Python risch.frac_in函数代码示例

def test_frac_in():
    assert frac_in(Poly((x + 1) / x * t, t), x) == (Poly(t * x + t, x), Poly(x, x))
    assert frac_in((x + 1) / x * t, x) == (Poly(t * x + t, x), Poly(x, x))
    assert frac_in((Poly((x + 1) / x * t, t), Poly(t + 1, t)), x) == (Poly(t * x + t, x), Poly((1 + t) * x, x))
    raises(ValueError, lambda: frac_in((x + 1) / log(x) * t, x))
    assert frac_in(Poly((2 + 2 * x + x * (1 + x)) / (1 + x) ** 2, t), x, cancel=True) == (
        Poly(x + 2, x),
        Poly(x + 1, x),
    )

def cancel_exp(b, c, n, DE):
    """
    Poly Risch Differential Equation - Cancellation: Hyperexponential case.

    Given a derivation D on k[t], n either an integer or +oo, b in k, and
    c in k[t] with Dt/t in k and b != 0, either raise
    NonElementaryIntegralException, in which case the equation Dq + b*q == c
    has no solution of degree at most n in k[t], or a solution q in k[t] of
    this equation with deg(q) <= n.
    """
    from sympy.integrals.prde import parametric_log_deriv

    eta = DE.d.quo(Poly(DE.t, DE.t)).as_expr()

    with DecrementLevel(DE):
        etaa, etad = frac_in(eta, DE.t)
        ba, bd = frac_in(b, DE.t)
        A = parametric_log_deriv(ba, bd, etaa, etad, DE)
        if A is not None:
            a, m, z = A
            if a == 1:
                raise NotImplementedError("is_deriv_in_field() is required to "
                    "solve this problem.")
                # if c*z*t**m == Dp for p in k<t> and q = p/(z*t**m) in k[t] and
                # deg(q) <= n:
                #     return q
                # else:
                #     raise NonElementaryIntegralException

    if c.is_zero:
        return c  # return 0

    if n < c.degree(DE.t):
        raise NonElementaryIntegralException

    q = Poly(0, DE.t)
    while not c.is_zero:
        m = c.degree(DE.t)
        if n < m:
            raise NonElementaryIntegralException
        # a1 = b + m*Dt/t
        a1 = b.as_expr()
        with DecrementLevel(DE):
            # TODO: Write a dummy function that does this idiom
            a1a, a1d = frac_in(a1, DE.t)
            a1a = a1a*etad + etaa*a1d*Poly(m, DE.t)
            a1d = a1d*etad

            a2a, a2d = frac_in(c.LC(), DE.t)

            sa, sd = rischDE(a1a, a1d, a2a, a2d, DE)
        stm = Poly(sa.as_expr()/sd.as_expr()*DE.t**m, DE.t, expand=False)
        q += stm
        n = m - 1
        c -= b*stm + derivation(stm, DE)  # deg(c) becomes smaller
    return q

def cancel_primitive(b, c, n, DE):
    """
    Poly Risch Differential Equation - Cancellation: Primitive case.

    Given a derivation D on k[t], n either an integer or +oo, b in k, and
    c in k[t] with Dt in k and b != 0, either raise
    NonElementaryIntegralException, in which case the equation Dq + b*q == c
    has no solution of degree at most n in k[t], or a solution q in k[t] of
    this equation with deg(q) <= n.
    """
    from sympy.integrals.prde import is_log_deriv_k_t_radical_in_field

    with DecrementLevel(DE):
        ba, bd = frac_in(b, DE.t)
        A = is_log_deriv_k_t_radical_in_field(ba, bd, DE)
        if A is not None:
            n, z = A
            if n == 1:  # b == Dz/z
                raise NotImplementedError("is_deriv_in_field() is required to "
                    " solve this problem.")
                # if z*c == Dp for p in k[t] and deg(p) <= n:
                #     return p/z
                # else:
                #     raise NonElementaryIntegralException

    if c.is_zero:
        return c  # return 0

    if n < c.degree(DE.t):
        raise NonElementaryIntegralException

    q = Poly(0, DE.t)
    while not c.is_zero:
        m = c.degree(DE.t)
        if n < m:
            raise NonElementaryIntegralException
        with DecrementLevel(DE):
            a2a, a2d = frac_in(c.LC(), DE.t)
            sa, sd = rischDE(ba, bd, a2a, a2d, DE)
        stm = Poly(sa.as_expr()/sd.as_expr()*DE.t**m, DE.t, expand=False)
        q += stm
        n = m - 1
        c -= b*stm + derivation(stm, DE)

    return q

def prde_cancel_liouvillian(b, Q, n, DE):
    """
    Pg, 237.
    """
    H = []

