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 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_transform_2(t):
# transform(transform(M, x, I), x, I^-1) == M
x = SR.var("x")
M = randpolym(x, 2)
T = randpolym(x, 2)
invT = T.inverse()
M1 = transform(M, x, T)
M2 = transform(M1, x, invT)
t.assertEqual(M2.simplify_rational(), M)
def assert_fuchsify_by_blocks_works(test, m, b, x, eps):
test.assertFalse(is_fuchsian(m, x))
mt, t = fuchsify_off_diagonal_blocks(m, x, eps, b=b)
test.assertTrue(
(mt - transform(m, x, t)).simplify_rational().is_zero())
test.assertTrue(is_fuchsian(mt, x))
def test_factorize_1(t):
x = SR.var("x")
e = SR.var("epsilon")
M = matrix([[1 / x, 0, 0], [0, 2 / x, 0], [0, 0, 3 / x]]) * e
M = transform(M, x, matrix([[1, 1, 0], [0, 1, 0], [1 + 2 * e, 0, e]]))
F, T = factorize(M, x, e)
F = F.simplify_rational()
for f in F.list():
t.assertEqual(limit_fixed(f, e, 0), 0)
def test_balance_transform_1(t):
x = SR.var("x")
M = randpolym(x, 2)
P = matrix([[1, 1], [0, 0]])
x1 = randint(-10, 10)
x2 = randint(20, 30)
b1 = balance(P, x1, x2, x)
M1 = balance_transform(M, P, x1, x2, x)
M2 = transform(M, x, balance(P, x1, x2, x))
t.assertEqual(M1.simplify_rational(), M2.simplify_rational())
M1 = balance_transform(M, P, x1, oo, x)
M2 = transform(M, x, balance(P, x1, oo, x))
t.assertEqual(M1.simplify_rational(), M2.simplify_rational())
M1 = balance_transform(M, P, oo, x2, x)
M2 = transform(M, x, balance(P, oo, x2, x))
t.assertEqual(M1.simplify_rational(), M2.simplify_rational())
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 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 test_block_triangular_form_4(t):
M = matrix([
[1, 2, 3, 0, 0, 0],
[4, 5, 6, 0, 0, 0],
[7, 8, 9, 0, 0, 0],
[2, 0, 0, 1, 2, 0],
[0, 2, 0, 3, 4, 0],
[0, 0, 2, 0, 0, 1]
])
x = SR.var("dummy")
T = matrix.identity(6)[random.sample(xrange(6), 6),:]
M = transform(M, x, T)
MM, T, B = block_triangular_form(M)
t.assertEqual(MM, transform(M, x, T))
t.assertEqual(sorted(s for o, s in B), [1, 2, 3])
for o, s in B:
for i in xrange(s):
for j in xrange(s):
MM[o + i, o + j] = 0
for i in xrange(6):
for j in xrange(i):
MM[i, j] = 0
t.assertEqual(MM, matrix(6))
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 test_block_triangular_form_3(t):
m = matrix([
[1, 0, 1, 0],
[0, 1, 0, 1],
[1, 0, 1, 0],
[0, 0, 0, 1]
])
mt, tt, b = block_triangular_form(m)
x = SR.var("dummy")
t.assertEqual(mt, transform(m, x, tt))
t.assertEqual(matrix([
[1, 1, 0, 0],
[1, 1, 0, 0],
[0, 0, 1, 0],
[0, 0, 1, 1]
]), mt)
def test_simplify_by_jordanification(t):
x = SR.var("x")
M = matrix(
[[4 / (x + 1), -1 / (6 * x * (x + 1)), -1 / (3 * x * (x + 1))],
[
6 * (13 * x + 6) / (x * (x + 1)),
-5 * (x + 3) / (3 * x * (x + 1)),
2 * (x - 6) / (3 * x * (x + 1))
],
[
-63 * (x - 1) / (x * (x + 1)),
(5 * x - 9) / (6 * x * (x + 1)), -(x - 18) / (3 * x * (x + 1))
]]).simplify_rational()
MM, T = simplify_by_jordanification(M, x)
MM = MM.simplify_rational()
t.assertEqual(MM, transform(M, x, T).simplify_rational())
t.assertLess(matrix_complexity(MM), matrix_complexity(M))
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 assertIsTriangular(t, M1, M2, x, T, B):
t.assertEqual(M2, transform(M1, x, T))
def assertTransformation(t, m1_path, x_name, t_path, m2_path):
M1 = fuchsia.import_matrix_from_file(m1_path)
T = fuchsia.import_matrix_from_file(t_path)
M2 = fuchsia.import_matrix_from_file(m2_path)
t.assertEqual(M2.simplify_rational(),
fuchsia.transform(M1, SR.var(x_name), T).simplify_rational())
def test_transform_1(t):
# transform(M, x, I) == M
x = SR.var("x")
M = randpolym(x, 3)
MM = transform(M, x, identity_matrix(M.nrows()))
t.assertEqual(MM, M)
def test_pap_3_52_slow(t):
x, eps = SR.var("x eps")
M = import_matrix_from_file("test/data/pap_3_52.mtx")
N, T = normalize(M, x, eps)
N = N.simplify_rational()
t.assertEqual(N, transform(M, x, T).simplify_rational())
