def _verifySolve(self, x, y, batch_dims=None):
for np_type in [np.float32, np.float64, np.complex64, np.complex128]:
if np_type == np.float32 or np_type == np.complex64:
tol = 1e-5
else:
tol = 1e-12
for adjoint in False, True:
if np_type is [np.float32, np.float64]:
a = x.real().astype(np_type)
b = y.real().astype(np_type)
else:
a = x.astype(np_type)
b = y.astype(np_type)
a_np = np.conj(np.transpose(a)) if adjoint else a
if batch_dims is not None:
a = np.tile(a, batch_dims + [1, 1])
a_np = np.tile(a_np, batch_dims + [1, 1])
b = np.tile(b, batch_dims + [1, 1])
np_ans = np.linalg.solve(a_np, b)
for use_placeholder in False, True:
with self.test_session(use_gpu=True) as sess:
if use_placeholder:
a_ph = array_ops.placeholder(dtypes.as_dtype(np_type))
b_ph = array_ops.placeholder(dtypes.as_dtype(np_type))
tf_ans = linalg_ops.matrix_solve(a_ph, b_ph, adjoint=adjoint)
out = sess.run(tf_ans, {a_ph: a, b_ph: b})
else:
tf_ans = linalg_ops.matrix_solve(a, b, adjoint=adjoint)
out = tf_ans.eval()
self.assertEqual(tf_ans.get_shape(), out.shape)
self.assertEqual(np_ans.shape, out.shape)
self.assertAllClose(np_ans, out, atol=tol, rtol=tol)
def _verifySolve(self, x, y, batch_dims=None):
for np_type in self.float_types & {np.float32, np.float64}:
if np_type == np.float32:
tol = 1e-4
else:
tol = 1e-12
for adjoint in False, True:
a = x.astype(np_type)
b = y.astype(np_type)
a_np = np.transpose(a) if adjoint else a
if batch_dims is not None:
a = np.tile(a, batch_dims + [1, 1])
a_np = np.tile(a_np, batch_dims + [1, 1])
b = np.tile(b, batch_dims + [1, 1])
np_ans = np.linalg.solve(a_np, b)
for use_placeholder in False, True:
with self.session() as sess:
if use_placeholder:
with self.test_scope():
a_ph = array_ops.placeholder(
dtypes.as_dtype(np_type))
b_ph = array_ops.placeholder(
dtypes.as_dtype(np_type))
tf_ans = linalg_ops.matrix_solve(
a_ph, b_ph, adjoint=adjoint)
out = sess.run(tf_ans, {a_ph: a, b_ph: b})
else:
with self.test_scope():
tf_ans = linalg_ops.matrix_solve(
a, b, adjoint=adjoint)
out = sess.run(tf_ans)
self.assertEqual(tf_ans.get_shape(), out.shape)
self.assertEqual(np_ans.shape, out.shape)
self.assertAllClose(np_ans, out, atol=tol, rtol=tol)
def testNonSquareMatrix(self):
# When the solve of a non-square matrix is attempted we should return
# an error
with self.test_session():
with self.assertRaises(ValueError):
matrix = constant_op.constant([[1., 2., 3.], [3., 4., 5.]])
linalg_ops.matrix_solve(matrix, matrix)
def testWrongDimensions(self):
# The matrix and right-hand sides should have the same number of rows.
with self.test_session():
matrix = constant_op.constant([[1., 0.], [0., 1.]])
rhs = constant_op.constant([[1., 0.]])
with self.assertRaises(ValueError):
linalg_ops.matrix_solve(matrix, rhs)
def testWrongDimensions(self):
# The matrix and right-hand sides should have the same number of rows.
with self.test_session():
matrix = constant_op.constant([[1., 0.], [0., 1.]])
rhs = constant_op.constant([[1., 0.]])
with self.assertRaises(ValueError):
linalg_ops.matrix_solve(matrix, rhs)
def _verifySolve(self, x, y, batch_dims=None):
for np_type in [np.float32, np.float64, np.complex64, np.complex128]:
if np_type == np.float32 or np_type == np.complex64:
tol = 1e-5
else:
tol = 1e-12
for adjoint in False, True:
if np_type is [np.float32, np.float64]:
a = x.real().astype(np_type)
b = y.real().astype(np_type)
else:
a = x.astype(np_type)
b = y.astype(np_type)
a_np = np.conj(np.transpose(a)) if adjoint else a
if batch_dims is not None:
a = np.tile(a, batch_dims + [1, 1])
a_np = np.tile(a_np, batch_dims + [1, 1])
b = np.tile(b, batch_dims + [1, 1])
np_ans = np.linalg.solve(a_np, b)
for use_placeholder in False, True:
with self.test_session(use_gpu=True) as sess:
if use_placeholder:
a_ph = array_ops.placeholder(dtypes.as_dtype(np_type))
b_ph = array_ops.placeholder(dtypes.as_dtype(np_type))
tf_ans = linalg_ops.matrix_solve(a_ph, b_ph, adjoint=adjoint)
out = sess.run(tf_ans, {a_ph: a, b_ph: b})
else:
tf_ans = linalg_ops.matrix_solve(a, b, adjoint=adjoint)
out = tf_ans.eval()
self.assertEqual(tf_ans.get_shape(), out.shape)
self.assertEqual(np_ans.shape, out.shape)
self.assertAllClose(np_ans, out, atol=tol, rtol=tol)
def testNonSquareMatrix(self):
# When the solve of a non-square matrix is attempted we should return
# an error
with self.test_session():
with self.assertRaises(ValueError):
matrix = constant_op.constant([[1., 2., 3.], [3., 4., 5.]])
