def test_lstsq_returns_expected_values(self, arrays):
m_ss, m_sb, m_bb, m_bs = arrays
hy.assume(hn.wide(m_sb))
cond_bs = np.linalg.cond(m_bs).max()
cond_sb = np.linalg.cond(m_sb).max()
# overconstrained
x_sb = gfl.lstsq(m_bs, m_bb)
m_bst = la.dagger(m_bs)
# with self.subTest(msg='lstsq(over)'):
self.assertArrayAllClose(m_bst @ m_bs @ x_sb,
m_bst @ m_bb,
cond=cond_bs)
x_bs = gfl.rlstsq(m_bb, m_sb)
m_sbt = la.dagger(m_sb)
# with self.subTest(msg='rlstsq(over)'):
self.assertArrayAllClose(x_bs @ m_sb @ m_sbt,
m_bb @ m_sbt,
cond=cond_sb)
# underconstrained
x_bs = gfl.lstsq(m_sb, m_ss)
# with self.subTest(msg='lstsq(under)'):
self.assertArrayAllClose(m_sb @ x_bs, m_ss, cond=cond_sb)
x_sb = gfl.rlstsq(m_ss, m_bs)
# with self.subTest(msg='rlstsq(under)'):
self.assertArrayAllClose(x_sb @ m_bs, m_ss, cond=cond_bs)
def test_lq_complete_returns_expected_values(self, m_sb):
hy.assume(hn.wide(m_sb))
hy.assume(hn.all_well_behaved(m_sb))
cond = np.linalg.cond(m_sb).max()
left, unitary = gfl.lq_n(m_sb)
wide = left @ unitary
eye = unitary @ la.dagger(unitary)
eyet = la.dagger(unitary) @ unitary
id_b = np.identity(m_sb.shape[-1], m_sb.dtype)
# with self.subTest(msg='lq'):
self.assertArrayAllClose(wide, m_sb, cond=cond)
# with self.subTest(msg='Q Q^T'):
self.assertArrayAllClose(id_b, eye, cond=cond)
# with self.subTest(msg='Q^T Q'):
self.assertArrayAllClose(id_b, eyet, cond=cond)
def test_lq_returns_expected_values_with_tall(self, m_bs):
hy.assume(hn.tall(m_bs))
hy.assume(hn.all_well_behaved(m_bs))
cond = np.linalg.cond(m_bs).max()
left, unitary = gfl.lq_n(m_bs)
tall = left @ unitary
eye = unitary @ la.dagger(unitary)
eyet = la.dagger(unitary) @ unitary
id_s = np.identity(m_bs.shape[-1], m_bs.dtype)
# with self.subTest(msg='lq'):
self.assertArrayAllClose(tall, m_bs, cond=cond)
# with self.subTest(msg='Q Q^T'):
self.assertArrayAllClose(id_s, eye, cond=cond)
# with self.subTest(msg='Q^T Q'):
self.assertArrayAllClose(id_s, eyet, cond=cond)
def test_qr_complete_returns_expected_values(self, m_bs):
hy.assume(hn.tall(m_bs))
hy.assume(hn.all_well_behaved(m_bs))
cond = np.linalg.cond(m_bs).max()
unitary, right = gfl.qr_m(m_bs)
tall = unitary @ right
eye = la.dagger(unitary) @ unitary
eyet = unitary @ la.dagger(unitary)
id_b = np.identity(m_bs.shape[-2], m_bs.dtype)
# with self.subTest(msg='qr'):
self.assertArrayAllClose(tall, m_bs, cond=cond)
# with self.subTest(msg='Q^T Q'):
self.assertArrayAllClose(id_b, eye, cond=cond)
# with self.subTest(msg='Q Q^T'):
self.assertArrayAllClose(id_b, eyet, cond=cond)
def test_qr_returns_expected_values_with_wide(self, m_sb):
hy.assume(hn.wide(m_sb))
hy.assume(hn.all_well_behaved(m_sb))
cond = np.linalg.cond(m_sb).max()
unitary, right = gfl.qr_m(m_sb)
