Python densearith.dmp_max_norm函数代码示例

def dmp_zz_collins_resultant(f, g, u, K):
    """
    Collins's modular resultant algorithm in `Z[X]`.

    Examples
    ========

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

    >>> f = x + y + 2
    >>> g = 2*x*y + x + 3

    >>> R.dmp_zz_collins_resultant(f, g)
    -2*y**2 - 5*y + 1

    """

    n = dmp_degree(f, u)
    m = dmp_degree(g, u)

    if n < 0 or m < 0:
        return dmp_zero(u - 1)

    A = dmp_max_norm(f, u, K)
    B = dmp_max_norm(g, u, K)

    a = dmp_ground_LC(f, u, K)
    b = dmp_ground_LC(g, u, K)

    v = u - 1

    B = K(2)*K.factorial(K(n + m))*A**m*B**n
    r, p, P = dmp_zero(v), K.one, K.one

    while P <= B:
        p = K(nextprime(p))

        while not (a % p) or not (b % p):
            p = K(nextprime(p))

        F = dmp_ground_trunc(f, p, u, K)
        G = dmp_ground_trunc(g, p, u, K)

        try:
            R = dmp_zz_modular_resultant(F, G, p, u, K)
        except HomomorphismFailed:
            continue

        if K.is_one(P):
            r = R
        else:
            r = dmp_apply_pairs(r, R, _collins_crt, (P, p, K), v, K)

        P *= p

    return r

def dmp_zz_collins_resultant(f, g, u, K):
    """
    Collins's modular resultant algorithm in `Z[X]`.

    Examples
    ========

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

    >>> f = ZZ.map([[1], [1, 2]])
    >>> g = ZZ.map([[2, 1], [3]])

    >>> dmp_zz_collins_resultant(f, g, 1, ZZ)
    [-2, -5, 1]

    """

    n = dmp_degree(f, u)
    m = dmp_degree(g, u)

    if n < 0 or m < 0:
        return dmp_zero(u-1)

    A = dmp_max_norm(f, u, K)
    B = dmp_max_norm(g, u, K)

    a = dmp_ground_LC(f, u, K)
    b = dmp_ground_LC(g, u, K)

    v = u - 1

    B = K(2)*K.factorial(n+m)*A**m*B**n
    r, p, P = dmp_zero(v), K.one, K.one

    while P <= B:
        p = K(nextprime(p))

        while not (a % p) or not (b % p):
            p = K(nextprime(p))

        F = dmp_ground_trunc(f, p, u, K)
        G = dmp_ground_trunc(g, p, u, K)

        try:
            R = dmp_zz_modular_resultant(F, G, p, u, K)
        except HomomorphismFailed:
            continue

        if K.is_one(P):
            r = R
        else:
            r = dmp_apply_pairs(r, R, _collins_crt, (P, p, K), v, K)

        P *= p

    return r

def dmp_zz_mignotte_bound(f, u, K):
    """Mignotte bound for multivariate polynomials in `K[X]`. """
    a = dmp_max_norm(f, u, K)
    b = abs(dmp_ground_LC(f, u, K))
    n = sum(dmp_degree_list(f, u))

    return K.sqrt(K(n+1))*2**n*a*b

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 max_norm(f):
     """Returns maximum norm of `f`. """
     return dmp_max_norm(f.rep, f.lev, f.dom)

def test_dmp_max_norm():
    assert dmp_max_norm([[[]]], 2, ZZ) == 0
    assert dmp_max_norm([[[1]]], 2, ZZ) == 1

    assert dmp_max_norm(f_0, 2, ZZ) == 6

def dmp_factor_list(f, u, K0):
    """Factor multivariate polynomials into irreducibles in `K[X]`. """
    if not u:
        return dup_factor_list(f, K0)

    J, f = dmp_terms_gcd(f, u, K0)
    cont, f = dmp_ground_primitive(f, u, K0)

    if K0.is_FiniteField:  # pragma: no cover
        coeff, factors = dmp_gf_factor(f, u, K0)
    elif K0.is_Algebraic:
        coeff, factors = dmp_ext_factor(f, u, K0)
    else:
        if not K0.is_Exact:
            K0_inexact, K0 = K0, K0.get_exact()
            f = dmp_convert(f, u, K0_inexact, K0)
        else:
            K0_inexact = None

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

            denom, f = dmp_clear_denoms(f, u, K0, K)
            f = dmp_convert(f, u, K0, K)
        else:
            K = K0

        if K.is_ZZ:
            levels, f, v = dmp_exclude(f, u, K)
            coeff, factors = dmp_zz_factor(f, v, K)

            for i, (f, k) in enumerate(factors):
                factors[i] = (dmp_include(f, levels, v, K), k)
        elif K.is_Poly:
            f, v = dmp_inject(f, u, K)

            coeff, factors = dmp_factor_list(f, v, K.dom)

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

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

        if K0.is_Field:
            for i, (f, k) in enumerate(factors):
                factors[i] = (dmp_convert(f, u, K, K0), k)

            coeff = K0.convert(coeff, K)
            coeff = K0.quo(coeff, denom)

            if K0_inexact:
                for i, (f, k) in enumerate(factors):
                    max_norm = dmp_max_norm(f, u, K0)
                    f = dmp_quo_ground(f, max_norm, u, K0)
                    f = dmp_convert(f, u, K0, K0_inexact)
                    factors[i] = (f, k)
                    coeff = K0.mul(coeff, K0.pow(max_norm, k))

                coeff = K0_inexact.convert(coeff, K0)
                K0 = K0_inexact

    for i, j in enumerate(reversed(J)):
        if not j:
            continue

        term = {(0,)*(u - i) + (1,) + (0,)*i: K0.one}
        factors.insert(0, (dmp_from_dict(term, u, K0), j))

    return coeff*cont, _sort_factors(factors)