linalg_ops.matrix_solve(matrix, matrix)
def testNotInvertible(self):
# The input should be invertible.
with self.test_session():
with self.assertRaisesOpError("Input matrix is not invertible."):
# All rows of the matrix below add to zero
matrix = constant_op.constant([[1., 0., -1.], [-1., 1., 0.],
[0., -1., 1.]])
linalg_ops.matrix_solve(matrix, matrix).eval()
def testNotInvertible(self):
# The input should be invertible.
with self.test_session():
with self.assertRaisesOpError("Input matrix is not invertible."):
# All rows of the matrix below add to zero
matrix = constant_op.constant([[1., 0., -1.], [-1., 1., 0.],
[0., -1., 1.]])
linalg_ops.matrix_solve(matrix, matrix).eval()
def testConcurrent(self, adjoint):
with self.session() as sess:
lhs1 = random_ops.random_normal([3, 3], seed=42)
lhs2 = random_ops.random_normal([3, 3], seed=42)
rhs1 = random_ops.random_normal([3, 3], seed=42)
rhs2 = random_ops.random_normal([3, 3], seed=42)
with self.test_scope():
s1 = linalg_ops.matrix_solve(lhs1, rhs1, adjoint=adjoint)
s2 = linalg_ops.matrix_solve(lhs2, rhs2, adjoint=adjoint)
self.assertAllEqual(*sess.run([s1, s2]))
def testWrongDimensions(self):
# The matrix and right-hand sides should have the same number of rows.
matrix = constant_op.constant([[1., 0.], [0., 1.]])
rhs = constant_op.constant([[1., 0.]])
with self.assertRaises((ValueError, errors_impl.InvalidArgumentError)):
self.evaluate(linalg_ops.matrix_solve(matrix, rhs))
# The matrix and right-hand side should have the same batch dimensions
matrix = np.random.normal(size=(2, 6, 2, 2))
rhs = np.random.normal(size=(2, 3, 2, 2))
with self.assertRaises((ValueError, errors_impl.InvalidArgumentError)):
self.evaluate(linalg_ops.matrix_solve(matrix, rhs))
def _EigGrad(op, grad_e, grad_v):
"""Gradient for Eig.
Based on eq. 4.77 from paper by
Christoph Boeddeker et al.
https://arxiv.org/abs/1701.00392
See also
"Computation of eigenvalue and eigenvector derivatives
for a general complex-valued eigensystem" by Nico van der Aa.
As for now only distinct eigenvalue case is considered.
"""
e = op.outputs[0]
compute_v = op.get_attr("compute_v")
# a = op.inputs[0], which satisfies
# a[...,:,:] * v[...,:,i] = e[...,i] * v[...,i]
with ops.control_dependencies([grad_e, grad_v]):
if compute_v:
v = op.outputs[1]
vt = _linalg.adjoint(v)
