コード例 #1
0
ファイル: specialpolys.py プロジェクト: Acebulf/sympy
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
コード例 #2
0
ファイル: specialpolys.py プロジェクト: hridog00/Proyecto
def dmp_fateman_poly_F_1(n, K):
    """Fateman's GCD benchmark: trivial GCD """
    u = [K(1), K(0)]

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

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

    for i in range(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
コード例 #3
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ファイル: polyclasses.py プロジェクト: fxkr/sympy
    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)
コード例 #4
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    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)
コード例 #5
0
    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)
コード例 #6
0
ファイル: polyclasses.py プロジェクト: fxkr/sympy
    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)
コード例 #7
0
ファイル: polyclasses.py プロジェクト: Ryzh/sympy
    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)
コード例 #8
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    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)
コード例 #9
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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)
コード例 #10
0
ファイル: densetools.py プロジェクト: asmeurer/sympy
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
コード例 #11
0
ファイル: densetools.py プロジェクト: msgoff/sympy
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
コード例 #12
0
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_quo(dmp_mul(f, g, u, K), dmp_gcd(f, g, u, K), u, K)

    return dmp_mul_ground(h, c, u, K)
コード例 #13
0
ファイル: euclidtools.py プロジェクト: addisonc/sympy
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)
コード例 #14
0
ファイル: euclidtools.py プロジェクト: AdrianPotter/sympy
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)
コード例 #15
0
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
コード例 #16
0
ファイル: densetools.py プロジェクト: jenshnielsen/sympy
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
コード例 #17
0
ファイル: specialpolys.py プロジェクト: yuri-karadzhov/sympy
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
コード例 #18
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    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
コード例 #19
0
ファイル: polyclasses.py プロジェクト: Ryzh/sympy
    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
コード例 #20
0
ファイル: specialpolys.py プロジェクト: addisonc/sympy
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
コード例 #21
0
ファイル: specialpolys.py プロジェクト: Acebulf/sympy
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
コード例 #22
0
ファイル: specialpolys.py プロジェクト: hridog00/Proyecto
def dmp_fateman_poly_F_2(n, K):
    """Fateman's GCD benchmark: linearly dense quartic inputs """
    u = [K(1), K(0)]

    for i in range(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
コード例 #23
0
ファイル: factortools.py プロジェクト: tuhina/sympy
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
コード例 #24
0
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
コード例 #25
0
ファイル: factortools.py プロジェクト: tuhina/sympy
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
コード例 #26
0
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
コード例 #27
0
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
コード例 #28
0
ファイル: densetools.py プロジェクト: msgoff/sympy
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
コード例 #29
0
ファイル: densetools.py プロジェクト: jenshnielsen/sympy
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
コード例 #30
0
ファイル: densetools.py プロジェクト: asmeurer/sympy
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
コード例 #31
0
def dmp_ff_lcm(f, g, u, K):
    """
    Computes polynomial LCM over a field in ``K[X]``.

    **Examples**

    >>> from sympy.polys.domains import QQ
    >>> from sympy.polys.euclidtools import dmp_ff_lcm

    >>> f = [[QQ(1,4)], [QQ(1), QQ(0)], [QQ(1), QQ(0), QQ(0)]]
    >>> g = [[QQ(1,2)], [QQ(1), QQ(0)], []]

    >>> dmp_ff_lcm(f, g, 1, QQ)
    [[1/1], [4/1, 0/1], [4/1, 0/1, 0/1], []]

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

    return dmp_ground_monic(h, u, K)
コード例 #32
0
ファイル: euclidtools.py プロジェクト: addisonc/sympy
def dmp_ff_lcm(f, g, u, K):
    """
    Computes polynomial LCM over a field in ``K[X]``.

    **Examples**

    >>> from sympy.polys.domains import QQ
    >>> from sympy.polys.euclidtools import dmp_ff_lcm

    >>> f = [[QQ(1,4)], [QQ(1), QQ(0)], [QQ(1), QQ(0), QQ(0)]]
    >>> g = [[QQ(1,2)], [QQ(1), QQ(0)], []]

    >>> dmp_ff_lcm(f, g, 1, QQ)
    [[1/1], [4/1, 0/1], [4/1, 0/1, 0/1], []]

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

    return dmp_ground_monic(h, u, K)
コード例 #33
0
ファイル: euclidtools.py プロジェクト: mattpap/sympy
def dmp_ff_lcm(f, g, u, K):
    """
    Computes polynomial LCM over a field in `K[X]`.

    Examples
    ========

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

    >>> f = QQ(1,4)*x**2 + x*y + y**2
    >>> g = QQ(1,2)*x**2 + x*y

    >>> R.dmp_ff_lcm(f, g)
    x**3 + 4*x**2*y + 4*x*y**2

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

    return dmp_ground_monic(h, u, K)
コード例 #34
0
def dmp_ff_lcm(f, g, u, K):
    """
    Computes polynomial LCM over a field in `K[X]`.

    Examples
    ========

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

    >>> f = QQ(1,4)*x**2 + x*y + y**2
    >>> g = QQ(1,2)*x**2 + x*y

    >>> R.dmp_ff_lcm(f, g)
    x**3 + 4*x**2*y + 4*x*y**2

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

    return dmp_ground_monic(h, u, K)
コード例 #35
0
def dmp_zz_modular_resultant(f, g, p, u, K):
    """
    Compute resultant of `f` and `g` modulo a prime `p`.

    Examples
    ========

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

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

    >>> dmp_zz_modular_resultant(f, g, ZZ(5), 1, ZZ)
    [-2, 0, 1]

    """
    if not u:
        return gf_int(dup_prs_resultant(f, g, K)[0] % p, p)

    v = u - 1

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

    N = dmp_degree_in(f, 1, u)
    M = dmp_degree_in(g, 1, u)

    B = n * M + m * N

    D, a = [K.one], -K.one
    r = dmp_zero(v)

    while dup_degree(D) <= B:
        while True:
            a += K.one

            if a == p:
                raise HomomorphismFailed('no luck')

            F = dmp_eval_in(f, gf_int(a, p), 1, u, K)

            if dmp_degree(F, v) == n:
                G = dmp_eval_in(g, gf_int(a, p), 1, u, K)

                if dmp_degree(G, v) == m:
                    break

        R = dmp_zz_modular_resultant(F, G, p, v, K)
        e = dmp_eval(r, a, v, K)

        if not v:
            R = dup_strip([R])
            e = dup_strip([e])
        else:
            R = [R]
            e = [e]

        d = K.invert(dup_eval(D, a, K), p)
        d = dup_mul_ground(D, d, K)
        d = dmp_raise(d, v, 0, K)

        c = dmp_mul(d, dmp_sub(R, e, v, K), v, K)
        r = dmp_add(r, c, v, K)

        r = dmp_ground_trunc(r, p, v, K)

