(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 sympy.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]_
Heuristic polynomial GCD in `Z[x]`.
Heuristic polynomial GCD in Z [ x ] .
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 sympy.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 xrange(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')
def dup_zz_heu_gcd ( f , g , K ) : 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 xrange ( 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' )