# Construct the matrix f(i,j) = (i != j ? 1 / (e_i - e_j) : 0).
# Notice that because of the term involving f, the gradient becomes
# infinite (or NaN in practice) when eigenvalues are not unique.
# Mathematically this should not be surprising, since for (k-fold)
# degenerate eigenvalues, the corresponding eigenvectors are only defined
# up to arbitrary rotation in a (k-dimensional) subspace.
f = array_ops.matrix_set_diag(
_SafeReciprocal(
array_ops.expand_dims(e, -2) -
array_ops.expand_dims(e, -1)), array_ops.zeros_like(e))
f = math_ops.conj(f)
vgv = math_ops.matmul(vt, grad_v)
mid = array_ops.matrix_diag(grad_e)
diag_grad_part = array_ops.matrix_diag(
array_ops.matrix_diag_part(
math_ops.cast(math_ops.real(vgv), vgv.dtype)))
mid += f * (
vgv - math_ops.matmul(math_ops.matmul(vt, v), diag_grad_part))
# vt is formally invertible as long as the original matrix is
# diagonalizable. However, in practice, vt may
# be ill-conditioned when matrix original matrix is close to
# non-diagonalizable one
grad_a = linalg_ops.matrix_solve(vt, math_ops.matmul(mid, vt))
else:
_, v = linalg_ops.eig(op.inputs[0])
vt = _linalg.adjoint(v)
# vt is formally invertible as long as the original matrix is
# diagonalizable. However, in practice, vt may
# be ill-conditioned when matrix original matrix is close to
# non-diagonalizable one
grad_a = linalg_ops.matrix_solve(
vt, math_ops.matmul(array_ops.matrix_diag(grad_e), vt))
return math_ops.cast(grad_a, op.inputs[0].dtype)
def testConcurrent(self):
with self.session(use_gpu=True) as sess:
all_ops = []
for adjoint_ in False, True:
lhs1 = random_ops.random_normal([3, 3], seed=42)
lhs2 = random_ops.random_normal([3, 3], seed=42)
rhs1 = random_ops.random_normal([3, 3], seed=42)
rhs2 = random_ops.random_normal([3, 3], seed=42)
s1 = linalg_ops.matrix_solve(lhs1, rhs1, adjoint=adjoint_)
s2 = linalg_ops.matrix_solve(lhs2, rhs2, adjoint=adjoint_)
all_ops += [s1, s2]
val = sess.run(all_ops)
self.assertAllEqual(val[0], val[1])
self.assertAllEqual(val[2], val[3])
def testConcurrent(self):
with self.test_session(use_gpu=True) as sess:
all_ops = []
for adjoint_ in False, True:
lhs1 = random_ops.random_normal([3, 3], seed=42)
lhs2 = random_ops.random_normal([3, 3], seed=42)
rhs1 = random_ops.random_normal([3, 3], seed=42)
rhs2 = random_ops.random_normal([3, 3], seed=42)
s1 = linalg_ops.matrix_solve(lhs1, rhs1, adjoint=adjoint_)
s2 = linalg_ops.matrix_solve(lhs2, rhs2, adjoint=adjoint_)
all_ops += [s1, s2]
val = sess.run(all_ops)
self.assertAllEqual(val[0], val[1])
self.assertAllEqual(val[2], val[3])
def benchmarkMatrixSolveOp(self):
run_gpu_test = test.is_gpu_available(True)
for adjoint in False, True:
for matrix_shape in self.matrix_shapes:
for num_rhs in 1, 2, matrix_shape[-1]:
with ops.Graph().as_default(), \
session.Session(config=benchmark.benchmark_config()) as sess, \
ops.device("/cpu:0"):
matrix, rhs = self._GenerateTestData(
matrix_shape, num_rhs)
x = linalg_ops.matrix_solve(matrix,
rhs,
adjoint=adjoint)
self.evaluate(variables.global_variables_initializer())
self.run_op_benchmark(
sess,
control_flow_ops.group(x),
min_iters=25,
store_memory_usage=False,
name=
("matrix_solve_cpu_shape_{matrix_shape}_num_rhs_{num_rhs}_"
"adjoint_{adjoint}").format(
matrix_shape=matrix_shape,
num_rhs=num_rhs,
adjoint=adjoint))
if run_gpu_test:
with ops.Graph().as_default(), \
session.Session(config=benchmark.benchmark_config()) as sess, \
ops.device("/gpu:0"):
matrix, rhs = self._GenerateTestData(
matrix_shape, num_rhs)
x = linalg_ops.matrix_solve(matrix,
rhs,
adjoint=adjoint)
self.evaluate(
variables.global_variables_initializer())
self.run_op_benchmark(
sess,
control_flow_ops.group(x),
min_iters=25,
store_memory_usage=False,
name=
("matrix_solve_gpu_shape_{matrix_shape}_num_rhs_"
"{num_rhs}_adjoint_{adjoint}").format(
matrix_shape=matrix_shape,
num_rhs=num_rhs,
adjoint=adjoint))
def test_broadcast_matmul_and_solve(self):
# These cannot be done in the automated (base test class) tests since they
# test shapes that tf.matmul cannot handle.
# In particular, tf.matmul does not broadcast.
with self.test_session() as sess:
x = random_ops.random_normal(shape=(2, 2, 3, 4))
# This LinearOperatorDiag will be broadcast to (2, 2, 3, 3) during solve
# and matmul with 'x' as the argument.
diag = random_ops.random_uniform(shape=(2, 1, 3))
operator = linalg.LinearOperatorDiag(diag, is_self_adjoint=True)
self.assertAllEqual((2, 1, 3, 3), operator.shape)
# Create a batch matrix with the broadcast shape of operator.
diag_broadcast = array_ops.concat((diag, diag), 1)
mat = array_ops.matrix_diag(diag_broadcast)
self.assertAllEqual((2, 2, 3, 3), mat.get_shape()) # being pedantic.
