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

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

    >>> R.dmp_zz_heu_gcd(f, g)
    (x + y, x + y, x)

    References
    ==========

    1. [Liao95]_
Heuristic polynomial GCD in `Z[X]`.
Heuristic polynomial GCD in Z [ X ] .
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 import ring, ZZ
    >>> R, x,y, = ring("x,y", ZZ)

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

    >>> R.dmp_zz_heu_gcd(f, g)
    (x + y, x + y, x)

    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

    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 = K(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):
                    h = dmp_mul_ground(h, gcd, u, K)
                    return h, cff_, cfg

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

    raise HeuristicGCDFailed('no luck')
def dmp_zz_heu_gcd ( f , g , u , K ) : 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 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 = K ( 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 ) : h = dmp_mul_ground ( h , gcd , u , K ) return h , cff_ , cfg x = 73794 * x * K . sqrt ( K . sqrt ( x ) ) // 27011 raise HeuristicGCDFailed ( 'no luck' )