    # Why use DecrementLevel? Below line answers that:
    # Assuming that we can solve such problems over 'k' (not k[t])
    if DE.case == 'primitive':
        with DecrementLevel(DE):
            ba, bd = frac_in(b, DE.t, field=True)

    for i in range(n, -1, -1):
        if DE.case == 'exp': # this re-checking can be avoided
            with DecrementLevel(DE):
                ba, bd = frac_in(b + i*derivation(DE.t, DE)/DE.t,
                                DE.t, field=True)
        with DecrementLevel(DE):
            Qy = [frac_in(q.nth(i), DE.t, field=True) for q in Q]
            fi, Ai = param_rischDE(ba, bd, Qy, DE)
        fi = [Poly(fa.as_expr()/fd.as_expr(), DE.t, field=True)
                for fa, fd in fi]

        ri = len(fi)

        if i == n:
            M = Ai
        else:
            M = Ai.col_join(M.row_join(zeros(M.rows, ri)))

        Fi, hi = [None]*ri, [None]*ri

        # from eq. on top of p.238 (unnumbered)
        for j in range(ri):
            hji = fi[j]*DE.t**i
            hi[j] = hji
            # building up Sum(djn*(D(fjn*t^n) - b*fjnt^n))
            Fi[j] = -(derivation(hji, DE) - b*hji)

        H += hi
        # in the next loop instead of Q it has
        # to be Q + Fi taking its place
        Q = Q + Fi

    return (H, M)

def prde_special_denom(a, ba, bd, G, DE, case='auto'):
    """
    Parametric Risch Differential Equation - Special part of the denominator.

    case is on of {'exp', 'tan', 'primitive'} for the hyperexponential,
    hypertangent, and primitive cases, respectively.  For the hyperexponential
    (resp. hypertangent) case, given a derivation D on k[t] and a in k[t],
    b in k<t>, and g1, ..., gm in k(t) with Dt/t in k (resp. Dt/(t**2 + 1) in
    k, sqrt(-1) not in k), a != 0, and gcd(a, t) == 1 (resp.
    gcd(a, t**2 + 1) == 1), return the tuple (A, B, GG, h) such that A, B, h in
    k[t], GG = [gg1, ..., ggm] in k(t)^m, and for any solution c1, ..., cm in
    Const(k) and q in k<t> of a*Dq + b*q == Sum(ci*gi, (i, 1, m)), r == q*h in
    k[t] satisfies A*Dr + B*r == Sum(ci*ggi, (i, 1, m)).

    For case == 'primitive', k<t> == k[t], so it returns (a, b, G, 1) in this
    case.
    """
    # TODO: Merge this with the very similar special_denom() in rde.py
    if case == 'auto':
        case = DE.case

    if case == 'exp':
        p = Poly(DE.t, DE.t)
    elif case == 'tan':
        p = Poly(DE.t**2 + 1, DE.t)
    elif case in ['primitive', 'base']:
        B = ba.quo(bd)
        return (a, B, G, Poly(1, DE.t))
    else:
        raise ValueError("case must be one of {'exp', 'tan', 'primitive', "
            "'base'}, not %s." % case)

    nb = order_at(ba, p, DE.t) - order_at(bd, p, DE.t)
    nc = min([order_at(Ga, p, DE.t) - order_at(Gd, p, DE.t) for Ga, Gd in G])
    n = min(0, nc - min(0, nb))
    if not nb:
        # Possible cancellation.
        if case == 'exp':
            dcoeff = DE.d.quo(Poly(DE.t, DE.t))
            with DecrementLevel(DE):  # We are guaranteed to not have problems,
                                      # because case != 'base'.
                alphaa, alphad = frac_in(-ba.eval(0)/bd.eval(0)/a.eval(0), DE.t)
                etaa, etad = frac_in(dcoeff, DE.t)
                A = parametric_log_deriv(alphaa, alphad, etaa, etad, DE)
                if A is not None:
                    a, m, z = A
                    if a == 1:
                        n = min(n, m)

