>> f = ZZ.map([[2, 6], [4, 12]])

    >>> dmp_content(f, 1, ZZ)
    [2, 6]

    """
    cont, v = dmp_LC(f, K), u-1

    if dmp_zero_p(f, u):
        return cont

    for c in f[1:]:
        cont = dmp_gcd(cont, c, v, K)

        if dmp_one_p(cont, v, K):
            break

    if K.is_negative(dmp_ground_LC(cont, v, K)):
        return dmp_neg(cont, v, K)
    else:
        return cont

def dmp_sqf_part(f, u, K):
    """
    Returns square-free part of a polynomial in ``K[X]``.

    Examples
    ========

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

    >>> R.dmp_sqf_part(x**3 + 2*x**2*y + x*y**2)
    x**2 + x*y

    """
    if not u:
        return dup_sqf_part(f, K)

    if K.is_FiniteField:
        return dmp_gf_sqf_part(f, u, K)

    if dmp_zero_p(f, u):
        return f

    if K.is_negative(dmp_ground_LC(f, u, K)):
        f = dmp_neg(f, u, K)

    gcd = dmp_gcd(f, dmp_diff(f, 1, u, K), u, K)
    sqf = dmp_quo(f, gcd, u, K)

    if K.has_Field:
        return dmp_ground_monic(sqf, u, K)
    else:
        return dmp_ground_primitive(sqf, u, K)[1]

def dmp_content(f, u, K):
    """
    Returns GCD of multivariate coefficients.

    Examples
    ========

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

    >>> R.dmp_content(2*x*y + 6*x + 4*y + 12)
    2*y + 6

    """
    cont, v = dmp_LC(f, K), u - 1

    if dmp_zero_p(f, u):
        return cont

    for c in f[1:]:
        cont = dmp_gcd(cont, c, v, K)

        if dmp_one_p(cont, v, K):
            break

    if K.is_negative(dmp_ground_LC(cont, v, K)):
        return dmp_neg(cont, v, K)
    else:
        return cont

def dmp_sqf_list(f, u, K, all=False):
    """
    Return square-free decomposition of a polynomial in ``K[X]``.

    Examples
    ========

    >>> from sympy.polys.domains import ZZ
    >>> from sympy.polys.sqfreetools import dmp_sqf_list

    >>> f = ZZ.map([[1], [2, 0], [1, 0, 0], [], [], []])

    >>> dmp_sqf_list(f, 1, ZZ)
    (1, [([[1], [1, 0]], 2), ([[1], []], 3)])

    >>> dmp_sqf_list(f, 1, ZZ, all=True)
    (1, [([[1]], 1), ([[1], [1, 0]], 2), ([[1], []], 3)])

    """
    if not u:
        return dup_sqf_list(f, K, all=all)

    if not K.has_CharacteristicZero:
        return dmp_gf_sqf_list(f, u, K, all=all)

    if K.has_Field or not K.is_Exact:
        coeff = dmp_ground_LC(f, u, K)
        f = dmp_ground_monic(f, u, K)
    else:
        coeff, f = dmp_ground_primitive(f, u, K)

        if K.is_negative(dmp_ground_LC(f, u, K)):
            f = dmp_neg(f, u, K)
            coeff = -coeff

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

    result, i = [], 1

    h = dmp_diff(f, 1, u, K)
    g, p, q = dmp_inner_gcd(f, h, u, K)

    while True:
        d = dmp_diff(p, 1, u, K)
        h = dmp_sub(q, d, u, K)

        if dmp_zero_p(h, u):
            result.append((p, i))
            break

        g, p, q = dmp_inner_gcd(p, h, u, K)

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

        i += 1

    return coeff, result

def dmp_sqf_list(f, u, K, all=False):
    """
    Return square-free decomposition of a polynomial in ``K[X]``.

    Examples
    ========

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

    >>> f = x**5 + 2*x**4*y + x**3*y**2

    >>> R.dmp_sqf_list(f)
    (1, [(x + y, 2), (x, 3)])
    >>> R.dmp_sqf_list(f, all=True)
    (1, [(1, 1), (x + y, 2), (x, 3)])

    """
    if not u:
        return dup_sqf_list(f, K, all=all)

    if K.is_FiniteField:
        return dmp_gf_sqf_list(f, u, K, all=all)

    if K.has_Field:
        coeff = dmp_ground_LC(f, u, K)
        f = dmp_ground_monic(f, u, K)
    else:
        coeff, f = dmp_ground_primitive(f, u, K)

        if K.is_negative(dmp_ground_LC(f, u, K)):
            f = dmp_neg(f, u, K)
            coeff = -coeff

