Python densearith.dmp_div函数代码示例

def test_dmp_div():
    f, g, q, r = [5,4,3,2,1], [1,2,3], [5,-6,0], [20,1]

    assert dmp_div(f, g, 0, ZZ) == (q, r)
    assert dmp_quo(f, g, 0, ZZ) == q
    assert dmp_rem(f, g, 0, ZZ) == r

    raises(ExactQuotientFailed, lambda: dmp_exquo(f, g, 0, ZZ))

    f, g, q, r = [[[1]]], [[[2]],[1]], [[[]]], [[[1]]]

    assert dmp_div(f, g, 2, ZZ) == (q, r)
    assert dmp_quo(f, g, 2, ZZ) == q
    assert dmp_rem(f, g, 2, ZZ) == r

    raises(ExactQuotientFailed, lambda: dmp_exquo(f, g, 2, ZZ))

def dmp_trial_division(f, factors, u, K):
    """Determine multiplicities of factors using trial division. """
    result = []

    for factor in factors:
        k = 0

        while True:
            q, r = dmp_div(f, factor, u, K)

            if dmp_zero_p(r, u):
                f, k = q, k+1
            else:
                break

        result.append((factor, k))

    return _sort_factors(result)

def dmp_zz_factor(f, u, K):
    """
    Factor (non square-free) polynomials in `Z[X]`.

    Given a multivariate 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
    Enhanced Extended Zassenhaus (EEZ) 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**2 - y**2)`::

        >>> from sympy.polys.factortools import dmp_zz_factor
        >>> from sympy.polys.domains import ZZ

        >>> dmp_zz_factor([[2], [], [-2, 0, 0]], 1, ZZ)
        (2, [([[1], [-1, 0]], 1), ([[1], [1, 0]], 1)])

    In result we got the following factorization::

                    f = 2 (x - y) (x + y)

    **References**

    1. [Gathen99]_

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

    if dmp_zero_p(f, u):
        return K.zero, []

    cont, g = dmp_ground_primitive(f, u, K)

    if dmp_ground_LC(g, u, K) < 0:
        cont, g = -cont, dmp_neg(g, u, K)

    if all([ d <= 0 for d in dmp_degree_list(g, u) ]):
        return cont, []

    G, g = dmp_primitive(g, u, K)

    factors = []

    if dmp_degree(g, u) > 0:
        g = dmp_sqf_part(g, u, K)
        H = dmp_zz_wang(g, u, K)

        for h in H:
            k = 0

            while True:
                q, r = dmp_div(f, h, u, K)

                if dmp_zero_p(r, u):
                    f, k = q, k+1
                else:
                    break

            factors.append((h, k))

    for g, k in dmp_zz_factor(G, u-1, K)[1]:
        factors.insert(0, ([g], k))

    return cont, _sort_factors(factors)

def dmp_zz_heu_gcd(f, g, u, 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 evaluation proces reduces f and g variable by
    variable into a large integer.  The final step  is to verify if the
    interpolated polynomial is the correct GCD. This gives cofactors of
    the input polynomials as a side effect.

    **Examples**

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

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

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

    **References**

    1. [Liao95]_

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

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

    if result is not None:
        return result

    df = dmp_degree(f, u)
    dg = dmp_degree(g, u)

    gcd, f, g = dmp_ground_extract(f, g, u, K)

    f_norm = dmp_max_norm(f, u, K)
    g_norm = dmp_max_norm(g, u, K)

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

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

    for i in xrange(0, HEU_GCD_MAX):
        ff = dmp_eval(f, x, u, K)
        gg = dmp_eval(g, x, u, K)

        v = u - 1

        if not (dmp_zero_p(ff, v) or dmp_zero_p(gg, v)):
            h, cff, cfg = dmp_zz_heu_gcd(ff, gg, v, K)

            h = _dmp_zz_gcd_interpolate(h, x, v, K)
            h = dmp_ground_primitive(h, u, K)[1]

            cff_, r = dmp_div(f, h, u, K)

            if dmp_zero_p(r, u):
                cfg_, r = dmp_div(g, h, u, K)

                if dmp_zero_p(r, u):
                    h = dmp_mul_ground(h, gcd, u, K)
                    return h, cff_, cfg_

            cff = _dmp_zz_gcd_interpolate(cff, x, v, K)

            h, r = dmp_div(f, cff, u, K)

            if dmp_zero_p(r, u):
                cfg_, r = dmp_div(g, h, u, K)

                if dmp_zero_p(r, u):
                    h = dmp_mul_ground(h, gcd, u, K)
                    return h, cff, cfg_

            cfg = _dmp_zz_gcd_interpolate(cfg, x, v, K)

            h, r = dmp_div(g, cfg, u, K)

            if dmp_zero_p(r, u):
                cff_, r = dmp_div(f, h, u, K)

                if dmp_zero_p(r, u):
#.........这里部分代码省略.........

 def div(f, g):
     """Polynomial division with remainder of `f` and `g`. """
     lev, dom, per, F, G = f.unify(g)
     q, r = dmp_div(F, G, lev, dom)
     return per(q), per(r)