operator_matmul = operator.matmul(x)
mat_matmul = math_ops.matmul(mat, x)
self.assertAllEqual(operator_matmul.get_shape(), mat_matmul.get_shape())
self.assertAllClose(*sess.run([operator_matmul, mat_matmul]))
operator_solve = operator.solve(x)
mat_solve = linalg_ops.matrix_solve(mat, x)
self.assertAllEqual(operator_solve.get_shape(), mat_solve.get_shape())
self.assertAllClose(*sess.run([operator_solve, mat_solve]))
def _MatrixSolveGrad(op, grad): """Gradients for MatrixSolve.""" a = op.inputs[0] c = op.outputs[0] grad_b = linalg_ops.matrix_solve(a, grad, adjoint=True) grad_a = -math_ops.matmul(grad_b, c, transpose_b=True) return (grad_a, grad_b)
def _verifySolve(self, x, y, batch_dims=None):
for adjoint in False, True:
for np_type in [np.float32, np.float64, np.complex64, np.complex128]:
if np_type is [np.float32, np.float64]:
a = x.real().astype(np_type)
b = y.real().astype(np_type)
else:
a = x.astype(np_type)
b = y.astype(np_type)
if adjoint:
a_np = np.conj(np.transpose(a))
else:
a_np = a
if batch_dims is not None:
a = np.tile(a, batch_dims + [1, 1])
a_np = np.tile(a_np, batch_dims + [1, 1])
b = np.tile(b, batch_dims + [1, 1])
np_ans = np.linalg.solve(a_np, b)
with self.test_session():
tf_ans = linalg_ops.matrix_solve(a, b, adjoint=adjoint)
out = tf_ans.eval()
self.assertEqual(tf_ans.get_shape(), out.shape)
self.assertEqual(np_ans.shape, out.shape)
self.assertAllClose(np_ans, out)
def sqrt_solve(self, x):
"""Computes `solve(self, x)`.
Doesn't actually do the sqrt! Named as such to agree with API.
To compute (M + V D V.T), we use the Woodbury matrix identity:
inv(M + V D V.T) = inv(M) - inv(M) V inv(C) V.T inv(M)
where,
C = inv(D) + V.T inv(M) V.
See: https://en.wikipedia.org/wiki/Woodbury_matrix_identity
Args:
x: `Tensor`
Returns:
inv_of_self_times_x: `Tensor`
"""
minv_x = linalg_ops.matrix_triangular_solve(self._m, x)
vt_minv_x = math_ops.matmul(self._v, minv_x, transpose_a=True)
cinv_vt_minv_x = linalg_ops.matrix_solve(
self._woodbury_sandwiched_term(), vt_minv_x)
v_cinv_vt_minv_x = math_ops.matmul(self._v, cinv_vt_minv_x)
minv_v_cinv_vt_minv_x = linalg_ops.matrix_triangular_solve(
self._m, v_cinv_vt_minv_x)
return minv_x - minv_v_cinv_vt_minv_x
def test_static_dims_broadcast_rhs_has_extra_dims_dynamic(self):
# Since the second arg has extra dims, and the domain dim of the first arg
# is larger than the number of linear equations, code will "flip" the extra
# dims of the first arg to the far right, making extra linear equations
# (then call the matrix function, then flip back).
# We have verified that this optimization indeed happens. How? We stepped
# through with a debugger.
# batch_shape = [2]
matrix = rng.rand(3, 3)
rhs = rng.rand(2, 3, 2)
matrix_broadcast = matrix + np.zeros((2, 1, 1))
matrix_ph = array_ops.placeholder(dtypes.float64, shape=[None, None])
rhs_ph = array_ops.placeholder(dtypes.float64, shape=[None, None, None])
with self.cached_session():
result = linear_operator_util.matrix_solve_with_broadcast(matrix_ph,
rhs_ph)
self.assertAllEqual(3, result.shape.ndims)
expected = linalg_ops.matrix_solve(matrix_broadcast, rhs)
self.assertAllClose(
expected.eval(),
result.eval(feed_dict={
matrix_ph: matrix,
rhs_ph: rhs
}))
def test_broadcast_matmul_and_solve(self):
# These cannot be done in the automated (base test class) tests since they
# test shapes that tf.matmul cannot handle.
# In particular, tf.matmul does not broadcast.
with self.cached_session() as sess:
x = random_ops.random_normal(shape=(2, 2, 3, 4))
# This LinearOperatorDiag will be broadcast to (2, 2, 3, 3) during solve
# and matmul with 'x' as the argument.
diag = random_ops.random_uniform(shape=(2, 1, 3))
operator = linalg.LinearOperatorDiag(diag, is_self_adjoint=True)
self.assertAllEqual((2, 1, 3, 3), operator.shape)
# Create a batch matrix with the broadcast shape of operator.
diag_broadcast = array_ops.concat((diag, diag), 1)
mat = array_ops.matrix_diag(diag_broadcast)
self.assertAllEqual((2, 2, 3, 3), mat.shape) # being pedantic.