        elif case == 'tan':
            dcoeff = DE.d.quo(Poly(DE.t**2 + 1, DE.t))
            with DecrementLevel(DE):  # We are guaranteed to not have problems,
                                      # because case != 'base'.
                betaa, alphaa, alphad =  real_imag(ba, bd*a, DE.t)
                betad = alphad
                etaa, etad = frac_in(dcoeff, DE.t)
                if recognize_log_derivative(2*betaa, betad, DE):
                    A = parametric_log_deriv(alphaa, alphad, etaa, etad, DE)
                    B = parametric_log_deriv(betaa, betad, etaa, etad, DE)
                    if A is not None and B is not None:
                        a, s, z = A
                        if a == 1:
                             n = min(n, s/2)

    N = max(0, -nb)
    pN = p**N
    pn = p**-n  # This is 1/h

    A = a*pN
    B = ba*pN.quo(bd) + Poly(n, DE.t)*a*derivation(p, DE).quo(p)*pN
    G = [(Ga*pN*pn).cancel(Gd, include=True) for Ga, Gd in G]
    h = pn

    # (a*p**N, (b + n*a*Dp/p)*p**N, g1*p**(N - n), ..., gm*p**(N - n), p**-n)
    return (A, B, G, h)

def is_log_deriv_k_t_radical_in_field(fa, fd, DE, case='auto', z=None):
    """
    Checks if f can be written as the logarithmic derivative of a k(t)-radical.

    f in k(t) can be written as the logarithmic derivative of a k(t) radical if
    there exist n in ZZ and u in k(t) with n, u != 0 such that n*f == Du/u.
    Either returns (n, u) or None, which means that f cannot be written as the
    logarithmic derivative of a k(t)-radical.

    case is one of {'primitive', 'exp', 'tan', 'auto'} for the primitive,
    hyperexponential, and hypertangent cases, respectively.  If case it 'auto',
    it will attempt to determine the type of the derivation automatically.
    """
    fa, fd = fa.cancel(fd, include=True)

    # f must be simple
    n, s = splitfactor(fd, DE)
    if not s.is_one:
        pass
        #return None

    z = z or Dummy('z')
    H, b = residue_reduce(fa, fd, DE, z=z)
    if not b:
        # I will have to verify, but I believe that the answer should be
        # None in this case. This should never happen for the
        # functions given when solving the parametric logarithmic
        # derivative problem when integration elementary functions (see
        # Bronstein's book, page 255), so most likely this indicates a bug.
        return None

    roots = [(i, i.real_roots()) for i, _ in H]
    if not all(len(j) == i.degree() and all(k.is_Rational for k in j) for
               i, j in roots):
        # If f is the logarithmic derivative of a k(t)-radical, then all the
        # roots of the resultant must be rational numbers.
        return None

    # [(a, i), ...], where i*log(a) is a term in the log-part of the integral
    # of f
    respolys, residues = zip(*roots) or [[], []]
    # Note: this might be empty, but everything below should work find in that
    # case (it should be the same as if it were [[1, 1]])
    residueterms = [(H[j][1].subs(z, i), i) for j in xrange(len(H)) for
        i in residues[j]]

    # TODO: finish writing this and write tests

    p = cancel(fa.as_expr()/fd.as_expr() - residue_reduce_derivation(H, DE, z))

    p = p.as_poly(DE.t)
    if p is None:
        # f - Dg will be in k[t] if f is the logarithmic derivative of a k(t)-radical
        return None

    if p.degree(DE.t) >= max(1, DE.d.degree(DE.t)):
        return None

    if case == 'auto':
        case = DE.case

    if case == 'exp':
        wa, wd = derivation(DE.t, DE).cancel(Poly(DE.t, DE.t), include=True)
        with DecrementLevel(DE):
            pa, pd = frac_in(p, DE.t, cancel=True)
            wa, wd = frac_in((wa, wd), DE.t)
            A = parametric_log_deriv(pa, pd, wa, wd, DE)
        if A is None:
            return None
        n, e, u = A
        u *= DE.t**e
#        raise NotImplementedError("The hyperexponential case is "
#            "not yet completely implemented for is_log_deriv_k_t_radical_in_field().")