        D = dup_mul(D, [K.one, -a], K)
        D = dup_trunc(D, p, K)

    return r
コード例 #36
0
def test_dup_sqf():
    assert dup_sqf_part([], ZZ) == []
    assert dup_sqf_p([], ZZ) == True

    assert dup_sqf_part([7], ZZ) == [1]
    assert dup_sqf_p([7], ZZ) == True

    assert dup_sqf_part([2,2], ZZ) == [1,1]
    assert dup_sqf_p([2,2], ZZ) == True

    assert dup_sqf_part([1,0,1,1], ZZ) == [1,0,1,1]
    assert dup_sqf_p([1,0,1,1], ZZ) == True

    assert dup_sqf_part([-1,0,1,1], ZZ) == [1,0,-1,-1]
    assert dup_sqf_p([-1,0,1,1], ZZ) == True

    assert dup_sqf_part([2,3,0,0], ZZ) == [2,3,0]
    assert dup_sqf_p([2,3,0,0], ZZ) == False

    assert dup_sqf_part([-2,3,0,0], ZZ) == [2,-3,0]
    assert dup_sqf_p([-2,3,0,0], ZZ) == False

    assert dup_sqf_list([], ZZ) == (0, [])
    assert dup_sqf_list([1], ZZ) == (1, [])

    assert dup_sqf_list([1,0], ZZ) == (1, [([1,0], 1)])
    assert dup_sqf_list([2,0,0], ZZ) == (2, [([1,0], 2)])
    assert dup_sqf_list([3,0,0,0], ZZ) == (3, [([1,0], 3)])

    assert dup_sqf_list([ZZ(2),ZZ(4),ZZ(2)], ZZ) == \
        (ZZ(2), [([ZZ(1),ZZ(1)], 2)])
    assert dup_sqf_list([QQ(2),QQ(4),QQ(2)], QQ) == \
        (QQ(2), [([QQ(1),QQ(1)], 2)])

    assert dup_sqf_list([-1,1,0,0,1,-1], ZZ) == \
        (-1, [([1,1,1,1], 1), ([1,-1], 2)])
    assert dup_sqf_list([1,0,6,0,12,0,8,0,0], ZZ) == \
        (1, [([1,0], 2), ([1,0,2], 3)])

    K = FF(2)
    f = map(K, [1,0,1])

    assert dup_sqf_list(f, K) == \
        (K(1), [([K(1),K(1)], 2)])

    K = FF(3)
    f = map(K, [1,0,0,2,0,0,2,0,0,1,0])

    assert dup_sqf_list(f, K) == \
        (K(1), [([K(1), K(0)], 1),
                ([K(1), K(1)], 3),
                ([K(1), K(2)], 6)])

    f = [1,0,0,1]
    g = map(K, f)

    assert dup_sqf_part(f, ZZ) == f
    assert dup_sqf_part(g, K) == [K(1), K(1)]

    assert dup_sqf_p(f, ZZ) == True
    assert dup_sqf_p(g, K) == False

    A = [[1],[],[-3],[],[6]]
    D = [[1],[],[-5],[],[5],[],[4]]

    f, g = D, dmp_sub(A, dmp_mul(dmp_diff(D, 1, 1, ZZ), [[1,0]], 1, ZZ), 1, ZZ)

    res = dmp_resultant(f, g, 1, ZZ)

    assert dup_sqf_list(res, ZZ) == (45796, [([4,0,1], 3)])

    assert dup_sqf_list_include([DMP([1, 0, 0, 0], ZZ), DMP([], ZZ), DMP([], ZZ)], ZZ[x]) == \
        [([DMP([1, 0, 0, 0], ZZ)], 1), ([DMP([1], ZZ), DMP([], ZZ)], 2)]
コード例 #37
0
ファイル: test_densearith.py プロジェクト: vperic/sympy
def test_dmp_mul():
    assert dmp_mul([ZZ(5)], [ZZ(7)], 0, ZZ) == \
           dup_mul([ZZ(5)], [ZZ(7)], ZZ)
    assert dmp_mul([QQ(5,7)], [QQ(3,7)], 0, QQ) == \
           dup_mul([QQ(5,7)], [QQ(3,7)], QQ)

    assert dmp_mul([[[]]], [[[]]], 2, ZZ) == [[[]]]
    assert dmp_mul([[[ZZ(1)]]], [[[]]], 2, ZZ) == [[[]]]
    assert dmp_mul([[[]]], [[[ZZ(1)]]], 2, ZZ) == [[[]]]
    assert dmp_mul([[[ZZ(2)]]], [[[ZZ(1)]]], 2, ZZ) == [[[ZZ(2)]]]
    assert dmp_mul([[[ZZ(1)]]], [[[ZZ(2)]]], 2, ZZ) == [[[ZZ(2)]]]

    assert dmp_mul([[[]]], [[[]]], 2, QQ) == [[[]]]
    assert dmp_mul([[[QQ(1,2)]]], [[[]]], 2, QQ) == [[[]]]
    assert dmp_mul([[[]]], [[[QQ(1,2)]]], 2, QQ) == [[[]]]
    assert dmp_mul([[[QQ(2,7)]]], [[[QQ(1,3)]]], 2, QQ) == [[[QQ(2,21)]]]
    assert dmp_mul([[[QQ(1,7)]]], [[[QQ(2,3)]]], 2, QQ) == [[[QQ(2,21)]]]

    K = FF(6)

    assert dmp_mul([[K(2)],[K(1)]], [[K(3)],[K(4)]], 1, K) == [[K(5)],[K(4)]]
コード例 #38
0
def dmp_inner_subresultants(f, g, u, K):
    """
    Subresultant PRS algorithm in `K[X]`.

    Examples
    ========

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

    >>> f = ZZ.map([[3, 0], [], [-1, 0, 0, -4]])
    >>> g = ZZ.map([[1], [1, 0, 0, 0], [-9]])

    >>> a = [[3, 0, 0, 0, 0], [1, 0, -27, 4]]
    >>> b = [[-3, 0, 0, -12, 1, 0, -54, 8, 729, -216, 16]]

    >>> R = ZZ.map([f, g, a, b])
    >>> B = ZZ.map([[-1], [1], [9, 0, 0, 0, 0, 0, 0, 0, 0]])
    >>> D = ZZ.map([0, 1, 1])

    >>> dmp_inner_subresultants(f, g, 1, ZZ) == (R, B, D)
    True

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

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

    if n < m:
        f, g = g, f
        n, m = m, n

    R = [f, g]
    d = n - m
    v = u - 1

    b = dmp_pow(dmp_ground(-K.one, v), d + 1, v, K)
    c = dmp_ground(-K.one, v)

    B, D = [b], [d]

    if dmp_zero_p(f, u) or dmp_zero_p(g, u):
        return R, B, D

    h = dmp_prem(f, g, u, K)
    h = dmp_mul_term(h, b, 0, u, K)

    while not dmp_zero_p(h, u):
        k = dmp_degree(h, u)
        R.append(h)

        lc = dmp_LC(g, K)

        p = dmp_pow(dmp_neg(lc, v, K), d, v, K)

        if not d:
            q = c
        else:
            q = dmp_pow(c, d - 1, v, K)

        c = dmp_quo(p, q, v, K)
        b = dmp_mul(dmp_neg(lc, v, K), dmp_pow(c, m - k, v, K), v, K)

        f, g, m, d = g, h, k, m - k

        B.append(b)
        D.append(d)

        h = dmp_prem(f, g, u, K)
        h = [dmp_quo(ch, b, v, K) for ch in h]

    return R, B, D
コード例 #39
0
def dmp_prs_resultant(f, g, u, K):
    """
    Resultant algorithm in `K[X]` using subresultant PRS.

