Python densearith.dmp_mul函数代码示例

def dmp_fateman_poly_F_1(n, K):
    """Fateman's GCD benchmark: trivial GCD """
    u = [K(1), K(0)]

    for i in xrange(0, n):
        u = [dmp_one(i, K), u]

    v = [K(1), K(0), K(0)]

    for i in xrange(0, n):
        v = [dmp_one(i, K), dmp_zero(i), v]

    m = n - 1

    U = dmp_add_term(u, dmp_ground(K(1), m), 0, n, K)
    V = dmp_add_term(u, dmp_ground(K(2), m), 0, n, K)

    f = [[-K(3), K(0)], [], [K(1), K(0), -K(1)]]

    W = dmp_add_term(v, dmp_ground(K(1), m), 0, n, K)
    Y = dmp_raise(f, m, 1, K)

    F = dmp_mul(U, V, n, K)
    G = dmp_mul(W, Y, n, K)

    H = dmp_one(n, K)

    return F, G, H

    def mul(f, g):
        """Multiply two multivariate fractions `f` and `g`. """
        if isinstance(g, DMP):
            lev, dom, per, (F_num, F_den), G = f.poly_unify(g)
            num, den = dmp_mul(F_num, G, lev, dom), F_den
        else:
            lev, dom, per, F, G = f.frac_unify(g)
            (F_num, F_den), (G_num, G_den) = F, G

            num = dmp_mul(F_num, G_num, lev, dom)
            den = dmp_mul(F_den, G_den, lev, dom)

        return per(num, den)

    def quo(f, g):
        """Computes quotient of fractions `f` and `g`. """
        if isinstance(g, DMP):
            lev, dom, per, (F_num, F_den), G = f.poly_unify(g)
            num, den = F_num, dmp_mul(F_den, G, lev, dom)
        else:
            lev, dom, per, F, G = f.frac_unify(g)
            (F_num, F_den), (G_num, G_den) = F, G

            num = dmp_mul(F_num, G_den, lev, dom)
            den = dmp_mul(F_den, G_num, lev, dom)

        return per(num, den)

    def sub(f, g):
        """Subtract two multivariate fractions ``f`` and ``g``. """
        if isinstance(g, DMP):
            lev, dom, per, (F_num, F_den), G = f.poly_unify(g)
            num, den = dmp_sub_mul(F_num, F_den, G, lev, dom), F_den
        else:
            lev, dom, per, F, G = f.frac_unify(g)
            (F_num, F_den), (G_num, G_den) = F, G

            num = dmp_sub(dmp_mul(F_num, G_den, lev, dom),
                          dmp_mul(F_den, G_num, lev, dom), lev, dom)
            den = dmp_mul(F_den, G_den, lev, dom)

        return per(num, den)

def dmp_compose(f, g, u, K):
    """
    Evaluate functional composition ``f(g)`` in ``K[X]``.

    Examples
    ========

    >>> from sympy.polys.domains import ZZ
    >>> from sympy.polys.densetools import dmp_compose

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

    >>> dmp_compose(f, g, 1, ZZ)
    [[1, 3, 0]]

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

    if dmp_zero_p(f, u):
        return f

    h = [f[0]]

    for c in f[1:]:
        h = dmp_mul(h, g, u, K)
        h = dmp_add_term(h, c, 0, u, K)

    return h

def dmp_compose(f, g, u, K):
    """
    Evaluate functional composition ``f(g)`` in ``K[X]``.

    Examples
    ========

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

    >>> R.dmp_compose(x*y + 2*x + y, y)
    y**2 + 3*y

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

    if dmp_zero_p(f, u):
        return f

    h = [f[0]]

    for c in f[1:]:
        h = dmp_mul(h, g, u, K)
        h = dmp_add_term(h, c, 0, u, K)

    return h

def dmp_rr_lcm(f, g, u, K):
    """
    Computes polynomial LCM over a ring in `K[X]`.

    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_rr_lcm(f, g)
    x**3 + 2*x**2*y + x*y**2

    """
    fc, f = dmp_ground_primitive(f, u, K)
    gc, g = dmp_ground_primitive(g, u, K)

    c = K.lcm(fc, gc)

    h = dmp_quo(dmp_mul(f, g, u, K),
                dmp_gcd(f, g, u, K), u, K)

    return dmp_mul_ground(h, c, u, K)

def dmp_rr_lcm(f, g, u, K):
    """
    Computes polynomial LCM over a ring in ``K[X]``.