    elif case == 'primitive':
        with DecrementLevel(DE):
            pa, pd = frac_in(p, DE.t)
            A = is_log_deriv_k_t_radical_in_field(pa, pd, DE, case='auto')
        if A is None:
            return None
        n, u = A

    elif case == 'base':
        # TODO: we can use more efficient residue reduction from ratint()
        if not fd.is_sqf or fa.degree() >= fd.degree():
            # f is the logarithmic derivative in the base case if and only if
            # f = fa/fd, fd is square-free, deg(fa) < deg(fd), and
            # gcd(fa, fd) == 1.  The last condition is handled by cancel() above.
            return None
        # Note: if residueterms = [], returns (1, 1)
        # f had better be 0 in that case.
        n = reduce(ilcm, [i.as_numer_denom()[1] for _, i in residueterms], S(1))
        u = Mul(*[Pow(i, j*n) for i, j in residueterms])
        return (n, u)

    elif case == 'tan':
        raise NotImplementedError("The hypertangent case is "
        "not yet implemented for is_log_deriv_k_t_radical_in_field()")

    elif case in ['other_linear', 'other_nonlinear']:
#.........这里部分代码省略.........

def bound_degree(a, b, cQ, DE, case='auto', parametric=False):
    """
    Bound on polynomial solutions.

    Given a derivation D on k[t] and a, b, c in k[t] with a != 0, return
    n in ZZ such that deg(q) <= n for any solution q in k[t] of
    a*Dq + b*q == c, when parametric=False, or deg(q) <= n for any solution
    c1, ..., cm in Const(k) and q in k[t] of a*Dq + b*q == Sum(ci*gi, (i, 1, m))
    when parametric=True.

    For parametric=False, cQ is c, a Poly; for parametric=True, cQ is Q ==
    [q1, ..., qm], a list of Polys.

    This constitutes step 3 of the outline given in the rde.py docstring.
    """
    from sympy.integrals.prde import (parametric_log_deriv, limited_integrate,
        is_log_deriv_k_t_radical_in_field)
    # TODO: finish writing this and write tests

    if case == 'auto':
        case = DE.case

    da = a.degree(DE.t)
    db = b.degree(DE.t)

    # The parametric and regular cases are identical, except for this part
    if parametric:
        dc = max([i.degree(DE.t) for i in cQ])
    else:
        dc = cQ.degree(DE.t)

    alpha = cancel(-b.as_poly(DE.t).LC().as_expr()/
        a.as_poly(DE.t).LC().as_expr())

    if case == 'base':
        n = max(0, dc - max(db, da - 1))
        if db == da - 1 and alpha.is_Integer:
            n = max(0, alpha, dc - db)

    elif case == 'primitive':
        if db > da:
            n = max(0, dc - db)
        else:
            n = max(0, dc - da + 1)

        etaa, etad = frac_in(DE.d, DE.T[DE.level - 1])

        t1 = DE.t
        with DecrementLevel(DE):
            alphaa, alphad = frac_in(alpha, DE.t)
            if db == da - 1:
                # if alpha == m*Dt + Dz for z in k and m in ZZ:
                try:
                    (za, zd), m = limited_integrate(alphaa, alphad, [(etaa, etad)],
                        DE)
                except NonElementaryIntegralException:
                    pass
                else:
                    assert len(m) == 1
                    n = max(n, m[0])

            elif db == da:
                # if alpha == Dz/z for z in k*:
                    # beta = -lc(a*Dz + b*z)/(z*lc(a))
                    # if beta == m*Dt + Dw for w in k and m in ZZ:
                        # n = max(n, m)
                A = is_log_deriv_k_t_radical_in_field(alphaa, alphad, DE)
                if A is not None:
                    aa, z = A
                    if aa == 1:
                        beta = -(a*derivation(z, DE).as_poly(t1) +
                            b*z.as_poly(t1)).LC()/(z.as_expr()*a.LC())
                        betaa, betad = frac_in(beta, DE.t)
                        try:
                            (za, zd), m = limited_integrate(betaa, betad,
                                [(etaa, etad)], DE)
                        except NonElementaryIntegralException:
                            pass
                        else:
                            assert len(m) == 1
                            n = max(n, m[0])

    elif case == 'exp':
        n = max(0, dc - max(db, da))
        if da == db:
            etaa, etad = frac_in(DE.d.quo(Poly(DE.t, DE.t)), DE.T[DE.level - 1])
            with DecrementLevel(DE):
                alphaa, alphad = frac_in(alpha, DE.t)
                A = parametric_log_deriv(alphaa, alphad, etaa, etad, DE)
                if A is not None:
                    # if alpha == m*Dt/t + Dz/z for z in k* and m in ZZ:
                        # n = max(n, m)
                    a, m, z = A
                    if a == 1:
                        n = max(n, m)

    elif case in ['tan', 'other_nonlinear']:
        delta = DE.d.degree(DE.t)
        lam = DE.d.LC()
        alpha = cancel(alpha/lam)
#.........这里部分代码省略.........

