def assertReductionWorks(t, filename):
M = import_matrix_from_file(filename)
x, eps = SR.var("x eps")
t.assertIn(x, M.variables())
M_pranks = singularities(M, x).values()
t.assertNotEqual(M_pranks, [0] * len(M_pranks))
#1 Fuchsify
m, t1 = simplify_by_factorization(M, x)
Mf, t2 = fuchsify(m, x)
Tf = t1 * t2
t.assertTrue((Mf - transform(M, x, Tf)).simplify_rational().is_zero())
Mf_pranks = singularities(Mf, x).values()
t.assertEqual(Mf_pranks, [0] * len(Mf_pranks))
#2 Normalize
t.assertFalse(is_normalized(Mf, x, eps))
m, t1 = simplify_by_factorization(Mf, x)
Mn, t2 = normalize(m, x, eps)
Tn = t1 * t2
t.assertTrue((Mn - transform(Mf, x, Tn)).simplify_rational().is_zero())
t.assertTrue(is_normalized(Mn, x, eps))
#3 Factorize
t.assertIn(eps, Mn.variables())
m, t1 = simplify_by_factorization(Mn, x)
Mc, t2 = factorize(m, x, eps, seed=3)
Tc = t1 * t2
t.assertTrue((Mc - transform(Mn, x, Tc)).simplify_rational().is_zero())
t.assertNotIn(eps, (Mc / eps).simplify_rational().variables())
def test_reduce_at_one_point_1(t):
x = SR.var("x")
M0 = matrix([[1 / x, 4, 0, 5], [0, 2 / x, 0, 0], [0, 0, 3 / x, 6],
[0, 0, 0, 4 / x]])
u = matrix([[0, Rational((3, 5)),
Rational((4, 5)), 0],
[Rational((5, 13)), 0, 0,
Rational((12, 13))]])
M1 = transform(M0, x, balance(u.transpose() * u, 0, 1, x))
M1 = M1.simplify_rational()
u = matrix([[8, 0, 15, 0]]) / 17
M2 = transform(M1, x, balance(u.transpose() * u, 0, 2, x))
M2 = M2.simplify_rational()
M2_sing = singularities(M2, x)
t.assertIn(0, M2_sing)
t.assertEqual(M2_sing[0], 2)
M3, T23 = reduce_at_one_point(M2, x, 0, 2)
M3 = M3.simplify_rational()
t.assertEqual(M3, transform(M2, x, T23).simplify_rational())
M3_sing = singularities(M3, x)
t.assertIn(0, M3_sing)
t.assertEqual(M3_sing[0], 1)
M4, T34 = reduce_at_one_point(M3, x, 0, 1)
M4 = M4.simplify_rational()
t.assertEqual(M4, transform(M3, x, T34).simplify_rational())
M4_sing = singularities(M4, x)
t.assertIn(0, M4_sing)
t.assertEqual(M4_sing[0], 0)
def assertIsReduced(t, m_path, x_name, eps_name):
M = fuchsia.import_matrix_from_file(m_path)
x = SR.var(x_name)
eps = SR.var(eps_name)
pranks = fuchsia.singularities(M, x).values()
t.assertEqual(pranks, [0]*len(pranks))
t.assertTrue(eps not in (M/eps).simplify_rational().variables())
def test_normalize_5(t):
# An unnormalizable example by A. A. Bolibrukh
x, e = SR.var("x eps")
b = import_matrix_from_file("test/data/bolibrukh.mtx")
f, ft = fuchsify(b, x)
f_pranks = singularities(f, x).values()
t.assertEqual(f_pranks, [0] * len(f_pranks))
with t.assertRaises(FuchsiaError):
n, nt = normalize(f, x, e)
def test_normalize_1(t):
