def test_issue924():
    sage.var("a x")
    log = sage.log
    i = sympy.integrate(log(x)/a, (x, a, a+1))
    i2 = sympy.simplify(i)
    s = sage.SR(i2)
    assert s == (a*log(1+a) - a*log(a) + log(1+a) - 1)/a

def symbolic_sis(n, alpha, q, m=None, epsilon=None):
    if epsilon is None:
        epsilon = var("epsilon")
        assume(epsilon>0)
        assume(epsilon<ZZ(1)/2)
    delta_0 = var("delta_0")
    assume(delta_0>=1.0)

    e = alpha*q/sqrt(2*pi)

    if m is None:
        m = sqrt(n*log(q)/log(delta_0))

    v = e * delta_0**m * q**(n/m)  # norm of the vector

    # epsilon = exp(-pi*(|v|^2/q^2))
    f = log(1/epsilon)/pi == (v/q)**2

    # solve
    f = 2* q**2 * m * f * pi
    f = f.simplify_full()
    f = f.solve(delta_0**(2*m))[0]
    f = f.log().canonicalize_radical()
    f = f.solve(log(delta_0))[0]
    f = f.simplify_log()
    return f

  def nice_matrix(self):
      Mnice = Matrix(SR, self.M.nrows(), self.M.ncols())
      for edge in range(1, self.num_edges()+1):
          Mnice[0,edge] = self.M[0,edge]
      for v in range(1, self.num_vertices()+1):
          kappas = 1
          for expon, coef in self.M[v,0].dict().items():
              if expon[0]==0:
                  Mnice[v,0] += coef
              else:
                  kappas *= var("ka{0}".format(expon[0]))**coef
          if kappas != 1:
              Mnice[v,0] += kappas

          for edge in range(1, self.num_edges()+1):
              psis = 1
              for expon, coef in self.M[v,edge].dict().items():
                  if expon[0]==0:
                      Mnice[v,edge] += coef
                  elif expon[0]==1:
                      psis *= var(StrataGraph.ps_name)**coef
                  elif expon[0]==2:
                      psis *= var(StrataGraph.ps2_name)**coef
              if psis != 1:
                  Mnice[v,edge] += psis
      return Mnice

def check_expression(expr, var_symbols):
    """Does eval(expr) both in Sage and SymPy and does other checks."""

    # evaluate the expression in the context of Sage:
    sage.var(var_symbols)
    a = globals().copy()
    # safety checks...
    assert not "sin" in a
    a.update(sage.__dict__)
    assert "sin" in a
    e_sage = eval(expr, a)
    assert not isinstance(e_sage, sympy.Basic)

    # evaluate the expression in the context of SymPy:
    sympy.var(var_symbols)
    b = globals().copy()
    assert not "sin" in b
    b.update(sympy.__dict__)
    assert "sin" in b
    b.update(sympy.__dict__)
    e_sympy = eval(expr, b)
    assert isinstance(e_sympy, sympy.Basic)

    # Do the actual checks:
    assert sympy.S(e_sage) == e_sympy
    assert e_sage == sage.SR(e_sympy)

def symbolic_modulus_switching(n, alpha, q, h, m=None, epsilon=None):
    if epsilon is None:
        epsilon = var("epsilon")
        assume(epsilon>0)
        assume(epsilon<ZZ(1)/2)

    delta_0 = var("delta_0")
    assume(delta_0>=1.0)

    if m is None:
        m = sqrt(n*log(q)/log(delta_0))

    e = alpha*q/sqrt(2*pi)

    c = e * sqrt(m-n)/sqrt(h)

    v = delta_0**m * (q/c)**(n/m)  # norm of the vector
    v_ = v**2/m  # variance of each component

    v_r = (m-n) * e**2 *v_   # noise contribution
    v_l = h * v_ * c**2  # nose contribution of rounding noise

    # epsilon = exp(-pi*(|v|^2/q^2))

    f = log(1/epsilon)/pi == (v_l + v_r)/q**2

    # solve
    f = 2* q**2 * m * f * pi
    f = f.simplify_full()
    f = f.solve(delta_0**(2*m))[0]
    f = f.log().canonicalize_radical()
    f = f.solve(log(delta_0))[0]
    f = f.simplify_log()
    return f

 def q_expansion(self,n=20):
     r"""
     Give the q-expansion of the quotient.
     """
     var('q')
     et = qexp_eta(ZZ[['q']],n)
     etA= et.subs(q=q**self._arg_num).power_series(ZZ[['q']])
     etB= et.subs(q=q**self._arg_den).power_series(ZZ[['q']])
     res = etA**(self._exp_num)/etB**(self._exp_den)
     return res

