示例#1
0
def ratint_logpart(f, g, x, t=None):
    """Lazard-Rioboo-Trager algorithm.

       Given a field K and polynomials f and g in K[x], such that f and g
       are coprime, deg(f) < deg(g) and g is square-free, returns a list
       of tuples (s_i, q_i) of polynomials, for i = 1..n, such that s_i
       in K[t, x] and q_i in K[t], and:
                               ___    ___
                     d  f   d  \  `   \  `
                     -- - = --  )      )   a log(s_i(a, x))
                     dx g   dx /__,   /__,
                              i=1..n a | q_i(a) = 0

    """
    f, g = Poly(f, x), Poly(g, x)

    t = t or Dummy('t')
    a, b = g, f - g.diff()*Poly(t, x)

    R = subresultants(a, b)
    res = Poly(resultant(a, b), t)

    R_map, H = {}, []

    for r in R:
        R_map[r.degree()] = r

    def _include_sign(c, sqf):
        if c < 0:
            h, k = sqf[0]
            sqf[0] = h*c, k

    C, res_sqf = res.sqf_list()
    _include_sign(C, res_sqf)

    for q, i in res_sqf:
        _, q = q.primitive()

        if g.degree() == i:
            H.append((g, q))
        else:
            h = R_map[i]
            h_lc = Poly(h.LC(), t, field=True)

            c, h_lc_sqf = h_lc.sqf_list(all=True)
            _include_sign(c, h_lc_sqf)

            for a, j in h_lc_sqf:
                h = h.exquo(Poly(a.gcd(q)**j, x))

            inv, coeffs = h_lc.invert(q), [S(1)]

            for coeff in h.coeffs()[1:]:
                T = (inv*coeff).rem(q)
                coeffs.append(T.as_basic())

            h = Poly(dict(zip(h.monoms(), coeffs)), x)

            H.append((h, q))

    return H
示例#2
0
def ratint_logpart(f, g, x, t=None):
    """
    Lazard-Rioboo-Trager algorithm.

    Given a field K and polynomials f and g in K[x], such that f and g
    are coprime, deg(f) < deg(g) and g is square-free, returns a list
    of tuples (s_i, q_i) of polynomials, for i = 1..n, such that s_i
    in K[t, x] and q_i in K[t], and:
                           ___    ___
                 d  f   d  \  `   \  `
                 -- - = --  )      )   a log(s_i(a, x))
                 dx g   dx /__,   /__,
                          i=1..n a | q_i(a) = 0

    Examples
    ========

        >>> from sympy.integrals.rationaltools import ratint_logpart
        >>> from sympy.abc import x
        >>> from sympy import Poly
        >>> ratint_logpart(Poly(1, x, domain='ZZ'),
        ... Poly(x**2 + x + 1, x, domain='ZZ'), x)
        [(Poly(x + 3*_t/2 + 1/2, x, domain='QQ[_t]'),
        ...Poly(3*_t**2 + 1, _t, domain='ZZ'))]
        >>> ratint_logpart(Poly(12, x, domain='ZZ'),
        ... Poly(x**2 - x - 2, x, domain='ZZ'), x)
        [(Poly(x - 3*_t/8 - 1/2, x, domain='QQ[_t]'),
        ...Poly(-_t**2 + 16, _t, domain='ZZ'))]

    """
    f, g = Poly(f, x), Poly(g, x)

    t = t or Dummy('t')
    a, b = g, f - g.diff()*Poly(t, x)

    R = subresultants(a, b)

    res = Poly(resultant(a, b), t, composite=False)

    R_map, H = {}, []

    for r in R:
        R_map[r.degree()] = r

    def _include_sign(c, sqf):
        if c < 0:
            h, k = sqf[0]
            sqf[0] = h*c, k

    C, res_sqf = res.sqf_list()
    _include_sign(C, res_sqf)

    for q, i in res_sqf:
        _, q = q.primitive()

        if g.degree() == i:
            H.append((g, q))
        else:
            h = R_map[i]
            h_lc = Poly(h.LC(), t, field=True)

            c, h_lc_sqf = h_lc.sqf_list(all=True)
            _include_sign(c, h_lc_sqf)

            for a, j in h_lc_sqf:
                h = h.quo(Poly(a.gcd(q)**j, x))

            inv, coeffs = h_lc.invert(q), [S(1)]

            for coeff in h.coeffs()[1:]:
                T = (inv*coeff).rem(q)
                coeffs.append(T.as_expr())

            h = Poly(dict(zip(h.monoms(), coeffs)), x)

            H.append((h, q))

    return H
示例#3
0
def ratint_logpart(f, g, x, t=None):
    """Lazard-Rioboo-Trager algorithm.