    Examples
    ========

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

    >>> f = ZZ.map([[3, 0], [], [-1, 0, 0, -4]])
    >>> g = ZZ.map([[1], [1, 0, 0, 0], [-9]])

    >>> a = ZZ.map([[3, 0, 0, 0, 0], [1, 0, -27, 4]])
    >>> b = ZZ.map([[-3, 0, 0, -12, 1, 0, -54, 8, 729, -216, 16]])

    >>> dmp_prs_resultant(f, g, 1, ZZ) == (b[0], [f, g, a, b])
    True

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

    if dmp_zero_p(f, u) or dmp_zero_p(g, u):
        return (dmp_zero(u - 1), [])

    R, B, D = dmp_inner_subresultants(f, g, u, K)

    if dmp_degree(R[-1], u) > 0:
        return (dmp_zero(u - 1), R)
    if dmp_one_p(R[-2], u, K):
        return (dmp_LC(R[-1], K), R)

    s, i, v = 1, 1, u - 1

    p = dmp_one(v, K)
    q = dmp_one(v, K)

    for b, d in list(zip(B, D))[:-1]:
        du = dmp_degree(R[i - 1], u)
        dv = dmp_degree(R[i], u)
        dw = dmp_degree(R[i + 1], u)

        if du % 2 and dv % 2:
            s = -s

        lc, i = dmp_LC(R[i], K), i + 1

        p = dmp_mul(dmp_mul(p, dmp_pow(b, dv, v, K), v, K),
                    dmp_pow(lc, du - dw, v, K), v, K)
        q = dmp_mul(q, dmp_pow(lc, dv * (1 + d), v, K), v, K)

        _, p, q = dmp_inner_gcd(p, q, v, K)

    if s < 0:
        p = dmp_neg(p, v, K)

    i = dmp_degree(R[-2], u)

    res = dmp_pow(dmp_LC(R[-1], K), i, v, K)
    res = dmp_quo(dmp_mul(res, p, v, K), q, v, K)

    return res, R
コード例 #40
0
ファイル: test_euclidtools.py プロジェクト: vperic/sympy
def test_dmp_gcd():
    assert dmp_zz_heu_gcd([[]], [[]], 1, ZZ) == ([[]], [[]], [[]])
    assert dmp_rr_prs_gcd([[]], [[]], 1, ZZ) == ([[]], [[]], [[]])

    assert dmp_zz_heu_gcd([[2]], [[]], 1, ZZ) == ([[2]], [[1]], [[]])
    assert dmp_rr_prs_gcd([[2]], [[]], 1, ZZ) == ([[2]], [[1]], [[]])

    assert dmp_zz_heu_gcd([[-2]], [[]], 1, ZZ) == ([[2]], [[-1]], [[]])
    assert dmp_rr_prs_gcd([[-2]], [[]], 1, ZZ) == ([[2]], [[-1]], [[]])

    assert dmp_zz_heu_gcd([[]], [[-2]], 1, ZZ) == ([[2]], [[]], [[-1]])
    assert dmp_rr_prs_gcd([[]], [[-2]], 1, ZZ) == ([[2]], [[]], [[-1]])

    assert dmp_zz_heu_gcd([[]], [[2], [4]], 1, ZZ) == ([[2], [4]], [[]], [[1]])
    assert dmp_rr_prs_gcd([[]], [[2], [4]], 1, ZZ) == ([[2], [4]], [[]], [[1]])

    assert dmp_zz_heu_gcd([[2], [4]], [[]], 1, ZZ) == ([[2], [4]], [[1]], [[]])
    assert dmp_rr_prs_gcd([[2], [4]], [[]], 1, ZZ) == ([[2], [4]], [[1]], [[]])

    assert dmp_zz_heu_gcd([[2]], [[2]], 1, ZZ) == ([[2]], [[1]], [[1]])
    assert dmp_rr_prs_gcd([[2]], [[2]], 1, ZZ) == ([[2]], [[1]], [[1]])

    assert dmp_zz_heu_gcd([[-2]], [[2]], 1, ZZ) == ([[2]], [[-1]], [[1]])
    assert dmp_rr_prs_gcd([[-2]], [[2]], 1, ZZ) == ([[2]], [[-1]], [[1]])

    assert dmp_zz_heu_gcd([[2]], [[-2]], 1, ZZ) == ([[2]], [[1]], [[-1]])
    assert dmp_rr_prs_gcd([[2]], [[-2]], 1, ZZ) == ([[2]], [[1]], [[-1]])

    assert dmp_zz_heu_gcd([[-2]], [[-2]], 1, ZZ) == ([[2]], [[-1]], [[-1]])
    assert dmp_rr_prs_gcd([[-2]], [[-2]], 1, ZZ) == ([[2]], [[-1]], [[-1]])

    assert dmp_zz_heu_gcd([[1], [2], [1]], [[1]], 1,
                          ZZ) == ([[1]], [[1], [2], [1]], [[1]])
    assert dmp_rr_prs_gcd([[1], [2], [1]], [[1]], 1,
                          ZZ) == ([[1]], [[1], [2], [1]], [[1]])

    assert dmp_zz_heu_gcd([[1], [2], [1]], [[2]], 1,
                          ZZ) == ([[1]], [[1], [2], [1]], [[2]])
    assert dmp_rr_prs_gcd([[1], [2], [1]], [[2]], 1,
                          ZZ) == ([[1]], [[1], [2], [1]], [[2]])

    assert dmp_zz_heu_gcd([[2], [4], [2]], [[2]], 1,
                          ZZ) == ([[2]], [[1], [2], [1]], [[1]])
    assert dmp_rr_prs_gcd([[2], [4], [2]], [[2]], 1,
                          ZZ) == ([[2]], [[1], [2], [1]], [[1]])

    assert dmp_zz_heu_gcd([[2]], [[2], [4], [2]], 1,
                          ZZ) == ([[2]], [[1]], [[1], [2], [1]])
    assert dmp_rr_prs_gcd([[2]], [[2], [4], [2]], 1,
                          ZZ) == ([[2]], [[1]], [[1], [2], [1]])

    assert dmp_zz_heu_gcd([[2], [4], [2]], [[1], [1]], 1,
                          ZZ) == ([[1], [1]], [[2], [2]], [[1]])
    assert dmp_rr_prs_gcd([[2], [4], [2]], [[1], [1]], 1,
                          ZZ) == ([[1], [1]], [[2], [2]], [[1]])

    assert dmp_zz_heu_gcd([[1], [1]], [[2], [4], [2]], 1,
                          ZZ) == ([[1], [1]], [[1]], [[2], [2]])
    assert dmp_rr_prs_gcd([[1], [1]], [[2], [4], [2]], 1,
                          ZZ) == ([[1], [1]], [[1]], [[2], [2]])

    assert dmp_zz_heu_gcd([[[[1, 2, 1]]]], [[[[2, 2]]]], 3,
                          ZZ) == ([[[[1, 1]]]], [[[[1, 1]]]], [[[[2]]]])
    assert dmp_rr_prs_gcd([[[[1, 2, 1]]]], [[[[2, 2]]]], 3,
                          ZZ) == ([[[[1, 1]]]], [[[[1, 1]]]], [[[[2]]]])

    f, g = [[[[1, 2, 1], [1, 1], []]]], [[[[1, 2, 1]]]]
    h, cff, cfg = [[[[1, 1]]]], [[[[1, 1], [1], []]]], [[[[1, 1]]]]

    assert dmp_zz_heu_gcd(f, g, 3, ZZ) == (h, cff, cfg)
    assert dmp_rr_prs_gcd(f, g, 3, ZZ) == (h, cff, cfg)

    assert dmp_zz_heu_gcd(g, f, 3, ZZ) == (h, cfg, cff)
    assert dmp_rr_prs_gcd(g, f, 3, ZZ) == (h, cfg, cff)

    f, g, h = dmp_fateman_poly_F_1(2, ZZ)
    H, cff, cfg = dmp_zz_heu_gcd(f, g, 2, ZZ)

    assert H == h and dmp_mul(H, cff, 2, ZZ) == f \
                  and dmp_mul(H, cfg, 2, ZZ) == g

    H, cff, cfg = dmp_rr_prs_gcd(f, g, 2, ZZ)

    assert H == h and dmp_mul(H, cff, 2, ZZ) == f \
                  and dmp_mul(H, cfg, 2, ZZ) == g

    f, g, h = dmp_fateman_poly_F_1(4, ZZ)
    H, cff, cfg = dmp_zz_heu_gcd(f, g, 4, ZZ)

    assert H == h and dmp_mul(H, cff, 4, ZZ) == f \
                  and dmp_mul(H, cfg, 4, ZZ) == g

    f, g, h = dmp_fateman_poly_F_1(6, ZZ)
    H, cff, cfg = dmp_zz_heu_gcd(f, g, 6, ZZ)

    assert H == h and dmp_mul(H, cff, 6, ZZ) == f \
                  and dmp_mul(H, cfg, 6, ZZ) == g

    f, g, h = dmp_fateman_poly_F_1(8, ZZ)