    **Examples**

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

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

    >>> dmp_rr_lcm(f, g, 1, ZZ)
    [[1], [2, 0], [1, 0, 0], []]

    """
    fc, f = dmp_ground_primitive(f, u, K)
    gc, g = dmp_ground_primitive(g, u, K)

    c = K.lcm(fc, gc)

    h = dmp_exquo(dmp_mul(f, g, u, K),
                  dmp_gcd(f, g, u, K), u, K)

    return dmp_mul_ground(h, c, u, K)

def dmp_fateman_poly_F_3(n, K):
    """Fateman's GCD benchmark: sparse inputs (deg f ~ vars f) """
    u = dup_from_raw_dict({n+1: K.one}, K)

    for i in xrange(0, n-1):
        u = dmp_add_term([u], dmp_one(i, K), n+1, i+1, K)

    v = dmp_add_term(u, dmp_ground(K(2), n-2), 0, n, K)

    f = dmp_sqr(dmp_add_term([dmp_neg(v, n-1, K)], dmp_one(n-1, K), n+1, n, K), n, K)
    g = dmp_sqr(dmp_add_term([v], dmp_one(n-1, K), n+1, n, K), n, K)

    v = dmp_add_term(u, dmp_one(n-2, K), 0, n-1, K)

    h = dmp_sqr(dmp_add_term([v], dmp_one(n-1, K), n+1, n, K), n, K)

    return dmp_mul(f, h, n, K), dmp_mul(g, h, n, K), h

    def quo(f, g):
        """Computes quotient of fractions ``f`` and ``g``. """
        if isinstance(g, DMP):
            lev, dom, per, (F_num, F_den), G = f.poly_unify(g)
            num, den = F_num, dmp_mul(F_den, G, lev, dom)
        else:
            lev, dom, per, F, G = f.frac_unify(g)
            (F_num, F_den), (G_num, G_den) = F, G

            num = dmp_mul(F_num, G_den, lev, dom)
            den = dmp_mul(F_den, G_num, lev, dom)

        res = per(num, den)
        if f.ring is not None and res not in f.ring:
            from sympy.polys.polyerrors import ExactQuotientFailed
            raise ExactQuotientFailed(f, g, f.ring)
        return res

def dmp_fateman_poly_F_2(n, K):
    """Fateman's GCD benchmark: linearly dense quartic inputs """
    u = [K(1), K(0)]

    for i in xrange(0, n - 1):
        u = [dmp_one(i, K), u]

    m = n - 1

    v = dmp_add_term(u, dmp_ground(K(2), m - 1), 0, n, K)

    f = dmp_sqr([dmp_one(m, K), dmp_neg(v, m, K)], n, K)
    g = dmp_sqr([dmp_one(m, K), v], n, K)

    v = dmp_add_term(u, dmp_one(m - 1, K), 0, n, K)

    h = dmp_sqr([dmp_one(m, K), v], n, K)

    return dmp_mul(f, h, n, K), dmp_mul(g, h, n, K), h

def dmp_zz_wang_lead_coeffs(f, T, cs, E, H, A, u, K):
    """Wang/EEZ: Compute correct leading coefficients. """
    C, J, v = [], [0]*len(E), u-1

    for h in H:
        c = dmp_one(v, K)
        d = dup_LC(h, K)*cs

        for i in reversed(xrange(len(E))):
            k, e, (t, _) = 0, E[i], T[i]

            while not (d % e):
                d, k = d//e, k+1

            if k != 0:
                c, J[i] = dmp_mul(c, dmp_pow(t, k, v, K), v, K), 1

        C.append(c)

    if any([ not j for j in J ]):
        raise ExtraneousFactors # pragma: no cover

    CC, HH = [], []

    for c, h in zip(C, H):
        d = dmp_eval_tail(c, A, v, K)
        lc = dup_LC(h, K)

        if K.is_one(cs):
            cc = lc//d
        else:
            g = K.gcd(lc, d)
            d, cc = d//g, lc//g
            h, cs = dup_mul_ground(h, d, K), cs//d

        c = dmp_mul_ground(c, cc, v, K)

        CC.append(c)
        HH.append(h)

    if K.is_one(cs):
        return f, HH, CC

    CCC, HHH = [], []

    for c, h in zip(CC, HH):
        CCC.append(dmp_mul_ground(c, cs, v, K))
        HHH.append(dmp_mul_ground(h, cs, 0, K))

    f = dmp_mul_ground(f, cs**(len(H)-1), u, K)

    return f, HHH, CCC

def dmp_zz_wang_hensel_lifting(f, H, LC, A, p, u, K):
    """Wang/EEZ: Parallel Hensel lifting algorithm. """
    S, n, v = [f], len(A), u-1