       Given a field K and polynomials f and g in K[x], such that f and g
       are coprime, deg(f) < deg(g) and g is square-free, returns a list
       of tuples (s_i, q_i) of polynomials, for i = 1..n, such that s_i
       in K[t, x] and q_i in K[t], and:
                               ___    ___
                     d  f   d  \  `   \  `
                     -- - = --  )      )   a log(s_i(a, x))
                     dx g   dx /__,   /__,
                              i=1..n a | q_i(a) = 0

    """
    f, g = Poly(f, x), Poly(g, x)

    t = t or Dummy('t')
    a, b = g, f - g.diff() * Poly(t, x)

    R = subresultants(a, b)
    res = Poly(resultant(a, b), t)

    R_map, H = {}, []

    for r in R:
        R_map[r.degree()] = r

    def _include_sign(c, sqf):
        if c < 0:
            h, k = sqf[0]
            sqf[0] = h * c, k

    C, res_sqf = res.sqf_list()
    _include_sign(C, res_sqf)

    for q, i in res_sqf:
        _, q = q.primitive()

        if g.degree() == i:
            H.append((g, q))
        else:
            h = R_map[i]
            h_lc = Poly(h.LC(), t, field=True)

            c, h_lc_sqf = h_lc.sqf_list(all=True)
            _include_sign(c, h_lc_sqf)

            for a, j in h_lc_sqf:
                h = h.exquo(Poly(a.gcd(q)**j, x))

            inv, coeffs = h_lc.invert(q), [S(1)]

            for coeff in h.coeffs()[1:]:
                T = (inv * coeff).rem(q)
                coeffs.append(T.as_basic())

            h = Poly(dict(zip(h.monoms(), coeffs)), x)

            H.append((h, q))

    return H
示例#4
0
def ratint_logpart(f, g, x, t=None):
    """
    Lazard-Rioboo-Trager algorithm.

    Given a field K and polynomials f and g in K[x], such that f and g
    are coprime, deg(f) < deg(g) and g is square-free, returns a list
    of tuples (s_i, q_i) of polynomials, for i = 1..n, such that s_i
    in K[t, x] and q_i in K[t], and:
                           ___    ___
                 d  f   d  \  `   \  `
                 -- - = --  )      )   a log(s_i(a, x))
                 dx g   dx /__,   /__,
                          i=1..n a | q_i(a) = 0

    Examples
    ========

        >>> from sympy.integrals.rationaltools import ratint_logpart
        >>> from sympy.abc import x
        >>> from sympy import Poly
        >>> ratint_logpart(Poly(1, x, domain='ZZ'),
        ... Poly(x**2 + x + 1, x, domain='ZZ'), x)
        [(Poly(x + 3*_t/2 + 1/2, x, domain='QQ[_t]'),
        ...Poly(3*_t**2 + 1, _t, domain='ZZ'))]
        >>> ratint_logpart(Poly(12, x, domain='ZZ'),
        ... Poly(x**2 - x - 2, x, domain='ZZ'), x)
        [(Poly(x - 3*_t/8 - 1/2, x, domain='QQ[_t]'),
        ...Poly(-_t**2 + 16, _t, domain='ZZ'))]

    """
    f, g = Poly(f, x), Poly(g, x)

    t = t or Dummy('t')
    a, b = g, f - g.diff() * Poly(t, x)

    R = subresultants(a, b)

    res = Poly(resultant(a, b), t, composite=False)

    R_map, H = {}, []

    for r in R:
        R_map[r.degree()] = r

    def _include_sign(c, sqf):
        if c < 0:
            h, k = sqf[0]
            sqf[0] = h * c, k

    C, res_sqf = res.sqf_list()
    _include_sign(C, res_sqf)

    for q, i in res_sqf:
        _, q = q.primitive()

        if g.degree() == i:
            H.append((g, q))
        else:
            h = R_map[i]
            h_lc = Poly(h.LC(), t, field=True)

            c, h_lc_sqf = h_lc.sqf_list(all=True)
            _include_sign(c, h_lc_sqf)

            for a, j in h_lc_sqf:
                h = h.quo(Poly(a.gcd(q)**j, x))

            inv, coeffs = h_lc.invert(q), [S(1)]

            for coeff in h.coeffs()[1:]:
                T = (inv * coeff).rem(q)
                coeffs.append(T.as_expr())

            h = Poly(dict(zip(h.monoms(), coeffs)), x)

            H.append((h, q))

    return H