operator_matmul = operator.matmul(x)
mat_matmul = math_ops.matmul(mat, x)
self.assertAllEqual(operator_matmul.shape, mat_matmul.shape)
self.assertAllClose(*self.evaluate([operator_matmul, mat_matmul]))
operator_solve = operator.solve(x)
mat_solve = linalg_ops.matrix_solve(mat, x)
self.assertAllEqual(operator_solve.shape, mat_solve.shape)
self.assertAllClose(*self.evaluate([operator_solve, mat_solve]))
def _verifySolve(self, x, y, batch_dims=None):
for adjoint in False, True:
for np_type in [
np.float32, np.float64, np.complex64, np.complex128
]:
if np_type is [np.float32, np.float64]:
a = x.real().astype(np_type)
b = y.real().astype(np_type)
else:
a = x.astype(np_type)
b = y.astype(np_type)
if adjoint:
a_np = np.conj(np.transpose(a))
else:
a_np = a
if batch_dims is not None:
a = np.tile(a, batch_dims + [1, 1])
a_np = np.tile(a_np, batch_dims + [1, 1])
b = np.tile(b, batch_dims + [1, 1])
np_ans = np.linalg.solve(a_np, b)
with self.test_session():
tf_ans = linalg_ops.matrix_solve(a, b, adjoint=adjoint)
out = tf_ans.eval()
self.assertEqual(tf_ans.get_shape(), out.shape)
self.assertEqual(np_ans.shape, out.shape)
self.assertAllClose(np_ans, out)
def test_static_dims_broadcast_rhs_has_extra_dims_dynamic(self):
# Since the second arg has extra dims, and the domain dim of the first arg
# is larger than the number of linear equations, code will "flip" the extra
# dims of the first arg to the far right, making extra linear equations
# (then call the matrix function, then flip back).
# We have verified that this optimization indeed happens. How? We stepped
# through with a debugger.
# batch_shape = [2]
matrix = rng.rand(3, 3)
rhs = rng.rand(2, 3, 2)
matrix_broadcast = matrix + np.zeros((2, 1, 1))
matrix_ph = array_ops.placeholder(dtypes.float64, shape=[None, None])
rhs_ph = array_ops.placeholder(dtypes.float64,
shape=[None, None, None])
with self.cached_session():
result = linear_operator_util.matrix_solve_with_broadcast(
matrix_ph, rhs_ph)
self.assertAllEqual(3, result.shape.ndims)
expected = linalg_ops.matrix_solve(matrix_broadcast, rhs)
self.assertAllClose(
self.evaluate(expected),
result.eval(feed_dict={
matrix_ph: matrix,
rhs_ph: rhs
}))
def test_solve(self):
self._skip_if_tests_to_skip_contains("solve")
for use_placeholder in False, True:
for shape in self._shapes_to_test:
for dtype in self._dtypes_to_test:
for adjoint in False, True:
for adjoint_arg in False, True:
with self.test_session(graph=ops.Graph()) as sess:
sess.graph.seed = random_seed.DEFAULT_GRAPH_SEED
operator, mat, feed_dict = self._operator_and_mat_and_feed_dict(
shape, dtype, use_placeholder=use_placeholder)
rhs = self._make_rhs(operator, adjoint=adjoint)
# If adjoint_arg, solve A X = (rhs^H)^H = rhs.
if adjoint_arg:
op_solve = operator.solve(
linear_operator_util.matrix_adjoint(rhs),
adjoint=adjoint, adjoint_arg=adjoint_arg)
else:
op_solve = operator.solve(
rhs, adjoint=adjoint, adjoint_arg=adjoint_arg)
mat_solve = linalg_ops.matrix_solve(mat, rhs, adjoint=adjoint)
if not use_placeholder:
self.assertAllEqual(
op_solve.get_shape(), mat_solve.get_shape())
op_solve_v, mat_solve_v = sess.run([op_solve, mat_solve],
feed_dict=feed_dict)
self.assertAC(op_solve_v, mat_solve_v)
def test_solve(self):
self._skip_if_tests_to_skip_contains("solve")
for use_placeholder in self._use_placeholder_options:
for shape in self._shapes_to_test:
for dtype in self._dtypes_to_test:
for adjoint in self._adjoint_options:
for adjoint_arg in self._adjoint_arg_options:
with self.test_session(graph=ops.Graph()) as sess:
sess.graph.seed = random_seed.DEFAULT_GRAPH_SEED
operator, mat, feed_dict = self._operator_and_mat_and_feed_dict(
shape, dtype, use_placeholder=use_placeholder)
rhs = self._make_rhs(operator, adjoint=adjoint)
# If adjoint_arg, solve A X = (rhs^H)^H = rhs.
if adjoint_arg:
op_solve = operator.solve(
linalg.adjoint(rhs),
adjoint=adjoint,
adjoint_arg=adjoint_arg)
else:
op_solve = operator.solve(
rhs, adjoint=adjoint, adjoint_arg=adjoint_arg)
mat_solve = linalg_ops.matrix_solve(mat, rhs, adjoint=adjoint)
if not use_placeholder:
self.assertAllEqual(op_solve.get_shape(),
mat_solve.get_shape())
op_solve_v, mat_solve_v = sess.run(
[op_solve, mat_solve], feed_dict=feed_dict)
self.assertAC(op_solve_v, mat_solve_v)
def sqrt_solve(self, x):
"""Computes `solve(self, x)`.
Doesn't actually do the sqrt! Named as such to agree with API.
