Python densearith.dup_max_norm函数代码示例

def dup_zz_mignotte_bound(f, K):
    """Mignotte bound for univariate polynomials in `K[x]`. """
    a = dup_max_norm(f, K)
    b = abs(dup_LC(f, K))
    n = dup_degree(f)

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

def dup_factor_list(f, K0):
    """Factor univariate polynomials into irreducibles in `K[x]`. """
    j, f = dup_terms_gcd(f, K0)
    cont, f = dup_primitive(f, K0)

    if K0.is_FiniteField:
        coeff, factors = dup_gf_factor(f, K0)
    elif K0.is_Algebraic:
        coeff, factors = dup_ext_factor(f, K0)
    else:
        if not K0.is_Exact:
            K0_inexact, K0 = K0, K0.get_exact()
            f = dup_convert(f, K0_inexact, K0)
        else:
            K0_inexact = None

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

            denom, f = dup_clear_denoms(f, K0, K)
            f = dup_convert(f, K0, K)
        else:
            K = K0

        if K.is_ZZ:
            coeff, factors = dup_zz_factor(f, K)
        elif K.is_Poly:
            f, u = dmp_inject(f, 0, K)

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

            for i, (f, k) in enumerate(factors):
                factors[i] = (dmp_eject(f, u, 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] = (dup_convert(f, 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 = dup_max_norm(f, K0)
                    f = dup_quo_ground(f, max_norm, K0)
                    f = dup_convert(f, 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

    if j:
        factors.insert(0, ([K0.one, K0.zero], j))

    return coeff*cont, _sort_factors(factors)

#.........这里部分代码省略.........
    if mod is None:
        if u == 1:
            mod = 2
        else:
            mod = 1

    history, configs, A, r = set([]), [], [K.zero]*u, None

    try:
        cs, s, E = dmp_zz_wang_test_points(f, T, ct, A, u, K)

        _, H = dup_zz_factor_sqf(s, K)

        r = len(H)

        if r == 1:
            return [f]

        bad_points = set([tuple(A)])
        configs = [(s, cs, E, H, A)]
    except EvaluationFailed:
        pass

    eez_num_configs = query('EEZ_NUMBER_OF_CONFIGS')
    eez_num_tries = query('EEZ_NUMBER_OF_TRIES')
    eez_mod_step = query('EEZ_MODULUS_STEP')

    while len(configs) < eez_num_configs:
        for _ in xrange(eez_num_tries):
            A = [ K(randint(-mod, mod)) for _ in xrange(u) ]

            if tuple(A) not in history:
                history.add(tuple(A))
            else:
                continue

            try:
                cs, s, E = dmp_zz_wang_test_points(f, T, ct, A, u, K)
            except EvaluationFailed:
                continue

            _, H = dup_zz_factor_sqf(s, K)

            rr = len(H)

            if r is not None:
                if rr != r: # pragma: no cover
                    if rr < r:
                        configs, r = [], rr
                    else:
                        continue
            else:
                r = rr

            if r == 1:
                return [f]

            configs.append((s, cs, E, H, A))

            if len(configs) == eez_num_configs:
                break
        else:
            mod += eez_mod_step

    s_norm, s_arg, i = None, 0, 0

    for s, _, _, _, _ in configs:
        _s_norm = dup_max_norm(s, K)

        if s_norm is not None:
            if _s_norm < s_norm:
                s_norm = _s_norm
                s_arg = i
        else:
            s_norm = _s_norm

        i += 1

    _, cs, E, H, A = configs[s_arg]

    try:
        f, H, LC = dmp_zz_wang_lead_coeffs(f, T, cs, E, H, A, u, K)
        factors = dmp_zz_wang_hensel_lifting(f, H, LC, A, p, u, K)
    except ExtraneousFactors: # pragma: no cover
        if query('EEZ_RESTART_IF_NEEDED'):
            return dmp_zz_wang(f, u, K, mod+1)
        else:
            raise ExtraneousFactors("we need to restart algorithm with better parameters")

    negative, result = 0, []

    for f in factors:
        _, f = dmp_ground_primitive(f, u, K)