    H, cff, cfg = dmp_zz_heu_gcd(f, g, 8, ZZ)

    assert H == h and dmp_mul(H, cff, 8, ZZ) == f \
                  and dmp_mul(H, cfg, 8, ZZ) == g

    f, g, h = dmp_fateman_poly_F_2(2, ZZ)
    H, cff, cfg = dmp_zz_heu_gcd(f, g, 2, ZZ)

    assert H == h and dmp_mul(H, cff, 2, ZZ) == f \
                  and dmp_mul(H, cfg, 2, ZZ) == g

    H, cff, cfg = dmp_rr_prs_gcd(f, g, 2, ZZ)

    assert H == h and dmp_mul(H, cff, 2, ZZ) == f \
                  and dmp_mul(H, cfg, 2, ZZ) == g

    f, g, h = dmp_fateman_poly_F_3(2, ZZ)
    H, cff, cfg = dmp_zz_heu_gcd(f, g, 2, ZZ)

    assert H == h and dmp_mul(H, cff, 2, ZZ) == f \
                  and dmp_mul(H, cfg, 2, ZZ) == g

    H, cff, cfg = dmp_rr_prs_gcd(f, g, 2, ZZ)

    assert H == h and dmp_mul(H, cff, 2, ZZ) == f \
                  and dmp_mul(H, cfg, 2, ZZ) == g

    f, g, h = dmp_fateman_poly_F_3(4, ZZ)
    H, cff, cfg = dmp_inner_gcd(f, g, 4, ZZ)

    assert H == h and dmp_mul(H, cff, 4, ZZ) == f \
                  and dmp_mul(H, cfg, 4, ZZ) == g

    f = [[QQ(1, 2)], [QQ(1)], [QQ(1, 2)]]
    g = [[QQ(1, 2)], [QQ(1, 2)]]

    h = [[QQ(1)], [QQ(1)]]

    assert dmp_qq_heu_gcd(f, g, 1, QQ) == (h, g, [[QQ(1, 2)]])
    assert dmp_ff_prs_gcd(f, g, 1, QQ) == (h, g, [[QQ(1, 2)]])

    f = [[RR(2.1), RR(-2.2), RR(2.1)], []]
    g = [[RR(1.0)], [], [], []]

    assert dmp_ff_prs_gcd(f, g, 1, RR) == \
        ([[RR(1.0)], []], [[RR(2.1), RR(-2.2), RR(2.1)]], [[RR(1.0)], [], []])
コード例 #41
0
ファイル: euclidtools.py プロジェクト: z-campbell/sympy
def dmp_inner_subresultants(f, g, u, K):
    """
    Subresultant PRS algorithm in `K[X]`.

    Examples
    ========

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

    >>> f = 3*x**2*y - y**3 - 4
    >>> g = x**2 + x*y**3 - 9

    >>> a = 3*x*y**4 + y**3 - 27*y + 4
    >>> b = -3*y**10 - 12*y**7 + y**6 - 54*y**4 + 8*y**3 + 729*y**2 - 216*y + 16

    >>> prs = [f, g, a, b]
    >>> sres = [[1], [1], [3, 0, 0, 0, 0], [-3, 0, 0, -12, 1, 0, -54, 8, 729, -216, 16]]

    >>> R.dmp_inner_subresultants(f, g) == (prs, sres)
    True

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

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

    if n < m:
        f, g = g, f
        n, m = m, n

    if dmp_zero_p(f, u):
        return [], []

    v = u - 1
    if dmp_zero_p(g, u):
        return [f], [dmp_ground(K.one, v)]

    R = [f, g]
    d = n - m

    b = dmp_pow(dmp_ground(-K.one, v), d + 1, v, K)

    h = dmp_prem(f, g, u, K)
    h = dmp_mul_term(h, b, 0, u, K)

    lc = dmp_LC(g, K)
    c = dmp_pow(lc, d, v, K)

    S = [dmp_ground(K.one, v), c]
    c = dmp_neg(c, v, K)

    while not dmp_zero_p(h, u):
        k = dmp_degree(h, u)
        R.append(h)

        f, g, m, d = g, h, k, m - k

        b = dmp_mul(dmp_neg(lc, v, K),
                    dmp_pow(c, d, v, K), v, K)

        h = dmp_prem(f, g, u, K)
        h = [ dmp_quo(ch, b, v, K) for ch in h ]

        lc = dmp_LC(g, K)

        if d > 1:
            p = dmp_pow(dmp_neg(lc, v, K), d, v, K)
            q = dmp_pow(c, d - 1, v, K)
            c = dmp_quo(p, q, v, K)
        else:
            c = dmp_neg(lc, v, K)

        S.append(dmp_neg(c, v, K))

    return R, S
コード例 #42
0
ファイル: polyclasses.py プロジェクト: fxkr/sympy
 def mul(f, g):
     """Multiply two multivariate polynomials `f` and `g`. """
     lev, dom, per, F, G = f.unify(g)
     return per(dmp_mul(F, G, lev, dom))
コード例 #43
0
ファイル: euclidtools.py プロジェクト: AdrianPotter/sympy
def dmp_zz_modular_resultant(f, g, p, u, K):
    """
    Compute resultant of `f` and `g` modulo a prime `p`.

    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_modular_resultant(f, g, 5)
    -2*y**2 + 1

    """
    if not u:
        return gf_int(dup_prs_resultant(f, g, K)[0] % p, p)

    v = u - 1

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

    N = dmp_degree_in(f, 1, u)
    M = dmp_degree_in(g, 1, u)

    B = n*M + m*N

    D, a = [K.one], -K.one
    r = dmp_zero(v)

    while dup_degree(D) <= B:
        while True:
            a += K.one

            if a == p:
                raise HomomorphismFailed('no luck')

            F = dmp_eval_in(f, gf_int(a, p), 1, u, K)

            if dmp_degree(F, v) == n:
                G = dmp_eval_in(g, gf_int(a, p), 1, u, K)

                if dmp_degree(G, v) == m:
                    break

        R = dmp_zz_modular_resultant(F, G, p, v, K)
        e = dmp_eval(r, a, v, K)

        if not v:
            R = dup_strip([R])
            e = dup_strip([e])
        else:
            R = [R]
            e = [e]

        d = K.invert(dup_eval(D, a, K), p)
        d = dup_mul_ground(D, d, K)
        d = dmp_raise(d, v, 0, K)

        c = dmp_mul(d, dmp_sub(R, e, v, K), v, K)
        r = dmp_add(r, c, v, K)

        r = dmp_ground_trunc(r, p, v, K)