def eqn_list_to_curve_plot(L,rat_pts):
    xpoly_rng = PolynomialRing(QQ,'x')
    poly_tup = [xpoly_rng(tup) for tup in L]
    f = poly_tup[0]
    h = poly_tup[1]
    g = f+h**2/4
    if len(g.real_roots())==0 and g(0)<0:
        return text("$X(\mathbb{R})=\emptyset$",(1,1),fontsize=50)
    X0 = [real(z[0]) for z in g.base_extend(CC).roots()]+[real(z[0]) for z in g.derivative().base_extend(CC).roots()]
    a,b = inflate_interval(min(X0),max(X0),1.5)
    groots = [a]+g.real_roots()+[b]
    if b-a<1e-7:
        a=-3
        b=3
        groots=[a,b]
    ngints = len(groots)-1
    plotzones = []
    npts = 100
    for j in range(ngints):
        c = groots[j]
        d = groots[j+1]
        if g((c+d)/2)<0:
            continue
        (c,d) = inflate_interval(c,d,1.1)
        s = (d-c)/npts
        u = c
        yvals = []
        for i in range(npts+1):
            v = g(u)
            if v>0:
                v = sqrt(v)
                w = -h(u)/2
                yvals.append(w+v)
                yvals.append(w-v)
            u += s
        (m,M) = inflate_interval(min(yvals),max(yvals),1.2)
        plotzones.append((c,d,m,M))
    x = var('x')
    y = var('y')
    plot=sum(implicit_plot(y**2 + y*h(x) - f(x), (x,R[0],R[1]),(y,R[2],R[3]), aspect_ratio='automatic', plot_points=500, zorder=1) for R in plotzones)
    xmin=min([R[0] for R in plotzones])
    xmax=max([R[1] for R in plotzones])
    ymin=min([R[2] for R in plotzones])
    ymax=max([R[3] for R in plotzones])
    for P in rat_pts:
    	(x,y,z)=eval(P.replace(':',','))
        z=ZZ(z)
     	if z: # Do not attempt to plot points at infinity
      		x=ZZ(x)/z
      		y=ZZ(y)/z**3
      		if x >= xmin and x <= xmax and y >= ymin and y <= ymax:
       			plot += point((x,y),color='red',size=40,zorder=2)
    return plot

    def test_wt_20_eigen(self):
        N20 = KlingenEisensteinAndCuspForms(20)
        pl = N20.hecke_charpoly(2)
        x = pl.parent().gens()[0]
        pl1 = ((x + Integer(840960)) *
               (x ** Integer(2) - Integer(1378464) * x + Integer(328189501440)))
        self.assertTrue(pl == (x - Integer(119538120)) * pl1)

        x = var("x")

        f20_1 = N20.eigenform_with_eigenvalue_t2(alpha20_1)
        f20_2 = N20.eigenform_with_eigenvalue_t2(alpha20_2)
        f20_3 = N20.eigenform_with_eigenvalue_t2(alpha20_3)

        l = [f20_1, f20_2, f20_3]

        l = [f.normalize(f[(1, 1, 1)]) for f in l]

        cons20[-1] = coerce_pol(cons20[-1], alpha20_3.parent())
        self.assertTrue(cons20 == [f._construction for f in l])
        self.assertTrue(all(polynomial_to_form(c, 4) == f
                            for c, f in zip(cons20, l)))

        dcts = [f20_1_dct, f20_2_dct, f20_3_dct]

        self.assertTrue(all(coerce_dict(d, f.base_ring) == f.fc_dct for d, f in zip(dcts, l)))

def cohendiv():
    dig = DIG('../invgen/Traces/NLA/cohendiv.tcs')
    rs = dig.getInvs(inv_typ='eqt', seed=0, vs=dig.vs, deg=2)

    q,y,rvu,x,a,b = var('q,y,rvu,x,a,b')
    expected_rs = [-q*y - rvu + x == 0, -a*y + b == 0, -q*y - 2*b + x >= 0, -2*a*y + rvu >= 0]
    assert set(expected_rs) == set([r.inv for r in rs]), rs

def main2():
    t, n = var('t, n')
    G = symbolic_expression(t**n * (t - Integer(1))**n /
                            factorial(n) * e**t).function(t)
    T = symbolic_expression(G.integral(t, Integer(0), Integer(1))).function(n)
    print([T(j) for j in range(Integer(10))])
    print([T(j).n() for j in range(Integer(10))])

 def test_web_latex_split_on(self):
     r"""
     Checking utility: web_latex_split_on
     """
     x = var('x')
     self.assertEqual(web_latex_split_on("test string"), "test string")
     self.assertEqual(web_latex_split_on(x**2 + 1),
                      '\\( x^{2} \\) + \\(  1 \\)')

 def test_web_latex_split_on_re(self):
     r"""
     Checking utility: web_latex_split_on_re
     """
     x = var('x')
     f = x**2 + 1
     expected = '\\(x^{2} \\) \\(\\mathstrut+  1 \\)'
     self.assertEqual(web_latex_split_on_re(f), expected)