# Test with apparent singularities at 0 and oo, but not at 1.
x = SR.var("x")
M = matrix([[1 / x, 5 / (x - 1), 0, 6 / (x - 1)], [0, 2 / x, 0, 0],
[0, 0, 3 / x, 7 / (x - 1)], [6 / (x - 1), 0, 0, 1 / x]])
N, T = normalize(M, x, SR.var("epsilon"))
N = N.simplify_rational()
t.assertEqual(N, transform(M, x, T).simplify_rational())
for point, prank in singularities(N, x).iteritems():
R = matrix_c0(N, x, point, prank)
evlist = R.eigenvalues()
t.assertEqual(evlist, [0] * len(evlist))
def assertReductionWorks(test, filename, fuchsian=False):
m = import_matrix_from_file(filename)
x, eps = SR.var("x eps")
test.assertIn(x, m.variables())
if not fuchsian:
m_pranks = singularities(m, x).values()
test.assertNotEqual(m_pranks, [0]*len(m_pranks))
mt, t = epsilon_form(m, x, eps)
test.assertTrue((mt-transform(m, x, t)).simplify_rational().is_zero())
test.assertTrue(is_fuchsian(mt, x))
test.assertTrue(is_normalized(mt, x, eps))
test.assertNotIn(eps, (mt/eps).simplify_rational().variables())
def assertNormalizeBlocksWorks(test, filename):
x, eps = SR.var("x eps")
m = import_matrix_from_file(filename)
test.assertIn(x, m.variables())
test.assertIn(eps, m.variables())
test.assertFalse(is_normalized(m, x, eps))
m_pranks = singularities(m, x).values()
test.assertNotEqual(m_pranks, [0] * len(m_pranks))
m, t, b = block_triangular_form(m)
mt, tt = reduce_diagonal_blocks(m, x, eps, b=b)
t = t * tt
test.assertTrue(
(mt - transform(m, x, t)).simplify_rational().is_zero())
test.assertTrue(are_diagonal_blocks_reduced(mt, b, x, eps))
def test_fuchsify_2(t):
x = SR.var("x")
M = matrix([[0, 1 / x / (x - 1), 0, 0], [0, 0, 0, 0], [0, 0, 0, 0],
[0, 0, 0, 0]])
u = matrix([[6, 3, 2, 0]]) / 7
P = u.transpose() * u
M = balance_transform(M, P, 1, 0, x).simplify_rational()
M = balance_transform(M, P, 1, 0, x).simplify_rational()
M = balance_transform(M, P, 1, 0, x).simplify_rational()
M = balance_transform(M, P, 1, 0, x).simplify_rational()
M = balance_transform(M, P, 1, 0, x).simplify_rational()
MM, T = fuchsify(M, x)
MM = MM.simplify_rational()
t.assertEqual(MM, transform(M, x, T).simplify_rational())
pranks = singularities(MM, x).values()
t.assertEqual(pranks, [0] * len(pranks))
def test_fuchsify_1(t):
x = SR.var("x")
M = matrix([[1 / x, 5, 0, 6], [0, 2 / x, 0, 0], [0, 0, 3 / x, 7],
[0, 0, 0, 4 / x]])
u = matrix([[0, Rational((3, 5)),
Rational((4, 5)), 0],
[Rational((5, 13)), 0, 0,
Rational((12, 13))]])
M = transform(M, x, balance(u.transpose() * u, 0, 1, x))
M = M.simplify_rational()
u = matrix([[8, 0, 15, 0]]) / 17
M = transform(M, x, balance(u.transpose() * u, 0, 2, x))
M = M.simplify_rational()
Mx, T = fuchsify(M, x)
Mx = Mx.simplify_rational()
t.assertEqual(Mx, transform(M, x, T).simplify_rational())
pranks = singularities(Mx, x).values()
t.assertEqual(pranks, [0] * len(pranks))
def assertIsFuchsian(t, m_path, x_name):
M = fuchsia.import_matrix_from_file(m_path)
x = SR.var(x_name)
pranks = fuchsia.singularities(M, x).values()
t.assertEqual(pranks, [0]*len(pranks))