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

    result, i = [], 1

    h = dmp_diff(f, 1, u, K)
    g, p, q = dmp_inner_gcd(f, h, u, K)

    while True:
        d = dmp_diff(p, 1, u, K)
        h = dmp_sub(q, d, u, K)

        if dmp_zero_p(h, u):
            result.append((p, i))
            break

        g, p, q = dmp_inner_gcd(p, h, u, K)

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

        i += 1

    return coeff, result

def test_dmp_neg():
    assert dmp_neg([ZZ(-1)], 0, ZZ) == [ZZ(1)]
    assert dmp_neg([QQ(-1,2)], 0, QQ) == [QQ(1,2)]

    assert dmp_neg([[[]]], 2, ZZ) == [[[]]]
    assert dmp_neg([[[ZZ(1)]]], 2, ZZ) == [[[ZZ(-1)]]]
    assert dmp_neg([[[ZZ(-7)]]], 2, ZZ) == [[[ZZ(7)]]]

    assert dmp_neg([[[]]], 2, QQ) == [[[]]]
    assert dmp_neg([[[QQ(1,9)]]], 2, QQ) == [[[QQ(-1,9)]]]
    assert dmp_neg([[[QQ(-7,9)]]], 2, QQ) == [[[QQ(7,9)]]]

    def _parse(cls, rep, dom, lev=None):
        if type(rep) is tuple:
            num, den = rep

            if lev is not None:
                if type(num) is dict:
                    num = dmp_from_dict(num, lev, dom)

                if type(den) is dict:
                    den = dmp_from_dict(den, lev, dom)
            else:
                num, num_lev = dmp_validate(num)
                den, den_lev = dmp_validate(den)

                if num_lev == den_lev:
                    lev = num_lev
                else:
                    raise ValueError('inconsistent number of levels')

            if dmp_zero_p(den, lev):
                raise ZeroDivisionError('fraction denominator')

            if dmp_zero_p(num, lev):
                den = dmp_one(lev, dom)
            else:
                if dmp_negative_p(den, lev, dom):
                    num = dmp_neg(num, lev, dom)
                    den = dmp_neg(den, lev, dom)
        else:
            num = rep

            if lev is not None:
                if type(num) is dict:
                    num = dmp_from_dict(num, lev, dom)
                elif type(num) is not list:
                    num = dmp_ground(dom.convert(num), lev)
            else:
                num, lev = dmp_validate(num)

            den = dmp_one(lev, dom)

        return num, den, lev

def _dmp_rr_trivial_gcd(f, g, u, K):
    """Handle trivial cases in GCD algorithm over a ring. """
    zero_f = dmp_zero_p(f, u)
    zero_g = dmp_zero_p(g, u)

    if zero_f and zero_g:
        return tuple(dmp_zeros(3, u, K))
    elif zero_f:
        if K.is_nonnegative(dmp_ground_LC(g, u, K)):
            return g, dmp_zero(u), dmp_one(u, K)
        else:
            return dmp_neg(g, u, K), dmp_zero(u), dmp_ground(-K.one, u)
    elif zero_g:
        if K.is_nonnegative(dmp_ground_LC(f, u, K)):
            return f, dmp_one(u, K), dmp_zero(u)
        else:
            return dmp_neg(f, u, K), dmp_ground(-K.one, u), dmp_zero(u)
    elif query('USE_SIMPLIFY_GCD'):
        return _dmp_simplify_gcd(f, g, u, K)
    else:
        return None

def test_dmp_sqf():
    assert dmp_sqf_part([[]], 1, ZZ) == [[]]
    assert dmp_sqf_p([[]], 1, ZZ) == True

    assert dmp_sqf_part([[7]], 1, ZZ) == [[1]]
    assert dmp_sqf_p([[7]], 1, ZZ) == True

    assert dmp_sqf_p(f_0, 2, ZZ) == True
    assert dmp_sqf_p(dmp_sqr(f_0, 2, ZZ), 2, ZZ) == False
    assert dmp_sqf_p(f_1, 2, ZZ) == True
    assert dmp_sqf_p(dmp_sqr(f_1, 2, ZZ), 2, ZZ) == False
    assert dmp_sqf_p(f_2, 2, ZZ) == True
    assert dmp_sqf_p(dmp_sqr(f_2, 2, ZZ), 2, ZZ) == False
    assert dmp_sqf_p(f_3, 2, ZZ) == True
    assert dmp_sqf_p(dmp_sqr(f_3, 2, ZZ), 2, ZZ) == False
    assert dmp_sqf_p(f_5, 2, ZZ) == False
    assert dmp_sqf_p(dmp_sqr(f_5, 2, ZZ), 2, ZZ) == False

