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_normalize_4(t):
# Test with non-zero normalized eigenvalues
x, e = SR.var("x eps")
M = matrix([[1 / x / 2, 0], [0, 0]])
with t.assertRaises(FuchsiaError):
N, T = normalize(M, x, e)
def test_normalize_3(t):
# Test with non-zero normalized eigenvalues
x = SR.var("x")
e = SR.var("epsilon")
M = matrix([[(1 - e) / x, 0], [0, (1 + e) / 3 / x]])
with t.assertRaises(FuchsiaError):
N, T = normalize(M, x, e)
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 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())