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

        result.append(f)

    return result

def dup_zz_zassenhaus(f, K):
    """Factor primitive square-free polynomials in `Z[x]`. """
    n = dup_degree(f)

    if n == 1:
        return [f]

    A = dup_max_norm(f, K)
    b = dup_LC(f, K)
    B = int(abs(K.sqrt(K(n+1))*2**n*A*b))
    C = int((n+1)**(2*n)*A**(2*n-1))
    gamma = int(ceil(2*log(C, 2)))
    bound = int(2*gamma*log(gamma))

    for p in xrange(3, bound+1):
        if not isprime(p) or b % p == 0:
            continue

        p = K.convert(p)

        F = gf_from_int_poly(f, p)

        if gf_sqf_p(F, p, K):
            break

    l = int(ceil(log(2*B + 1, p)))

    modular = []

    for ff in gf_factor_sqf(F, p, K)[1]:
        modular.append(gf_to_int_poly(ff, p))

    g = dup_zz_hensel_lift(p, f, modular, l, K)

    T = set(range(len(g)))
    factors, s = [], 1

    while 2*s <= len(T):
        for S in subsets(T, s):
            G, H = [b], [b]

            S = set(S)

            for i in S:
                G = dup_mul(G, g[i], K)
            for i in T-S:
                H = dup_mul(H, g[i], K)

            G = dup_trunc(G, p**l, K)
            H = dup_trunc(H, p**l, K)

            G_norm = dup_l1_norm(G, K)
            H_norm = dup_l1_norm(H, K)

            if G_norm*H_norm <= B:
                T = T - S

                G = dup_primitive(G, K)[1]
                f = dup_primitive(H, K)[1]

                factors.append(G)
                b = dup_LC(f, K)

                break
        else:
            s += 1

    return factors + [f]

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

    >>> f = ZZ.map([1, 0, -1])
    >>> g = ZZ.map([1, -3, 2])

    >>> dup_zz_heu_gcd(f, g, ZZ)
    ([1, -1], [1, 1], [1, -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 = 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)
#.........这里部分代码省略.........

def test_dup_max_norm():
    assert dup_max_norm([], ZZ) == 0
    assert dup_max_norm([1], ZZ) == 1

    assert dup_max_norm([1,4,2,3], ZZ) == 4

def dup_zz_zassenhaus(f, K):
    """Factor primitive square-free polynomials in `Z[x]`. """
    n = dup_degree(f)

    if n == 1:
        return [f]

    fc = f[-1]
    A = dup_max_norm(f, K)
    b = dup_LC(f, K)
    B = int(abs(K.sqrt(K(n + 1))*2**n*A*b))
    C = int((n + 1)**(2*n)*A**(2*n - 1))
    gamma = int(_ceil(2*_log(C, 2)))
    bound = int(2*gamma*_log(gamma))
    a = []
    # choose a prime number `p` such that `f` be square free in Z_p
    # if there are many factors in Z_p, choose among a few different `p`
    # the one with fewer factors
    for px in range(3, bound + 1):
        if not isprime(px) or b % px == 0:
            continue

        px = K.convert(px)

        F = gf_from_int_poly(f, px)

        if not gf_sqf_p(F, px, K):
            continue
        fsqfx = gf_factor_sqf(F, px, K)[1]
        a.append((px, fsqfx))
        if len(fsqfx) < 15 or len(a) > 4:
            break
    p, fsqf = min(a, key=lambda x: len(x[1]))

    l = int(_ceil(_log(2*B + 1, p)))

    modular = [gf_to_int_poly(ff, p) for ff in fsqf]

    g = dup_zz_hensel_lift(p, f, modular, l, K)

    sorted_T = range(len(g))
    T = set(sorted_T)
    factors, s = [], 1
    pl = p**l