    assert dmp_sqf_p(f_4, 2, ZZ) == True
    assert dmp_sqf_part(f_4, 2, ZZ) == dmp_neg(f_4, 2, ZZ)
    assert dmp_sqf_p(f_6, 3, ZZ) == True
    assert dmp_sqf_part(f_6, 3, ZZ) == f_6

    assert dmp_sqf_part(f_5, 2, ZZ) == [[[1]], [[1], [-1, 0]]]

    assert dup_sqf_list([], ZZ) == (ZZ(0), [])
    assert dup_sqf_list_include([], ZZ) == [([], 1)]

    assert dmp_sqf_list([[ZZ(3)]], 1, ZZ) == (ZZ(3), [])
    assert dmp_sqf_list_include([[ZZ(3)]], 1, ZZ) == [([[ZZ(3)]], 1)]

    f = [-1,1,0,0,1,-1]

    assert dmp_sqf_list(f, 0, ZZ) == \
        (-1, [([1,1,1,1], 1), ([1,-1], 2)])
    assert dmp_sqf_list_include(f, 0, ZZ) == \
        [([-1,-1,-1,-1], 1), ([1,-1], 2)]

    f = [[-1],[1],[],[],[1],[-1]]

    assert dmp_sqf_list(f, 1, ZZ) == \
        (-1, [([[1],[1],[1],[1]], 1), ([[1],[-1]], 2)])
    assert dmp_sqf_list_include(f, 1, ZZ) == \
        [([[-1],[-1],[-1],[-1]], 1), ([[1],[-1]], 2)]

    K = FF(2)

    f = [[-1], [2], [-1]]

    assert dmp_sqf_list_include(f, 1, ZZ) == \
        [([[-1]], 1), ([[1], [-1]], 2)]

    raises(DomainError, "dmp_sqf_list([[K(1), K(0), K(1)]], 1, K)")

def _dmp_zz_gcd_interpolate(h, x, v, K):
    """Interpolate polynomial GCD from integer GCD. """
    f = []

    while not dmp_zero_p(h, v):
        g = dmp_ground_trunc(h, x, v, K)
        f.insert(0, g)

        h = dmp_sub(h, g, v, K)
        h = dmp_exquo_ground(h, x, v, K)

    if K.is_negative(dmp_ground_LC(f, v+1, K)):
        return dmp_neg(f, v+1, K)
    else:
        return f

def dmp_fateman_poly_F_3(n, K):
    """Fateman's GCD benchmark: sparse inputs (deg f ~ vars f) """
    u = dup_from_raw_dict({n+1: K.one}, K)

    for i in xrange(0, n-1):
        u = dmp_add_term([u], dmp_one(i, K), n+1, i+1, K)

    v = dmp_add_term(u, dmp_ground(K(2), n-2), 0, n, K)

    f = dmp_sqr(dmp_add_term([dmp_neg(v, n-1, K)], dmp_one(n-1, K), n+1, n, K), n, K)
    g = dmp_sqr(dmp_add_term([v], dmp_one(n-1, K), n+1, n, K), n, K)

    v = dmp_add_term(u, dmp_one(n-2, K), 0, n-1, K)

    h = dmp_sqr(dmp_add_term([v], dmp_one(n-1, K), n+1, n, K), n, K)

    return dmp_mul(f, h, n, K), dmp_mul(g, h, n, K), h

def dmp_rr_prs_gcd(f, g, u, 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 sympy.polys.domains import ZZ
    >>> from sympy.polys.euclidtools import dmp_rr_prs_gcd

    >>> f = ZZ.map([[1], [2, 0], [1, 0, 0]])
    >>> g = ZZ.map([[1], [1, 0], []])

    >>> dmp_rr_prs_gcd(f, g, 1, ZZ)
    ([[1], [1, 0]], [[1], [1, 0]], [[1], []])

    """
    if not u:
        return dup_rr_prs_gcd(f, g, K)

    result = _dmp_rr_trivial_gcd(f, g, u, K)

    if result is not None:
        return result

    fc, F = dmp_primitive(f, u, K)
    gc, G = dmp_primitive(g, u, K)

    h = dmp_subresultants(F, G, u, K)[-1]
    c, _, _ = dmp_rr_prs_gcd(fc, gc, u-1, K)

    if K.is_negative(dmp_ground_LC(h, u, K)):
        h = dmp_neg(h, u, K)

    _, h = dmp_primitive(h, u, K)
    h = dmp_mul_term(h, c, 0, u, K)

    cff = dmp_quo(f, h, u, K)
    cfg = dmp_quo(g, h, u, K)

    return h, cff, cfg