def special_denom(a, ba, bd, ca, cd, DE, case='auto'):
    """
    Special part of the denominator.

    case is one of {'exp', 'tan', 'primitive'} for the hyperexponential,
    hypertangent, and primitive cases, respectively.  For the
    hyperexponential (resp. hypertangent) case, given a derivation D on
    k[t] and a in k[t], b, c, in k<t> with Dt/t in k (resp. Dt/(t**2 + 1) in
    k, sqrt(-1) not in k), a != 0, and gcd(a, t) == 1 (resp.
    gcd(a, t**2 + 1) == 1), return the quadruplet (A, B, C, 1/h) such that
    A, B, C, h in k[t] and for any solution q in k<t> of a*Dq + b*q == c,
    r = qh in k[t] satisfies A*Dr + B*r == C.

    For case == 'primitive', k<t> == k[t], so it returns (a, b, c, 1) in
    this case.

    This constitutes step 2 of the outline given in the rde.py docstring.
    """
    from sympy.integrals.prde import parametric_log_deriv
    # TODO: finish writing this and write tests

    if case == 'auto':
        case = DE.case

    if case == 'exp':
        p = Poly(DE.t, DE.t)
    elif case == 'tan':
        p = Poly(DE.t**2 + 1, DE.t)
    elif case in ['primitive', 'base']:
        B = ba.to_field().quo(bd)
        C = ca.to_field().quo(cd)
        return (a, B, C, Poly(1, DE.t))
    else:
        raise ValueError("case must be one of {'exp', 'tan', 'primitive', "
            "'base'}, not %s." % case)
    # assert a.div(p)[1]

    nb = order_at(ba, p, DE.t) - order_at(bd, p, DE.t)
    nc = order_at(ca, p, DE.t) - order_at(cd, p, DE.t)

    n = min(0, nc - min(0, nb))
    if not nb:
        # Possible cancellation.

        if case == 'exp':
            dcoeff = DE.d.quo(Poly(DE.t, DE.t))
            with DecrementLevel(DE):  # We are guaranteed to not have problems,
                                      # because case != 'base'.
                alphaa, alphad = frac_in(-ba.eval(0)/bd.eval(0)/a.eval(0), DE.t)
                etaa, etad = frac_in(dcoeff, DE.t)
                A = parametric_log_deriv(alphaa, alphad, etaa, etad, DE)
                if A is not None:
                    a, m, z = A
                    if a == 1:
                        n = min(n, m)
        else:
            raise NotImplementedError("Tangent case not implemented yet for "
                "RDE special_denom().")
        #     if alpha == m*Dt/t + Dz/z # parametric logarithmic derivative problem
        #         n = min(n, m)
        # elif case == 'tan':
        #     alpha*sqrt(-1) + beta = (-b/a).rem(p) == -b(sqrt(-1))/a(sqrt(-1))
        #     eta = derivation(t, DE).quo(Poly(t**2 + 1, t)) # eta in k
        #     if 2*beta == Dv/v for some v in k* (see pg. 176) and \
        #     alpha*sqrt(-1) + beta == 2*m*eta*sqrt(-1) + Dz/z:
        #     # parametric logarithmic derivative problem
        #         n = min(n, m)

    N = max(0, -nb, n - nc)
    pN = p**N
    pn = p**-n

    A = a*pN
    B = ba*pN.quo(bd) + Poly(n, DE.t)*a*derivation(p, DE).quo(p)*pN
    C = (ca*pN*pn).quo(cd)
    h = pn

    # (a*p**N, (b + n*a*Dp/p)*p**N, c*p**(N - n), p**-n)
    return (A, B, C, h)

def prde_no_cancel_b_small(b, Q, n, DE):
    """
    Parametric Poly Risch Differential Equation - No cancellation: deg(b) small enough.