        D = dup_mul(D, [K.one, -a], K)
        D = dup_trunc(D, p, K)

    return r
コード例 #44
0
 def mul(f, g):
     """Multiply two multivariate polynomials `f` and `g`. """
     lev, dom, per, F, G = f.unify(g)
     return per(dmp_mul(F, G, lev, dom))
コード例 #45
0
ファイル: test_euclidtools.py プロジェクト: addisonc/sympy
def test_dmp_gcd():
    assert dmp_zz_heu_gcd([[]], [[]], 1, ZZ) == ([[]], [[]], [[]])
    assert dmp_rr_prs_gcd([[]], [[]], 1, ZZ) == ([[]], [[]], [[]])

    assert dmp_zz_heu_gcd([[2]], [[]], 1, ZZ) == ([[2]], [[1]], [[]])
    assert dmp_rr_prs_gcd([[2]], [[]], 1, ZZ) == ([[2]], [[1]], [[]])

    assert dmp_zz_heu_gcd([[-2]], [[]], 1, ZZ) == ([[2]], [[-1]], [[]])
    assert dmp_rr_prs_gcd([[-2]], [[]], 1, ZZ) == ([[2]], [[-1]], [[]])

    assert dmp_zz_heu_gcd([[]], [[-2]], 1, ZZ) == ([[2]], [[]], [[-1]])
    assert dmp_rr_prs_gcd([[]], [[-2]], 1, ZZ) == ([[2]], [[]], [[-1]])

    assert dmp_zz_heu_gcd([[]], [[2],[4]], 1, ZZ) == ([[2],[4]], [[]], [[1]])
    assert dmp_rr_prs_gcd([[]], [[2],[4]], 1, ZZ) == ([[2],[4]], [[]], [[1]])

    assert dmp_zz_heu_gcd([[2],[4]], [[]], 1, ZZ) == ([[2],[4]], [[1]], [[]])
    assert dmp_rr_prs_gcd([[2],[4]], [[]], 1, ZZ) == ([[2],[4]], [[1]], [[]])

    assert dmp_zz_heu_gcd([[2]], [[2]], 1, ZZ) == ([[2]], [[1]], [[1]])
    assert dmp_rr_prs_gcd([[2]], [[2]], 1, ZZ) == ([[2]], [[1]], [[1]])

    assert dmp_zz_heu_gcd([[-2]], [[2]], 1, ZZ) == ([[2]], [[-1]], [[1]])
    assert dmp_rr_prs_gcd([[-2]], [[2]], 1, ZZ) == ([[2]], [[-1]], [[1]])

    assert dmp_zz_heu_gcd([[2]], [[-2]], 1, ZZ) == ([[2]], [[1]], [[-1]])
    assert dmp_rr_prs_gcd([[2]], [[-2]], 1, ZZ) == ([[2]], [[1]], [[-1]])

    assert dmp_zz_heu_gcd([[-2]], [[-2]], 1, ZZ) == ([[2]], [[-1]], [[-1]])
    assert dmp_rr_prs_gcd([[-2]], [[-2]], 1, ZZ) == ([[2]], [[-1]], [[-1]])

    assert dmp_zz_heu_gcd([[1],[2],[1]], [[1]], 1, ZZ) == ([[1]], [[1], [2], [1]], [[1]])
    assert dmp_rr_prs_gcd([[1],[2],[1]], [[1]], 1, ZZ) == ([[1]], [[1], [2], [1]], [[1]])

    assert dmp_zz_heu_gcd([[1],[2],[1]], [[2]], 1, ZZ) == ([[1]], [[1], [2], [1]], [[2]])
    assert dmp_rr_prs_gcd([[1],[2],[1]], [[2]], 1, ZZ) == ([[1]], [[1], [2], [1]], [[2]])

    assert dmp_zz_heu_gcd([[2],[4],[2]], [[2]], 1, ZZ) == ([[2]], [[1], [2], [1]], [[1]])
    assert dmp_rr_prs_gcd([[2],[4],[2]], [[2]], 1, ZZ) == ([[2]], [[1], [2], [1]], [[1]])

    assert dmp_zz_heu_gcd([[2]], [[2],[4],[2]], 1, ZZ) == ([[2]], [[1]], [[1], [2], [1]])
    assert dmp_rr_prs_gcd([[2]], [[2],[4],[2]], 1, ZZ) == ([[2]], [[1]], [[1], [2], [1]])

    assert dmp_zz_heu_gcd([[2],[4],[2]], [[1],[1]], 1, ZZ) == ([[1], [1]], [[2], [2]], [[1]])
    assert dmp_rr_prs_gcd([[2],[4],[2]], [[1],[1]], 1, ZZ) == ([[1], [1]], [[2], [2]], [[1]])

    assert dmp_zz_heu_gcd([[1],[1]], [[2],[4],[2]], 1, ZZ) == ([[1], [1]], [[1]], [[2], [2]])
    assert dmp_rr_prs_gcd([[1],[1]], [[2],[4],[2]], 1, ZZ) == ([[1], [1]], [[1]], [[2], [2]])

    assert dmp_zz_heu_gcd([[[[1,2,1]]]], [[[[2,2]]]], 3, ZZ) == ([[[[1,1]]]], [[[[1,1]]]], [[[[2]]]])
    assert dmp_rr_prs_gcd([[[[1,2,1]]]], [[[[2,2]]]], 3, ZZ) == ([[[[1,1]]]], [[[[1,1]]]], [[[[2]]]])

    f, g = [[[[1,2,1],[1,1],[]]]], [[[[1,2,1]]]]
    h, cff, cfg = [[[[1,1]]]], [[[[1,1],[1],[]]]], [[[[1,1]]]]

    assert dmp_zz_heu_gcd(f, g, 3, ZZ) == (h, cff, cfg)
    assert dmp_rr_prs_gcd(f, g, 3, ZZ) == (h, cff, cfg)

    assert dmp_zz_heu_gcd(g, f, 3, ZZ) == (h, cfg, cff)
    assert dmp_rr_prs_gcd(g, f, 3, ZZ) == (h, cfg, cff)

    f, g, h = dmp_fateman_poly_F_1(2, ZZ)
    H, cff, cfg = dmp_zz_heu_gcd(f, g, 2, ZZ)

    assert H == h and dmp_mul(H, cff, 2, ZZ) == f \
                  and dmp_mul(H, cfg, 2, ZZ) == g

    H, cff, cfg = dmp_rr_prs_gcd(f, g, 2, ZZ)

    assert H == h and dmp_mul(H, cff, 2, ZZ) == f \
                  and dmp_mul(H, cfg, 2, ZZ) == g

    f, g, h = dmp_fateman_poly_F_1(4, ZZ)
    H, cff, cfg = dmp_zz_heu_gcd(f, g, 4, ZZ)

    assert H == h and dmp_mul(H, cff, 4, ZZ) == f \
                  and dmp_mul(H, cfg, 4, ZZ) == g

    f, g, h = dmp_fateman_poly_F_1(6, ZZ)
    H, cff, cfg = dmp_zz_heu_gcd(f, g, 6, ZZ)

    assert H == h and dmp_mul(H, cff, 6, ZZ) == f \
                  and dmp_mul(H, cfg, 6, ZZ) == g

    f, g, h = dmp_fateman_poly_F_1(8, ZZ)

    H, cff, cfg = dmp_zz_heu_gcd(f, g, 8, ZZ)

    assert H == h and dmp_mul(H, cff, 8, ZZ) == f \
                  and dmp_mul(H, cfg, 8, ZZ) == g

    f, g, h = dmp_fateman_poly_F_2(2, ZZ)
    H, cff, cfg = dmp_zz_heu_gcd(f, g, 2, ZZ)

    assert H == h and dmp_mul(H, cff, 2, ZZ) == f \
                  and dmp_mul(H, cfg, 2, ZZ) == g

    H, cff, cfg = dmp_rr_prs_gcd(f, g, 2, ZZ)

    assert H == h and dmp_mul(H, cff, 2, ZZ) == f \
                  and dmp_mul(H, cfg, 2, ZZ) == g

    f, g, h = dmp_fateman_poly_F_3(2, ZZ)
    H, cff, cfg = dmp_zz_heu_gcd(f, g, 2, ZZ)

    assert H == h and dmp_mul(H, cff, 2, ZZ) == f \
                  and dmp_mul(H, cfg, 2, ZZ) == g

    H, cff, cfg = dmp_rr_prs_gcd(f, g, 2, ZZ)

    assert H == h and dmp_mul(H, cff, 2, ZZ) == f \
                  and dmp_mul(H, cfg, 2, ZZ) == g

    f, g, h = dmp_fateman_poly_F_3(4, ZZ)
    H, cff, cfg = dmp_inner_gcd(f, g, 4, ZZ)

    assert H == h and dmp_mul(H, cff, 4, ZZ) == f \
                  and dmp_mul(H, cfg, 4, ZZ) == g

    f = [[QQ(1,2)],[QQ(1)],[QQ(1,2)]]
    g = [[QQ(1,2)],[QQ(1,2)]]

    h = [[QQ(1)],[QQ(1)]]

    assert dmp_qq_heu_gcd(f, g, 1, QQ) == (h, g, [[QQ(1,2)]])
    assert dmp_ff_prs_gcd(f, g, 1, QQ) == (h, g, [[QQ(1,2)]])
コード例 #46
0
def dmp_inner_subresultants(f, g, u, K):
    """
    Subresultant PRS algorithm in `K[X]`.

