def _test_evecs_from_schur(dtype):
rand = _Random()
a = rand.randmat(5, 5, dtype)
t, q, ev = schur(a)
vl, vr = evecs_from_schur(t, q, select=None, left=True, right=True)
assert_array_almost_equal(dtype, np.dot(vr, np.dot(np.diag(ev),
np.linalg.inv(vr))), a)
assert_array_almost_equal(dtype, np.dot(np.linalg.inv(np.conj(vl.T)),
np.dot(np.diag(ev), np.conj(vl.T))),
a)
select = np.array([True, True, False, False, False], dtype=bool)
vl, vr = evecs_from_schur(t, q, select, left=True, right=True)
assert_equal(vr.shape[1], 2)
assert_equal(vl.shape[1], 2)
assert_array_almost_equal(dtype, np.dot(a, vr),
np.dot(vr, np.diag(ev[select])))
assert_array_almost_equal(dtype, np.dot(vl.T.conj(), a),
np.dot(np.diag(ev[select]), vl.T.conj()))
vl, vr = evecs_from_schur(t, q, lambda i: i<2, left=True, right=True)
assert_equal(vr.shape[1], 2)
assert_equal(vl.shape[1], 2)
assert_array_almost_equal(dtype, np.dot(a, vr),
np.dot(vr, np.diag(ev[select])))
assert_array_almost_equal(dtype, np.dot(vl.T.conj(), a),
np.dot(np.diag(ev[select]), vl.T.conj()))
def _test_schur(dtype):
rand = _Random()
a = rand.randmat(5, 5, dtype)
t, q = schur(a)[:2]
assert_array_almost_equal(dtype, np.dot(np.dot(q, t), np.conj(q.T)), a)
def _test_gen_schur(dtype):
rand = _Random()
a = rand.randmat(5, 5, dtype)
b = rand.randmat(5, 5, dtype)
s, t, q, z = gen_schur(a, b)[:4]
assert_array_almost_equal(dtype, np.dot(np.dot(q, s), z.T.conj()), a)
assert_array_almost_equal(dtype, np.dot(np.dot(q, t), z.T.conj()), b)
def _test_convert_r2c_schur(dtype):
rand = _Random()
a = rand.randmat(10, 10, dtype)
t, q = schur(a)[:2]
t2, q2 = convert_r2c_schur(t, q)
assert_array_almost_equal(dtype, np.dot(np.dot(q, t), np.conj(q.T)), a)
assert_array_almost_equal(dtype, np.dot(np.dot(q2, t2), np.conj(q2.T)),
a)
def _test_evecs_from_gen_schur(dtype):
rand = _Random()
a = rand.randmat(5, 5, dtype)
b = rand.randmat(5, 5, dtype)
s, t, q, z, alpha, beta = gen_schur(a, b)
vl, vr = evecs_from_gen_schur(s, t, q, z , select=None,
left=True, right=True)
assert_array_almost_equal(dtype, np.dot(a, np.dot(vr, np.diag(beta))),
np.dot(b, np.dot(vr, np.diag(alpha))))
assert_array_almost_equal(dtype,
np.dot(np.dot(np.diag(beta), vl.T.conj()),
a),
np.dot(np.dot(np.diag(alpha), vl.T.conj()),
b))
select = np.array([True, True, False, False, False], dtype=bool)
vl, vr = evecs_from_gen_schur(s, t, q, z, select,
left=True, right=True)
assert_equal(vr.shape[1], 2)
assert_equal(vl.shape[1], 2)
assert_array_almost_equal(dtype,
np.dot(a, np.dot(vr,
np.diag(beta[select]))),
np.dot(b, np.dot(vr,
np.diag(alpha[select]))))
assert_array_almost_equal(dtype,
np.dot(np.dot(np.diag(beta[select]),