wide = unitary @ right
eye = la.dagger(unitary) @ unitary
eyet = unitary @ la.dagger(unitary)
id_s = np.identity(m_sb.shape[-2], m_sb.dtype)
# with self.subTest(msg='qr'):
self.assertArrayAllClose(wide, m_sb, cond=cond)
# with self.subTest(msg='Q^T Q'):
self.assertArrayAllClose(id_s, eye, cond=cond)
# with self.subTest(msg='Q Q^T'):
self.assertArrayAllClose(id_s, eyet, cond=cond)
def test_rlstsq_qr_returns_expected_values_with_tall(self, arrays, fun):
m_ss, m_sb, m_bb, m_bs = arrays
hy.assume(hn.wide(m_sb))
hy.assume(hn.all_well_behaved(m_bs))
cond = np.linalg.cond(m_bs).max()
# underconstrained
x0_sb = gfl.rlstsq(m_ss, m_bs)
# underconstrained
x_sb, x_f, tau = fun(m_ss, m_bs)
# with self.subTest('rlstsq_qr(over,' + suffix):
self.assertArrayAllClose(x_sb, x0_sb, cond=cond)
# underconstrained
xx_sb = gfl.rqr_lstsq(m_ss, x_f, tau)
# with self.subTest('rqr_rlstsq(over,' + suffix):
self.assertArrayAllClose(xx_sb, x0_sb, cond=cond)
# overconstrained
y_sb = gfl.qr_lstsq(x_f, tau, m_bb)
m_bst = la.dagger(m_bs)
# with self.subTest('qr_rlstsq(under,' + suffix):
self.assertArrayAllClose(m_bst @ m_bs @ y_sb, m_bst @ m_bb, cond=cond)
def test_qr_rawn_returns_expected_values(self, m_bs):
hy.assume(hn.tall(m_bs))
hy.assume(hn.all_well_behaved(m_bs))
cond = np.linalg.cond(m_bs).max()
rrr = gfl.qr_n(m_bs)[1]
num = rrr.shape[-1]
ht_bs, tau = gfl.qr_rawn(m_bs)
h_bs = la.transpose(ht_bs)
vecs = np.tril(h_bs, -1)
vecs[(..., ) + np.diag_indices(num)] = 1
vnorm = gfb.norm(la.row(tau) * vecs, axis=-2)**2
right = np.triu(h_bs)
# with self.subTest(msg='raw_n'):
self.assertArrayAllClose(right[..., :num, :], rrr, cond=cond)
self.assertArrayAllClose(vnorm, 2 * tau.real, cond=cond)
for k in range(num):
vvv = vecs[..., num - k - 1:num - k]
ttt = la.scalar(tau[..., -k - 1])
right -= ttt * vvv * (la.dagger(vvv) @ right)
# with self.subTest(msg='h_n'):
self.assertArrayAllClose(right, m_bs, cond=cond)
def test_lq_rawn_returns_expected_values(self, m_bs):
hy.assume(hn.tall(m_bs))
hy.assume(hn.all_well_behaved(m_bs))
cond = np.linalg.cond(m_bs).max()
llo = gfl.lq_n(m_bs)[0]
num = llo.shape[-1]
ht_bs, tau = gfl.lq_rawn(m_bs)
h_bs = la.transpose(ht_bs)
vecs = np.triu(h_bs, 1)
vecs[(..., ) + np.diag_indices(num)] = 1
vnorm = gfb.norm(la.col(tau) * vecs[..., :num, :], axis=-1)**2
left = np.tril(h_bs)
# with self.subTest(msg='raw_n'):
self.assertArrayAllClose(left, llo, cond=cond)
# with self.subTest(msg='tau_n'):
self.assertArrayAllClose(vnorm, 2 * tau.real, cond=cond)
for k in range(num):
vvv = vecs[..., num - k - 1:num - k, :]
ttt = la.scalar(tau[..., -k - 1])
left -= ttt.conj() * (left @ la.dagger(vvv)) * vvv
# with self.subTest(msg='h_n'):
self.assertArrayAllClose(left, m_bs, cond=cond)