    Examples
    ========

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

    >>> f = 3*x**2*y - y**3 - 4
    >>> g = x**2 + x*y**3 - 9

    >>> a = 3*x*y**4 + y**3 - 27*y + 4
    >>> b = -3*y**10 - 12*y**7 + y**6 - 54*y**4 + 8*y**3 + 729*y**2 - 216*y + 16

    >>> prs = [f, g, a, b]
    >>> beta = [[-1], [1], [9, 0, 0, 0, 0, 0, 0, 0, 0]]
    >>> delta = [0, 1, 1]

    >>> R.dmp_inner_subresultants(f, g) == (prs, beta, delta)
    True

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

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

    if n < m:
        f, g = g, f
        n, m = m, n

    R = [f, g]
    d = n - m
    v = u - 1

    b = dmp_pow(dmp_ground(-K.one, v), d + 1, v, K)
    c = dmp_ground(-K.one, v)

    B, D = [b], [d]

    if dmp_zero_p(f, u) or dmp_zero_p(g, u):
        return R, B, D

    h = dmp_prem(f, g, u, K)
    h = dmp_mul_term(h, b, 0, u, K)

    while not dmp_zero_p(h, u):
        k = dmp_degree(h, u)
        R.append(h)

        lc = dmp_LC(g, K)

        p = dmp_pow(dmp_neg(lc, v, K), d, v, K)

        if not d:
            q = c
        else:
            q = dmp_pow(c, d - 1, v, K)

        c = dmp_quo(p, q, v, K)
        b = dmp_mul(dmp_neg(lc, v, K),
                    dmp_pow(c, m - k, v, K), v, K)

        f, g, m, d = g, h, k, m - k

        B.append(b)
        D.append(d)

        h = dmp_prem(f, g, u, K)

        h = [ dmp_quo(ch, b, v, K) for ch in h ]

    return R, B, D
コード例 #47
0
def dmp_zz_diophantine(F, c, A, d, p, u, K):
    """Wang/EEZ: Solve multivariate Diophantine equations. """
    if not A:
        S = [ [] for _ in F ]
        n = dup_degree(c)

        for i, coeff in enumerate(c):
            if not coeff:
                continue

            T = dup_zz_diophantine(F, n-i, p, K)

            for j, (s, t) in enumerate(zip(S, T)):
                t = dup_mul_ground(t, coeff, K)
                S[j] = dup_trunc(dup_add(s, t, K), p, K)
    else:
        n = len(A)
        e = dmp_expand(F, u, K)

        a, A = A[-1], A[:-1]
        B, G = [], []

        for f in F:
            B.append(dmp_quo(e, f, u, K))
            G.append(dmp_eval_in(f, a, n, u, K))

        C = dmp_eval_in(c, a, n, u, K)

        v = u - 1

        S = dmp_zz_diophantine(G, C, A, d, p, v, K)
        S = [ dmp_raise(s, 1, v, K) for s in S ]

        for s, b in zip(S, B):
            c = dmp_sub_mul(c, s, b, u, K)

        c = dmp_ground_trunc(c, p, u, K)

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

        for k in xrange(0, d):
            if dmp_zero_p(c, u):
                break

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

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

                for i, t in enumerate(T):
                    T[i] = dmp_mul(dmp_raise(t, 1, v, K), M, u, K)

                for i, (s, t) in enumerate(zip(S, T)):
                    S[i] = dmp_add(s, t, u, K)

                for t, b in zip(T, B):
                    c = dmp_sub_mul(c, t, b, u, K)

                c = dmp_ground_trunc(c, p, u, K)

        S = [ dmp_ground_trunc(s, p, u, K) for s in S ]

    return S
コード例 #48
0
ファイル: euclidtools.py プロジェクト: AdrianPotter/sympy
def dmp_inner_subresultants(f, g, u, K):
    """
    Subresultant PRS algorithm in `K[X]`.

    Examples
    ========

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

    >>> f = 3*x**2*y - y**3 - 4
    >>> g = x**2 + x*y**3 - 9

    >>> a = 3*x*y**4 + y**3 - 27*y + 4
    >>> b = -3*y**10 - 12*y**7 + y**6 - 54*y**4 + 8*y**3 + 729*y**2 - 216*y + 16

    >>> prs = [f, g, a, b]
    >>> sres = [[1], [1], [3, 0, 0, 0, 0], [-3, 0, 0, -12, 1, 0, -54, 8, 729, -216, 16]]

    >>> R.dmp_inner_subresultants(f, g) == (prs, sres)
    True

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

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

    if n < m:
        f, g = g, f
        n, m = m, n

    if dmp_zero_p(f, u):
        return [], []

    v = u - 1
    if dmp_zero_p(g, u):
        return [f], [dmp_ground(K.one, v)]

    R = [f, g]
    d = n - m

    b = dmp_pow(dmp_ground(-K.one, v), d + 1, v, K)

    h = dmp_prem(f, g, u, K)
    h = dmp_mul_term(h, b, 0, u, K)

    lc = dmp_LC(g, K)
    c = dmp_pow(lc, d, v, K)

    S = [dmp_ground(K.one, v), c]
    c = dmp_neg(c, v, K)

    while not dmp_zero_p(h, u):
        k = dmp_degree(h, u)
        R.append(h)

        f, g, m, d = g, h, k, m - k

        b = dmp_mul(dmp_neg(lc, v, K),
                    dmp_pow(c, d, v, K), v, K)

        h = dmp_prem(f, g, u, K)
        h = [ dmp_quo(ch, b, v, K) for ch in h ]

        lc = dmp_LC(g, K)

        if d > 1:
            p = dmp_pow(dmp_neg(lc, v, K), d, v, K)
            q = dmp_pow(c, d - 1, v, K)
            c = dmp_quo(p, q, v, K)
        else:
            c = dmp_neg(lc, v, K)

        S.append(dmp_neg(c, v, K))

    return R, S
コード例 #49
0
ファイル: factortools.py プロジェクト: tuhina/sympy
def dmp_zz_diophantine(F, c, A, d, p, u, K):
    """Wang/EEZ: Solve multivariate Diophantine equations. """
    if not A:
        S = [[] for _ in F]
        n = dup_degree(c)

        for i, coeff in enumerate(c):
            if not coeff:
                continue

            T = dup_zz_diophantine(F, n - i, p, K)

            for j, (s, t) in enumerate(zip(S, T)):
                t = dup_mul_ground(t, coeff, K)
                S[j] = dup_trunc(dup_add(s, t, K), p, K)
    else:
        n = len(A)
        e = dmp_expand(F, u, K)

        a, A = A[-1], A[:-1]
        B, G = [], []

        for f in F:
            B.append(dmp_quo(e, f, u, K))
            G.append(dmp_eval_in(f, a, n, u, K))

        C = dmp_eval_in(c, a, n, u, K)

        v = u - 1

        S = dmp_zz_diophantine(G, C, A, d, p, v, K)
        S = [dmp_raise(s, 1, v, K) for s in S]

        for s, b in zip(S, B):
            c = dmp_sub_mul(c, s, b, u, K)

        c = dmp_ground_trunc(c, p, u, K)

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

        for k in xrange(0, d):
            if dmp_zero_p(c, u):
                break

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

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

                for i, t in enumerate(T):
                    T[i] = dmp_mul(dmp_raise(t, 1, v, K), M, u, K)

                for i, (s, t) in enumerate(zip(S, T)):
                    S[i] = dmp_add(s, t, u, K)

                for t, b in zip(T, B):
                    c = dmp_sub_mul(c, t, b, u, K)

                c = dmp_ground_trunc(c, p, u, K)

        S = [dmp_ground_trunc(s, p, u, K) for s in S]

    return S
コード例 #50
0
def dmp_prs_resultant(f, g, u, K):
    """
    Resultant algorithm in `K[X]` using subresultant PRS.