vl.T.conj()),
a),
np.dot(np.dot(np.diag(alpha[select]),
vl.T.conj()),
b))
vl, vr = evecs_from_gen_schur(s, t, q, z, lambda i: i<2, left=True,
right=True)
assert_equal(vr.shape[1], 2)
assert_equal(vl.shape[1], 2)
assert_array_almost_equal(dtype,
np.dot(a, np.dot(vr,
np.diag(beta[select]))),
np.dot(b, np.dot(vr,
np.diag(alpha[select]))))
assert_array_almost_equal(dtype,
np.dot(np.dot(np.diag(beta[select]),
vl.T.conj()),
a),
np.dot(np.dot(np.diag(alpha[select]),
vl.T.conj()),
b))
def _test_lu(dtype):
rand = _Random()
a = rand.randmat(4, 4, dtype)
bmat = rand.randmat(4, 4, dtype)
bvec = rand.randvec(4, dtype)
lu = lu_factor(a)
xmat = lu_solve(lu, bmat)
xvec = lu_solve(lu, bvec)
assert_array_almost_equal(dtype, np.dot(a, xmat), bmat)
assert_array_almost_equal(dtype, np.dot(a, xvec), bvec)
def _test_gen_eig(dtype):
rand = _Random()
a = rand.randmat(4, 4, dtype)
b = rand.randmat(4, 4, dtype)
(alpha, beta, vl, vr) = gen_eig(a, b, True, True)
assert_array_almost_equal(dtype, np.dot(np.dot(a, vr), beta),
np.dot(np.dot(b, vr), alpha))
assert_array_almost_equal(dtype,
np.dot(beta, np.dot(np.conj(vl.T), a)),
np.dot(alpha, np.dot(np.conj(vl.T), b)))
def _test_convert_r2c_gen_schur(dtype):
rand = _Random()
a = rand.randmat(10, 10, dtype)
b = rand.randmat(10, 10, dtype)
s, t, q, z = gen_schur(a, b)[:4]
s2, t2, q2, z2 = convert_r2c_gen_schur(s, t, q, z)
assert_array_almost_equal(dtype, np.dot(np.dot(q, s), z.T.conj()), a)
assert_array_almost_equal(dtype, np.dot(np.dot(q, t), z.T.conj()), b)
assert_array_almost_equal(dtype, np.dot(np.dot(q2, s2), z2.T.conj()),
a)
assert_array_almost_equal(dtype, np.dot(np.dot(q2, t2), z2.T.conj()),
b)
def _test_order_gen_schur(dtype):
rand = _Random()
a = rand.randmat(10, 10, dtype)
b = rand.randmat(10, 10, dtype)
s, t, q, z, alpha, beta = gen_schur(a, b)
s2, t2, q2, z2, alpha2, beta2 = order_gen_schur(lambda i: i>2 and i<7,
s, t, q, z)
assert_array_almost_equal(dtype, np.dot(np.dot(q, s), z.T.conj()), a)
assert_array_almost_equal(dtype, np.dot(np.dot(q, t), z.T.conj()), b)
assert_array_almost_equal(dtype, np.dot(np.dot(q2, s2), z2.T.conj()),
a)
assert_array_almost_equal(dtype, np.dot(np.dot(q2, t2), z2.T.conj()),
b)
#Sorting here is a bit tricky: For real matrices we expect
#for complex conjugated pairs identical real parts - however
#that seems messed up (only an error on the order of machine precision)
#in the division. The solution here is to sort and compare the real
#and imaginary parts separately. The only error that would not be
#catched in this comparison is if the real and imaginary parts would
#be assembled differently in the two arrays - an error that is highly
#unlikely.