To compute (M + V D V.T), we use the the Woodbury matrix identity:
inv(M + V D V.T) = inv(M) - inv(M) V inv(C) V.T inv(M)
where,
C = inv(D) + V.T inv(M) V.
See: https://en.wikipedia.org/wiki/Woodbury_matrix_identity
Args:
x: `Tensor`
Returns:
inv_of_self_times_x: `Tensor`
"""
minv_x = linalg_ops.matrix_triangular_solve(self._m, x)
vt_minv_x = math_ops.matmul(self._v, minv_x, transpose_a=True)
cinv_vt_minv_x = linalg_ops.matrix_solve(
self._woodbury_sandwiched_term(), vt_minv_x)
v_cinv_vt_minv_x = math_ops.matmul(self._v, cinv_vt_minv_x)
minv_v_cinv_vt_minv_x = linalg_ops.matrix_triangular_solve(
self._m, v_cinv_vt_minv_x)
return minv_x - minv_v_cinv_vt_minv_x
def testBatchResultSize(self): # 3x3x3 matrices, 3x3x1 right-hand sides. matrix = np.array([1., 2., 3., 4., 5., 6., 7., 8., 9.] * 3).reshape(3, 3, 3) rhs = np.array([1., 2., 3.] * 3).reshape(3, 3, 1) answer = linalg_ops.matrix_solve(matrix, rhs) ls_answer = linalg_ops.matrix_solve_ls(matrix, rhs) self.assertEqual(ls_answer.get_shape(), [3, 3, 1]) self.assertEqual(answer.get_shape(), [3, 3, 1])
def testBatchResultSize(self): # 3x3x3 matrices, 3x3x1 right-hand sides. matrix = np.array([1., 2., 3., 4., 5., 6., 7., 8., 9.] * 3).reshape(3, 3, 3) rhs = np.array([1., 2., 3.] * 3).reshape(3, 3, 1) answer = linalg_ops.matrix_solve(matrix, rhs) ls_answer = linalg_ops.matrix_solve_ls(matrix, rhs) self.assertEqual(ls_answer.get_shape(), [3, 3, 1]) self.assertEqual(answer.get_shape(), [3, 3, 1])
def testBatchResultSize(self): # 3x3x3 matrices, 3x3x1 right-hand sides. matrix = np.array([1., 0., 0., 0., 1., 0., 0., 0., 1.] * 3).reshape(3, 3, 3) # pylint: disable=too-many-function-args rhs = np.array([1., 2., 3.] * 3).reshape(3, 3, 1) # pylint: disable=too-many-function-args answer = linalg_ops.matrix_solve(matrix, rhs) ls_answer = linalg_ops.matrix_solve_ls(matrix, rhs) self.assertEqual(ls_answer.get_shape(), [3, 3, 1]) self.assertEqual(answer.get_shape(), [3, 3, 1])
def _MatrixSolveGrad(op, grad): """Gradients for MatrixSolve.""" a = op.inputs[0] c = op.outputs[0] # TODO(rmlarsen): Get rid of explicit transpose after adding # adjoint_a attribute to solver. grad_b = linalg_ops.matrix_solve(array_ops.transpose(a), grad) grad_a = -math_ops.matmul(grad_b, c, transpose_b=True) return (grad_a, grad_b)
def _solve(self, rhs, adjoint=False, adjoint_arg=False):
if self.is_square is False:
raise NotImplementedError(
"Solve is not yet implemented for non-square operators.")
rhs = linear_operator_util.matrix_adjoint(rhs) if adjoint_arg else rhs
if self._can_use_cholesky():
return linalg_ops.cholesky_solve(self._get_cached_chol(), rhs)
return linalg_ops.matrix_solve(
self._get_cached_dense_matrix(), rhs, adjoint=adjoint)
def _MatrixSolveGrad(op, grad):
"""Gradients for MatrixSolve."""
a = op.inputs[0]
c = op.outputs[0]