    H = list(H)

    for i, a in enumerate(reversed(A[1:])):
        s = dmp_eval_in(S[0], a, n-i, u-i, K)
        S.insert(0, dmp_ground_trunc(s, p, v-i, K))

    d = max(dmp_degree_list(f, u)[1:])

    for j, s, a in zip(xrange(2, n+2), S, A):
        G, w = list(H), j-1

        I, J = A[:j-2], A[j-1:]

        for i, (h, lc) in enumerate(zip(H, LC)):
            lc = dmp_ground_trunc(dmp_eval_tail(lc, J, v, K), p, w-1, K)
            H[i] = [lc] + dmp_raise(h[1:], 1, w-1, K)

        m = dmp_nest([K.one, -a], w, K)
        M = dmp_one(w, K)

        c = dmp_sub(s, dmp_expand(H, w, K), w, K)

        dj = dmp_degree_in(s, w, w)

        for k in xrange(0, dj):
            if dmp_zero_p(c, w):
                break

            M = dmp_mul(M, m, w, K)
            C = dmp_diff_eval_in(c, k+1, a, w, w, K)

            if not dmp_zero_p(C, w-1):
                C = dmp_quo_ground(C, K.factorial(k+1), w-1, K)
                T = dmp_zz_diophantine(G, C, I, d, p, w-1, K)

                for i, (h, t) in enumerate(zip(H, T)):
                    h = dmp_add_mul(h, dmp_raise(t, 1, w-1, K), M, w, K)
                    H[i] = dmp_ground_trunc(h, p, w, K)

                h = dmp_sub(s, dmp_expand(H, w, K), w, K)
                c = dmp_ground_trunc(h, p, w, K)

    if dmp_expand(H, u, K) != f:
        raise ExtraneousFactors # pragma: no cover
    else:
        return H

def dup_real_imag(f, K):
    """
    Return bivariate polynomials ``f1`` and ``f2``, such that ``f = f1 + f2*I``.

    Examples
    ========

    >>> from sympy.polys.domains import ZZ
    >>> from sympy.polys.densetools import dup_real_imag

    >>> dup_real_imag([ZZ(1), ZZ(1), ZZ(1), ZZ(1)], ZZ)
    ([[1], [1], [-3, 0, 1], [-1, 0, 1]], [[3, 0], [2, 0], [-1, 0, 1, 0]])

    """
    if not K.is_ZZ and not K.is_QQ:
        raise DomainError(
            "computing real and imaginary parts is not supported over %s" % K)

    f1 = dmp_zero(1)
    f2 = dmp_zero(1)

    if not f:
        return f1, f2

    g = [[[K.one, K.zero]], [[K.one], []]]
    h = dmp_ground(f[0], 2)

    for c in f[1:]:
        h = dmp_mul(h, g, 2, K)
        h = dmp_add_term(h, dmp_ground(c, 1), 0, 2, K)

    H = dup_to_raw_dict(h)

    for k, h in H.iteritems():
        m = k % 4

        if not m:
            f1 = dmp_add(f1, h, 1, K)
        elif m == 1:
            f2 = dmp_add(f2, h, 1, K)
        elif m == 2:
            f1 = dmp_sub(f1, h, 1, K)
        else:
            f2 = dmp_sub(f2, h, 1, K)

    return f1, f2

def dup_real_imag(f, K):
    """
    Return bivariate polynomials ``f1`` and ``f2``, such that ``f = f1 + f2*I``.

    Examples
    ========

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

    >>> R.dup_real_imag(x**3 + x**2 + x + 1)
    (x**3 + x**2 - 3*x*y**2 + x - y**2 + 1, 3*x**2*y + 2*x*y - y**3 + y)

    """
    if not K.is_ZZ and not K.is_QQ:
        raise DomainError("computing real and imaginary parts is not supported over %s" % K)

    f1 = dmp_zero(1)
    f2 = dmp_zero(1)

    if not f:
        return f1, f2

    g = [[[K.one, K.zero]], [[K.one], []]]
    h = dmp_ground(f[0], 2)

    for c in f[1:]:
        h = dmp_mul(h, g, 2, K)
        h = dmp_add_term(h, dmp_ground(c, 1), 0, 2, K)

    H = dup_to_raw_dict(h)

    for k, h in H.items():
        m = k % 4

        if not m:
            f1 = dmp_add(f1, h, 1, K)
        elif m == 1:
            f2 = dmp_add(f2, h, 1, K)
        elif m == 2:
            f1 = dmp_sub(f1, h, 1, K)
        else:
            f2 = dmp_sub(f2, h, 1, K)

    return f1, f2