    Examples
    ========

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

    >>> f = 3*x**2*y - y**3 - 4
    >>> g = x**2 + x*y**3 - 9

    >>> a = 3*x*y**4 + y**3 - 27*y + 4
    >>> b = -3*y**10 - 12*y**7 + y**6 - 54*y**4 + 8*y**3 + 729*y**2 - 216*y + 16

    >>> res, prs = R.dmp_prs_resultant(f, g)

    >>> res == b             # resultant has n-1 variables
    False
    >>> res == b.drop(x)
    True
    >>> prs == [f, g, a, b]
    True

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

    if dmp_zero_p(f, u) or dmp_zero_p(g, u):
        return (dmp_zero(u - 1), [])

    R, B, D = dmp_inner_subresultants(f, g, u, K)

    if dmp_degree(R[-1], u) > 0:
        return (dmp_zero(u - 1), R)
    if dmp_one_p(R[-2], u, K):
        return (dmp_LC(R[-1], K), R)

    s, i, v = 1, 1, u - 1

    p = dmp_one(v, K)
    q = dmp_one(v, K)

    for b, d in list(zip(B, D))[:-1]:
        du = dmp_degree(R[i - 1], u)
        dv = dmp_degree(R[i  ], u)
        dw = dmp_degree(R[i + 1], u)

        if du % 2 and dv % 2:
            s = -s

        lc, i = dmp_LC(R[i], K), i + 1

        p = dmp_mul(dmp_mul(p, dmp_pow(b, dv, v, K), v, K),
                    dmp_pow(lc, du - dw, v, K), v, K)
        q = dmp_mul(q, dmp_pow(lc, dv*(1 + d), v, K), v, K)

        _, p, q = dmp_inner_gcd(p, q, v, K)

    if s < 0:
        p = dmp_neg(p, v, K)

    i = dmp_degree(R[-2], u)

    res = dmp_pow(dmp_LC(R[-1], K), i, v, K)
    res = dmp_quo(dmp_mul(res, p, v, K), q, v, K)

    return res, R
コード例 #51
0
def dmp_prs_resultant(f, g, u, K):
    """
    Resultant algorithm in `K[X]` using subresultant PRS.

    Examples
    ========

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

    >>> f = 3*x**2*y - y**3 - 4
    >>> g = x**2 + x*y**3 - 9

    >>> a = 3*x*y**4 + y**3 - 27*y + 4
    >>> b = -3*y**10 - 12*y**7 + y**6 - 54*y**4 + 8*y**3 + 729*y**2 - 216*y + 16

    >>> res, prs = R.dmp_prs_resultant(f, g)

    >>> res == b             # resultant has n-1 variables
    False
    >>> res == b.drop(x)
    True
    >>> prs == [f, g, a, b]
    True

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

    if dmp_zero_p(f, u) or dmp_zero_p(g, u):
        return (dmp_zero(u - 1), [])

    R, B, D = dmp_inner_subresultants(f, g, u, K)

    if dmp_degree(R[-1], u) > 0:
        return (dmp_zero(u - 1), R)
    if dmp_one_p(R[-2], u, K):
        return (dmp_LC(R[-1], K), R)

    s, i, v = 1, 1, u - 1

    p = dmp_one(v, K)
    q = dmp_one(v, K)

    for b, d in list(zip(B, D))[:-1]:
        du = dmp_degree(R[i - 1], u)
        dv = dmp_degree(R[i  ], u)
        dw = dmp_degree(R[i + 1], u)

        if du % 2 and dv % 2:
            s = -s

        lc, i = dmp_LC(R[i], K), i + 1

        p = dmp_mul(dmp_mul(p, dmp_pow(b, dv, v, K), v, K),
                    dmp_pow(lc, du - dw, v, K), v, K)
        q = dmp_mul(q, dmp_pow(lc, dv*(1 + d), v, K), v, K)

        _, p, q = dmp_inner_gcd(p, q, v, K)

    if s < 0:
        p = dmp_neg(p, v, K)

    i = dmp_degree(R[-2], u)

    res = dmp_pow(dmp_LC(R[-1], K), i, v, K)
    res = dmp_quo(dmp_mul(res, p, v, K), q, v, K)

    return res, R
コード例 #52
0
def dmp_zz_modular_resultant(f, g, p, u, K):
    """
    Compute resultant of `f` and `g` modulo a prime `p`.

    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_modular_resultant(f, g, 5)
    -2*y**2 + 1

    """
    if not u:
        return gf_int(dup_prs_resultant(f, g, K)[0] % p, p)

    v = u - 1

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

    N = dmp_degree_in(f, 1, u)
    M = dmp_degree_in(g, 1, u)

    B = n*M + m*N

    D, a = [K.one], -K.one
    r = dmp_zero(v)

    while dup_degree(D) <= B:
        while True:
            a += K.one

            if a == p:
                raise HomomorphismFailed('no luck')

            F = dmp_eval_in(f, gf_int(a, p), 1, u, K)

            if dmp_degree(F, v) == n:
                G = dmp_eval_in(g, gf_int(a, p), 1, u, K)

                if dmp_degree(G, v) == m:
                    break

        R = dmp_zz_modular_resultant(F, G, p, v, K)
        e = dmp_eval(r, a, v, K)

        if not v:
            R = dup_strip([R])
            e = dup_strip([e])
        else:
            R = [R]
            e = [e]

        d = K.invert(dup_eval(D, a, K), p)
        d = dup_mul_ground(D, d, K)
        d = dmp_raise(d, v, 0, K)

        c = dmp_mul(d, dmp_sub(R, e, v, K), v, K)
        r = dmp_add(r, c, v, K)

        r = dmp_ground_trunc(r, p, v, K)

        D = dup_mul(D, [K.one, -a], K)
        D = dup_trunc(D, p, K)

    return r
コード例 #53
0
ファイル: euclidtools.py プロジェクト: addisonc/sympy
def dmp_prs_resultant(f, g, u, K):
    """
    Resultant algorithm in ``K[X]`` using subresultant PRS.