# TODO(rmlarsen): Get rid of explicit transpose after adding
# adjoint_a attribute to solver.
grad_b = linalg_ops.matrix_solve(array_ops.transpose(a), grad)
grad_a = -math_ops.matmul(grad_b, c, transpose_b=True)
return (grad_a, grad_b)
def testConcurrent(self):
seed = [42, 24]
matrix_shape = [3, 3]
all_ops = []
for adjoint_ in False, True:
lhs1 = stateless_random_ops.stateless_random_normal(matrix_shape,
seed=seed)
lhs2 = stateless_random_ops.stateless_random_normal(matrix_shape,
seed=seed)
rhs1 = stateless_random_ops.stateless_random_normal(matrix_shape,
seed=seed)
rhs2 = stateless_random_ops.stateless_random_normal(matrix_shape,
seed=seed)
s1 = linalg_ops.matrix_solve(lhs1, rhs1, adjoint=adjoint_)
s2 = linalg_ops.matrix_solve(lhs2, rhs2, adjoint=adjoint_)
all_ops += [s1, s2]
val = self.evaluate(all_ops)
for i in range(0, len(all_ops), 2):
self.assertAllEqual(val[i], val[i + 1])
def test_static_dims_broadcast_matrix_has_extra_dims(self):
# batch_shape = [2]
matrix = rng.rand(2, 3, 3)
rhs = rng.rand(3, 7)
rhs_broadcast = rhs + np.zeros((2, 1, 1))
result = linear_operator_util.matrix_solve_with_broadcast(matrix, rhs)
self.assertAllEqual((2, 3, 7), result.shape)
expected = linalg_ops.matrix_solve(matrix, rhs_broadcast)
self.assertAllClose(*self.evaluate([expected, result]))
def test_static_dims_broadcast(self):
# batch_shape = [2]
matrix = rng.rand(3, 3)
rhs = rng.rand(2, 3, 7)
matrix_broadcast = matrix + np.zeros((2, 1, 1))
with self.cached_session():
result = linear_operator_util.matrix_solve_with_broadcast(matrix, rhs)
self.assertAllEqual((2, 3, 7), result.get_shape())
expected = linalg_ops.matrix_solve(matrix_broadcast, rhs)
self.assertAllEqual(expected.eval(), result.eval())
def _MatrixSolveGrad(op, grad):
"""Gradient for MatrixSolve."""
a = op.inputs[0]
adjoint_a = op.get_attr("adjoint")
c = op.outputs[0]
grad_b = linalg_ops.matrix_solve(a, grad, adjoint=not adjoint_a)
if adjoint_a:
grad_a = -math_ops.matmul(c, grad_b, adjoint_b=True)
else:
grad_a = -math_ops.matmul(grad_b, c, adjoint_b=True)
return (grad_a, grad_b)
def _MatrixSolveGrad(op, grad):
"""Gradient for MatrixSolve."""
a = op.inputs[0]
adjoint_a = op.get_attr("adjoint")
c = op.outputs[0]
grad_b = linalg_ops.matrix_solve(a, grad, adjoint=not adjoint_a)
if adjoint_a:
grad_a = -math_ops.matmul(c, grad_b, adjoint_b=True) # pylint: disable=invalid-unary-operand-type
else:
grad_a = -math_ops.matmul(grad_b, c, adjoint_b=True) # pylint: disable=invalid-unary-operand-type
return (grad_a, grad_b)
def _MatrixSolveGrad(op, grad):
"""Gradient for MatrixSolve."""
a = op.inputs[0]
adjoint_a = op.get_attr("adjoint")
c = op.outputs[0]
grad_b = linalg_ops.matrix_solve(a, grad, adjoint=not adjoint_a)
if adjoint_a:
grad_a = -math_ops.matmul(c, grad_b, adjoint_b=True)
else:
grad_a = -math_ops.matmul(grad_b, c, adjoint_b=True)
return (grad_a, grad_b)
def _solve(self, rhs, adjoint=False, adjoint_arg=False):
"""Default implementation of _solve."""
if self.is_square is False:
raise NotImplementedError(
"Solve is not yet implemented for non-square operators.")
logging.warn(
"Using (possibly slow) default implementation of solve."
" Requires conversion to a dense matrix and O(N^3) operations.")
rhs = linear_operator_util.matrix_adjoint(rhs) if adjoint_arg else rhs
if self._can_use_cholesky():
return linalg_ops.cholesky_solve(self._get_cached_chol(), rhs)
return linalg_ops.matrix_solve(
self._get_cached_dense_matrix(), rhs, adjoint=adjoint)
def benchmarkMatrixSolveOp(self):
run_gpu_test = test.is_gpu_available(True)
for adjoint in False, True:
for matrix_shape in self.matrix_shapes:
for num_rhs in 1, 2, matrix_shape[-1]:
with ops.Graph().as_default(), \
session.Session() as sess, \
ops.device("/cpu:0"):
matrix, rhs = self._GenerateTestData(matrix_shape, num_rhs)
x = linalg_ops.matrix_solve(matrix, rhs, adjoint=adjoint)
variables.global_variables_initializer().run()
self.run_op_benchmark(
sess,
control_flow_ops.group(x),
min_iters=25,
store_memory_usage=False,
name=("matrix_solve_cpu_shape_{matrix_shape}_num_rhs_{num_rhs}_"
"adjoint_{adjoint}").format(
matrix_shape=matrix_shape,
num_rhs=num_rhs,
adjoint=adjoint))
if run_gpu_test:
with ops.Graph().as_default(), \
session.Session() as sess, \
ops.device("/gpu:0"):
matrix, rhs = self._GenerateTestData(matrix_shape, num_rhs)
x = linalg_ops.matrix_solve(matrix, rhs, adjoint=adjoint)
variables.global_variables_initializer().run()
self.run_op_benchmark(
sess,
control_flow_ops.group(x),
min_iters=25,
store_memory_usage=False,
name=("matrix_solve_gpu_shape_{matrix_shape}_num_rhs_"
"{num_rhs}_adjoint_{adjoint}").format(
matrix_shape=matrix_shape, num_rhs=num_rhs,
adjoint=adjoint))
def _solve(self, rhs, adjoint=False, adjoint_arg=False):
"""Default implementation of _solve."""
if self.is_square is False:
raise NotImplementedError(
"Solve is not yet implemented for non-square operators.")
logging.warn(
"Using (possibly slow) default implementation of solve."