    Given a derivation D on k[t], n in ZZ, and b, q1, ..., qm in k[t] with
    deg(b) < deg(D) - 1 and either D == d/dt or deg(D) >= 2, returns
    h1, ..., hr in k[t] and a matrix A with coefficients in Const(k) such that
    if c1, ..., cm in Const(k) and q in k[t] satisfy deg(q) <= n and
    Dq + b*q == Sum(ci*qi, (i, 1, m)) then q = Sum(dj*hj, (j, 1, r)) where
    d1, ..., dr in Const(k) and A*Matrix([[c1, ..., cm, d1, ..., dr]]).T == 0.
    """
    m = len(Q)
    H = [Poly(0, DE.t)]*m

    for N in range(n, 0, -1):  # [n, ..., 1]
        for i in range(m):
            si = Q[i].nth(N + DE.d.degree(DE.t) - 1)/(N*DE.d.LC())
            sitn = Poly(si*DE.t**N, DE.t)
            H[i] = H[i] + sitn
            Q[i] = Q[i] - derivation(sitn, DE) - b*sitn

    if b.degree(DE.t) > 0:
        for i in range(m):
            si = Poly(Q[i].nth(b.degree(DE.t))/b.LC(), DE.t)
            H[i] = H[i] + si
            Q[i] = Q[i] - derivation(si, DE) - b*si
        if all(qi.is_zero for qi in Q):
            dc = -1
            M = Matrix()
        else:
            dc = max([qi.degree(DE.t) for qi in Q])
            M = Matrix(dc + 1, m, lambda i, j: Q[j].nth(i))
        A, u = constant_system(M, zeros(dc + 1, 1), DE)
        c = eye(m)
        A = A.row_join(zeros(A.rows, m)).col_join(c.row_join(-c))
        return (H, A)

    # else: b is in k, deg(qi) < deg(Dt)

    t = DE.t
    if DE.case != 'base':
        with DecrementLevel(DE):
            t0 = DE.t  # k = k0(t0)
            ba, bd = frac_in(b, t0, field=True)
            Q0 = [frac_in(qi.TC(), t0, field=True) for qi in Q]
            f, B = param_rischDE(ba, bd, Q0, DE)

            # f = [f1, ..., fr] in k^r and B is a matrix with
            # m + r columns and entries in Const(k) = Const(k0)
            # such that Dy0 + b*y0 = Sum(ci*qi, (i, 1, m)) has
            # a solution y0 in k with c1, ..., cm in Const(k)
            # if and only y0 = Sum(dj*fj, (j, 1, r)) where
            # d1, ..., dr ar in Const(k) and
            # B*Matrix([c1, ..., cm, d1, ..., dr]) == 0.

        # Transform fractions (fa, fd) in f into constant
        # polynomials fa/fd in k[t].
        # (Is there a better way?)
        f = [Poly(fa.as_expr()/fd.as_expr(), t, field=True)
             for fa, fd in f]
    else:
        # Base case. Dy == 0 for all y in k and b == 0.
        # Dy + b*y = Sum(ci*qi) is solvable if and only if
        # Sum(ci*qi) == 0 in which case the solutions are
        # y = d1*f1 for f1 = 1 and any d1 in Const(k) = k.

        f = [Poly(1, t, field=True)]  # r = 1
        B = Matrix([[qi.TC() for qi in Q] + [S(0)]])
        # The condition for solvability is
        # B*Matrix([c1, ..., cm, d1]) == 0
        # There are no constraints on d1.

    # Coefficients of t^j (j > 0) in Sum(ci*qi) must be zero.
    d = max([qi.degree(DE.t) for qi in Q])
    if d > 0:
        M = Matrix(d, m, lambda i, j: Q[j].nth(i + 1))
        A, _ = constant_system(M, zeros(d, 1), DE)
    else:
        # No constraints on the hj.
        A = Matrix(0, m, [])

    # Solutions of the original equation are
    #    y = Sum(dj*fj, (j, 1, r) + Sum(ei*hi, (i, 1, m)),
    # where  ei == ci  (i = 1, ..., m),  when
    # A*Matrix([c1, ..., cm]) == 0 and
    # B*Matrix([c1, ..., cm, d1, ..., dr]) == 0

    # Build combined constraint matrix with m + r + m columns.

    r = len(f)
    I = eye(m)
    A = A.row_join(zeros(A.rows, r + m))
    B = B.row_join(zeros(B.rows, m))
    C = I.row_join(zeros(m, r)).row_join(-I)

    return f + H, A.col_join(B).col_join(C)