    **Examples**

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

    >>> f = ZZ.map([[3, 0], [], [-1, 0, 0, -4]])
    >>> g = ZZ.map([[1], [1, 0, 0, 0], [-9]])

    >>> a = ZZ.map([[3, 0, 0, 0, 0], [1, 0, -27, 4]])
    >>> b = ZZ.map([[-3, 0, 0, -12, 1, 0, -54, 8, 729, -216, 16]])

    >>> dmp_prs_resultant(f, g, 1, ZZ) == (b[0], [f, g, a, b])
    True

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

    if dmp_zero_p(f, u) or dmp_zero_p(g, u):
        return (dmp_zero(u-1), [])

    R, B, D = dmp_inner_subresultants(f, g, u, K)

    if dmp_degree(R[-1], u) > 0:
        return (dmp_zero(u-1), R)
    if dmp_one_p(R[-2], u, K):
        return (dmp_LC(R[-1], K), R)

    s, i, v = 1, 1, u-1

    p = dmp_one(v, K)
    q = dmp_one(v, K)

    for b, d in zip(B, D)[:-1]:
        du = dmp_degree(R[i-1], u)
        dv = dmp_degree(R[i  ], u)
        dw = dmp_degree(R[i+1], u)

        if du % 2 and dv % 2:
            s = -s

        lc, i = dmp_LC(R[i], K), i+1

        p = dmp_mul(dmp_mul(p, dmp_pow(b, dv, v, K), v, K),
                               dmp_pow(lc, du-dw, v, K), v, K)
        q = dmp_mul(q, dmp_pow(lc, dv*(1+d), v, K), v, K)

        _, p, q = dmp_inner_gcd(p, q, v, K)

    if s < 0:
        p = dmp_neg(p, v, K)

    i = dmp_degree(R[-2], u)

    res = dmp_pow(dmp_LC(R[-1], K), i, v, K)
    res = dmp_exquo(dmp_mul(res, p, v, K), q, v, K)

    return res, R
コード例 #54
0
ファイル: euclidtools.py プロジェクト: addisonc/sympy
def dmp_zz_modular_resultant(f, g, p, u, K):
    """
    Compute resultant of ``f`` and ``g`` modulo a prime ``p``.

    **Examples**

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

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

    >>> dmp_zz_modular_resultant(f, g, ZZ(5), 1, ZZ)
    [-2, 0, 1]

    """
    if not u:
        return gf_int(dup_prs_resultant(f, g, K)[0] % p, p)

    v = u - 1

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

    N = dmp_degree_in(f, 1, u)
    M = dmp_degree_in(g, 1, u)

    B = n*M + m*N

    D, a = [K.one], -K.one
    r = dmp_zero(v)

    while dup_degree(D) <= B:
        while True:
            a += K.one

            if a == p:
                raise HomomorphismFailed('no luck')

            F = dmp_eval_in(f, gf_int(a, p), 1, u, K)

            if dmp_degree(F, v) == n:
                G = dmp_eval_in(g, gf_int(a, p), 1, u, K)

                if dmp_degree(G, v) == m:
                    break

        R = dmp_zz_modular_resultant(F, G, p, v, K)
        e = dmp_eval(r, a, v, K)

        if not v:
            R = dup_strip([R])
            e = dup_strip([e])
        else:
            R = [R]
            e = [e]

        d = K.invert(dup_eval(D, a, K), p)
        d = dup_mul_ground(D, d, K)
        d = dmp_raise(d, v, 0, K)

        c = dmp_mul(d, dmp_sub(R, e, v, K), v, K)
        r = dmp_add(r, c, v, K)

        r = dmp_ground_trunc(r, p, v, K)

        D = dup_mul(D, [K.one, -a], K)
        D = dup_trunc(D, p, K)

    return r
コード例 #55
0
ファイル: test_densearith.py プロジェクト: vperic/sympy
def test_dmp_expand():
    assert dmp_expand((), 1, ZZ) == [[1]]
    assert dmp_expand(([[1],[2],[3]], [[1],[2]], [[7],[5],[4],[3]]), 1, ZZ) == \
        dmp_mul([[1],[2],[3]], dmp_mul([[1],[2]], [[7],[5],[4],[3]], 1, ZZ), 1, ZZ)
コード例 #56
0
ファイル: euclidtools.py プロジェクト: addisonc/sympy
def dmp_inner_subresultants(f, g, u, K):
    """
    Subresultant PRS algorithm in ``K[X]``.

    **Examples**

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

    >>> f = ZZ.map([[3, 0], [], [-1, 0, 0, -4]])
    >>> g = ZZ.map([[1], [1, 0, 0, 0], [-9]])

    >>> a = [[3, 0, 0, 0, 0], [1, 0, -27, 4]]
    >>> b = [[-3, 0, 0, -12, 1, 0, -54, 8, 729, -216, 16]]

    >>> R = ZZ.map([f, g, a, b])
    >>> B = ZZ.map([[-1], [1], [9, 0, 0, 0, 0, 0, 0, 0, 0]])
    >>> D = ZZ.map([0, 1, 1])

    >>> dmp_inner_subresultants(f, g, 1, ZZ) == (R, B, D)
    True

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

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

    if n < m:
        f, g = g, f
        n, m = m, n

    R = [f, g]
    d = n - m
    v = u - 1

    b = dmp_pow(dmp_ground(-K.one, v), d+1, v, K)
    c = dmp_ground(-K.one, v)

    B, D = [b], [d]

    if dmp_zero_p(f, u) or dmp_zero_p(g, u):
        return R, B, D

    h = dmp_prem(f, g, u, K)
    h = dmp_mul_term(h, b, 0, u, K)

    while not dmp_zero_p(h, u):
        k = dmp_degree(h, u)
        R.append(h)

        lc = dmp_LC(g, K)

        p = dmp_pow(dmp_neg(lc, v, K), d, v, K)

        if not d:
            q = c
        else:
            q = dmp_pow(c, d-1, v, K)

        c = dmp_exquo(p, q, v, K)
        b = dmp_mul(dmp_neg(lc, v, K),
                    dmp_pow(c, m-k, v, K), v, K)

        f, g, m, d = g, h, k, m-k

        B.append(b)
        D.append(d)

        h = dmp_prem(f, g, u, K)
        h = [ dmp_exquo(ch, b, v, K) for ch in h ]

    return R, B, D
コード例 #57
0
def dmp_inner_subresultants(f, g, u, K):
    """
    Subresultant PRS algorithm in `K[X]`.

    Examples
    ========

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

    >>> f = 3*x**2*y - y**3 - 4
    >>> g = x**2 + x*y**3 - 9

    >>> a = 3*x*y**4 + y**3 - 27*y + 4
    >>> b = -3*y**10 - 12*y**7 + y**6 - 54*y**4 + 8*y**3 + 729*y**2 - 216*y + 16

    >>> prs = [f, g, a, b]
    >>> beta = [[-1], [1], [9, 0, 0, 0, 0, 0, 0, 0, 0]]
    >>> delta = [0, 1, 1]

    >>> R.dmp_inner_subresultants(f, g) == (prs, beta, delta)
    True

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

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

    if n < m:
        f, g = g, f
        n, m = m, n

    R = [f, g]
    d = n - m
    v = u - 1

    b = dmp_pow(dmp_ground(-K.one, v), d + 1, v, K)
    c = dmp_ground(-K.one, v)

    B, D = [b], [d]

    if dmp_zero_p(f, u) or dmp_zero_p(g, u):
        return R, B, D

    h = dmp_prem(f, g, u, K)
    h = dmp_mul_term(h, b, 0, u, K)

    while not dmp_zero_p(h, u):
        k = dmp_degree(h, u)
        R.append(h)

        lc = dmp_LC(g, K)

        p = dmp_pow(dmp_neg(lc, v, K), d, v, K)

        if not d:
            q = c
        else:
            q = dmp_pow(c, d - 1, v, K)

        c = dmp_quo(p, q, v, K)
        b = dmp_mul(dmp_neg(lc, v, K),
                    dmp_pow(c, m - k, v, K), v, K)

        f, g, m, d = g, h, k, m - k

        B.append(b)
        D.append(d)

        h = dmp_prem(f, g, u, K)

        h = [ dmp_quo(ch, b, v, K) for ch in h ]

    return R, B, D