assert_array_almost_equal(dtype, np.sort((alpha/beta).real),
np.sort((alpha2/beta2).real))
assert_array_almost_equal(dtype, np.sort((alpha/beta).imag),
np.sort((alpha2/beta2).imag))
assert_array_almost_equal(dtype, np.sort((alpha[3:7]/beta[3:7]).real),
np.sort((alpha2[:4]/beta2[:4]).real))
assert_array_almost_equal(dtype, np.sort((alpha[3:7]/beta[3:7]).imag),
np.sort((alpha2[:4]/beta2[:4]).imag))
sel = [False, False, 0, True, True, True, 1, False, False, False]
s3, t3, q3, z3 = order_gen_schur(sel, s, t, q, z)[:4]
assert_array_almost_equal(dtype, np.dot(np.dot(q3, s3), z3.T.conj()),
a)
assert_array_almost_equal(dtype, np.dot(np.dot(q3, t3), z3.T.conj()),
b)
assert_array_almost_equal(dtype, s2, s3)
assert_array_almost_equal(dtype, t2, t3)
assert_array_almost_equal(dtype, q2, q3)
assert_array_almost_equal(dtype, z2, z3)
def _test_order_schur(dtype):
rand = _Random()
a = rand.randmat(10, 10, dtype)
t, q, ev = schur(a)
t2, q2, ev2 = order_schur(lambda i: i>2 and i<7, t, q)
assert_array_almost_equal(dtype, np.dot(np.dot(q, t), np.conj(q.T)), a)
assert_array_almost_equal(dtype, np.dot(np.dot(q2, t2), np.conj(q2.T)),
a)
assert_array_almost_equal(dtype, np.sort(ev), np.sort(ev2))
assert_array_almost_equal(dtype, np.sort(ev[3:7]), np.sort(ev2[:4]))
sel = [False, False, 0, True, True, True, 1, False, False, False]
t3, q3 = order_schur(sel, t, q)[:2]
assert_array_almost_equal(dtype, np.dot(np.dot(q3, t3), np.conj(q3.T)),
a)
assert_array_almost_equal(dtype, t2, t3)
assert_array_almost_equal(dtype, q2, q3)
def _test_lu_with_dense(dtype):
rand = _Random()
a = rand.randmat(5, 5, dtype)
bmat = rand.randmat(5, 5, dtype)
bvec = rand.randvec(5, dtype)
ctx = MUMPSContext()
ctx.factor(sp.coo_matrix(a))
xvec = ctx.solve(bvec)
xmat = ctx.solve(bmat)
assert_array_almost_equal(dtype, np.dot(a, xmat), bmat)
assert_array_almost_equal(dtype, np.dot(a, xvec), bvec)
# now "sparse" right hand side
xvec = ctx.solve(sp.csc_matrix(bvec.reshape(5,1)))
xmat = ctx.solve(sp.csc_matrix(bmat))
assert_array_almost_equal(dtype, np.dot(a, xmat), bmat)
assert_array_almost_equal(dtype, np.dot(a, xvec),
bvec.reshape(5,1))
def _test_rcond_from_lu(dtype):
rand = _Random()
a = rand.randmat(10, 10, dtype)
norm1_a = np.linalg.norm(a, 1)
normI_a = np.linalg.norm(a, np.inf)
lu = lu_factor(a)
rcond1 = rcond_from_lu(lu, norm1_a, '1')
rcondI = rcond_from_lu(lu, normI_a, 'I')
err1 = abs(rcond1 -
1/(norm1_a * np.linalg.norm(np.linalg.inv(a), 1)))
errI = abs(rcondI -
1/(normI_a * np.linalg.norm(np.linalg.inv(a), np.inf)))
#rcond_from_lu returns an estimate for the reciprocal
#condition number only; hence we shouldn't be too strict about
#the assertions here
#Note: in my experience the estimate is excellent for somewhat
#larger matrices
assert_true(err1/rcond1 < 0.1)
assert_true(errI/rcondI < 0.1)
def _test_schur_complement_with_dense(dtype):
rand = _Random()
a = rand.randmat(10, 10, dtype)
s = schur_complement(sp.coo_matrix(a), range(3))
assert_array_almost_equal(dtype, np.linalg.inv(s),
np.linalg.inv(a)[:3, :3])