 def test_web_latex(self):
     r"""
     Checking utility: web_latex
     """
     x = var('x')
     self.assertEqual(web_latex("test string"), "test string")
     self.assertEqual(web_latex(x**23 + 2*x + 1),
                      '\\( x^{23} + 2 \\, x + 1 \\)')

 def test_Heckemethods(self):
     from sage.all import NumberField, var
     x = var('x')
     k = NumberField(x**3-x**2+x+1,'a')
     modlabel, numlabel = '82.-5a0+1a2', '5.3.3'
     mod = WebHecke.label2ideal(k, modlabel)
     assert WebHecke.ideal2label(mod) == modlabel
     num = WebHecke.label2number(numlabel)
     assert WebHecke.number2label(num) == numlabel

    def test_Heckemethods(self):
        from sage.all import NumberField, var

        x = var("x")
        k = NumberField(x ** 3 - x ** 2 + x + 1, "a")
        modlabel, numlabel = "82.-5a0+1a2", "5.3.3"
        mod = WebHecke.label2ideal(k, modlabel)
        assert WebHecke.ideal2label(mod) == modlabel
        num = WebHecke.label2number(numlabel)
        assert WebHecke.number2label(num) == numlabel

def plotit(k):
    k = int(k[0])
    # FIXME there could be a filename collission
    fn = tempfile.mktemp(suffix=".png")
    x = var('x')
    p = plot(sin(k * x))
    p.save(filename=fn)
    data = file(fn).read()
    os.remove(fn)
    return data

def eqn_list_to_curve_plot(L):
    xpoly_rng = PolynomialRing(QQ,'x')
    poly_tup = [xpoly_rng(tup) for tup in L]
    f = poly_tup[0]
    h = poly_tup[1]
    g = f+h**2/4
    if len(g.real_roots())==0 and g(0)<0:
        return text("$X(\mathbb{R})=\emptyset$",(1,1),fontsize=50)
    X0 = [real(z[0]) for z in g.base_extend(CC).roots()]+[real(z[0]) for z in g.derivative().base_extend(CC).roots()]
    a,b = inflate_interval(min(X0),max(X0),1.5)
    groots = [a]+g.real_roots()+[b]
    if b-a<1e-7:
        a=-3
        b=3
        groots=[a,b]
    ngints = len(groots)-1
    plotzones = []
    npts = 100
    for j in range(ngints):
        c = groots[j]
        d = groots[j+1]
        if g((c+d)/2)<0:
            continue
        (c,d) = inflate_interval(c,d,1.1)
        s = (d-c)/npts
        u = c
        yvals = []
        for i in range(npts+1):
            v = g(u)
            if v>0:
                v = sqrt(v)
                w = -h(u)/2
                yvals.append(w+v)
                yvals.append(w-v)
            u += s
        (m,M) = inflate_interval(min(yvals),max(yvals),1.2)
        plotzones.append((c,d,m,M))
    x = var('x')
    y = var('y')
    return sum(implicit_plot(y**2 + y*h(x) - f(x), (x,R[0],R[1]),
        (y,R[2],R[3]), aspect_ratio='automatic', plot_points=500) for R in
        plotzones)

 def test_web_latex_ideal_fact(self):
     r"""
     Checking utility: web_latex_ideal_fact
     """
     from sage.all import NumberField
     x = var('x')
     K = NumberField(x**2 - 5, 'a')
     a = K.gen()
     I = K.ideal(2/(5+a)).factor()
     self.assertEqual(web_latex_ideal_fact(I),
                      '\\( \\left(-a\\right)^{-1} \\)')

 def test_pol_to_html(self):
     r"""
     Checking utility: pol_to_html
     """
     x = var('x')
     f1 = x**2 + 2*x + 1
     self.assertEqual(pol_to_html(f1),
                      '<i>x</i><sup>2</sup> + 2<i>x</i> + 1')
     f2 = 'x^3 + 1'
     self.assertEqual(pol_to_html(f2),
                      '<i>x</i><sup>3</sup> + 1')

def EC_R_plot(ainvs, xmin, xmax, ymin, ymax, colour, legend):
    x = var("x")
    y = var("y")
    c = (xmin + xmax) / 2
    d = xmax - xmin
    return implicit_plot(
        y ** 2 + ainvs[0] * x * y + ainvs[2] * y - x ** 3 - ainvs[1] * x ** 2 - ainvs[3] * x - ainvs[4],
        (x, xmin, xmax),
        (y, ymin, ymax),
        plot_points=1000,
        aspect_ratio="automatic",
        color=colour,
    ) + plot(
        0,
        xmin=c - 1e-5 * d,
        xmax=c + 1e-5 * d,
        ymin=ymin,
        ymax=ymax,
        aspect_ratio="automatic",
        color=colour,
        legend_label=legend,
    )  # Add an extra plot outside the visible frame because implicit plots are buggy: their legend does not show (http://trac.sagemath.org/ticket/15903)