    while 2*s <= len(T):
        for S in subsets(sorted_T, s):
            # lift the constant coefficient of the product `G` of the factors
            # in the subset `S`; if it is does not divide `fc`, `G` does
            # not divide the input polynomial

            if b == 1:
                q = 1
                for i in S:
                    q = q*g[i][-1]
                q = q % pl
                if not _test_pl(fc, q, pl):
                    continue
            else:
                G = [b]
                for i in S:
                    G = dup_mul(G, g[i], K)
                G = dup_trunc(G, pl, K)
                G = dup_primitive(G, K)[1]
                q = G[-1]
                if q and fc % q != 0:
                    continue

            H = [b]
            S = set(S)
            T_S = T - S

            if b == 1:
                G = [b]
                for i in S:
                    G = dup_mul(G, g[i], K)
                G = dup_trunc(G, pl, K)

            for i in T_S:
                H = dup_mul(H, g[i], K)

            H = dup_trunc(H, pl, K)

            G_norm = dup_l1_norm(G, K)
            H_norm = dup_l1_norm(H, K)

            if G_norm*H_norm <= B:
                T = T_S
                sorted_T = [i for i in sorted_T if i not in S]

                G = dup_primitive(G, K)[1]
                f = dup_primitive(H, K)[1]

                factors.append(G)
                b = dup_LC(f, K)

                break
        else:
            s += 1

#.........这里部分代码省略.........

#.........这里部分代码省略.........
    p = K(nextprime(b))

    eez_mod = args.get('mod', None)

    if eez_mod is None:
        if u == 1:
            eez_mod = 2
        else:
            eez_mod = 1

    history, configs, A, r = set([]), [], [K.zero]*u, None

    try:
        cs, s, E = dmp_zz_wang_test_points(f, T, ct, A, u, K)

        _, H = dup_zz_factor_sqf(s, K)

        r = len(H)

        if r == 1:
            return [f]

        bad_points = set([tuple(A)])
        configs = [(s, cs, E, H, A)]
    except EvaluationFailed:
        pass

    while len(configs) < EEZ_NUM_OK:
        for _ in xrange(EEZ_NUM_TRY):
            A = [ K(randint(-eez_mod, eez_mod)) for _ in xrange(u) ]

            if tuple(A) not in history:
                history.add(tuple(A))
            else:
                continue

            try:
                cs, s, E = dmp_zz_wang_test_points(f, T, ct, A, u, K)
            except EvaluationFailed:
                continue

            _, H = dup_zz_factor_sqf(s, K)

            rr = len(H)

            if r is not None:
                if rr != r: # pragma: no cover
                    if rr < r:
                        configs, r = [], rr
                    else:
                        continue
            else:
                r = rr

            if r == 1:
                return [f]

            configs.append((s, cs, E, H, A))

            if len(configs) == EEZ_NUM_OK:
                break
        else:
            eez_mod += EEZ_MOD_STEP

    s_norm, s_arg, i = None, 0, 0

    for s, _, _, _, _ in configs:
        _s_norm = dup_max_norm(s, K)

        if s_norm is not None:
            if _s_norm < s_norm:
                s_norm = _s_norm
                s_arg = i
        else:
            s_norm = _s_norm

        i += 1

    _, cs, E, H, A = configs[s_arg]

    try:
        f, H, LC = dmp_zz_wang_lead_coeffs(f, T, cs, E, H, A, u, K)
        factors = dmp_zz_wang_hensel_lifting(f, H, LC, A, p, u, K)
    except ExtraneousFactors: # pragma: no cover
        if args.get('restart', True):
            return dmp_zz_wang(f, u, K, mod=eez_mod+1)
        else:
            raise ExtraneousFactors("we need to restart algorithm with better parameters")

    negative, result = 0, []

    for f in factors:
        _, f = dmp_ground_primitive(f, u, K)

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

        result.append(f)

    return result