" Requires conversion to a dense matrix and O(N^3) operations.")
rhs = linalg.adjoint(rhs) if adjoint_arg else rhs
if self._can_use_cholesky():
return linalg_ops.cholesky_solve(
linalg_ops.cholesky(self.to_dense()), rhs)
return linalg_ops.matrix_solve(self.to_dense(), rhs, adjoint=adjoint)
def _verifySolve(self, x, y, adjoint):
for np_type in self.float_types & {np.float32, np.float64}:
tol = 1e-4 if np_type == np.float32 else 1e-12
a = x.astype(np_type)
b = y.astype(np_type)
np_ans = np.linalg.solve(np.swapaxes(a, -2, -1) if adjoint else a, b)
with self.session() as sess:
with self.test_scope():
tf_ans = linalg_ops.matrix_solve(a, b, adjoint=adjoint)
out = sess.run(tf_ans)
self.assertEqual(tf_ans.shape, out.shape)
self.assertEqual(np_ans.shape, out.shape)
self.assertAllClose(np_ans, out, atol=tol, rtol=tol)
def testSolve(self):
with self.test_session():
for batch_shape in [(), (
2,
3,)]:
for k in [1, 4]:
operator, mat = self._build_operator_and_mat(batch_shape, k)
# Work with 5 simultaneous systems. 5 is arbitrary.
x = self._rng.randn(*(batch_shape + (k, 5)))
self._compare_results(
expected=linalg_ops.matrix_solve(mat, x).eval(),
actual=operator.solve(x))
def testSolve(self):
with self.test_session():
for batch_shape in [(), (
2,
3,)]:
for k in [1, 4]:
operator, mat = self._build_operator_and_mat(batch_shape, k)
# Work with 5 simultaneous systems. 5 is arbitrary.
x = self._rng.randn(*(batch_shape + (k, 5)))
self._compare_results(
expected=linalg_ops.matrix_solve(mat, x).eval(),
actual=operator.solve(x))
def test_dynamic_dims_broadcast_64bit(self):
# batch_shape = [2, 2]
matrix = rng.rand(2, 3, 3)
rhs = rng.rand(2, 1, 3, 7)
matrix_broadcast = matrix + np.zeros((2, 2, 1, 1))
rhs_broadcast = rhs + np.zeros((2, 2, 1, 1))
matrix_ph = array_ops.placeholder_with_default(matrix, shape=None)
rhs_ph = array_ops.placeholder_with_default(rhs, shape=None)
result, expected = self.evaluate([
linear_operator_util.matrix_solve_with_broadcast(matrix_ph, rhs_ph),
linalg_ops.matrix_solve(matrix_broadcast, rhs_broadcast)
])
self.assertAllClose(expected, result)
def test_static_dims_broadcast_rhs_has_extra_dims(self):
# Since the second arg has extra dims, and the domain dim of the first arg
# is larger than the number of linear equations, code will "flip" the extra
# dims of the first arg to the far right, making extra linear equations
# (then call the matrix function, then flip back).
# We have verified that this optimization indeed happens. How? We stepped
# through with a debugger.
# batch_shape = [2]
matrix = rng.rand(3, 3)
rhs = rng.rand(2, 3, 2)
matrix_broadcast = matrix + np.zeros((2, 1, 1))
result = linear_operator_util.matrix_solve_with_broadcast(matrix, rhs)
self.assertAllEqual((2, 3, 2), result.shape)
expected = linalg_ops.matrix_solve(matrix_broadcast, rhs)
self.assertAllClose(*self.evaluate([expected, result]))
def matrix_solve_with_broadcast(matrix, rhs, adjoint=False, name=None):
"""Solve systems of linear equations."""
with ops.name_scope(name, "MatrixSolveWithBroadcast", [matrix, rhs]):
matrix = ops.convert_to_tensor(matrix, name="matrix")
rhs = ops.convert_to_tensor(rhs, name="rhs", dtype=matrix.dtype)
# If either matrix/rhs has extra dims, we can reshape to get rid of them.
matrix, rhs, reshape_inv, still_need_to_transpose = _reshape_for_efficiency(
matrix, rhs, adjoint_a=adjoint)
# This will broadcast by brute force if we still need to.
matrix, rhs = broadcast_matrix_batch_dims([matrix, rhs])
solution = linalg_ops.matrix_solve(
matrix, rhs, adjoint=adjoint and still_need_to_transpose)
return reshape_inv(solution)
def test_solve(self):
self._maybe_skip("solve")
with self.test_session() as sess:
for use_placeholder in False, True:
for shape in self._shapes_to_test:
for dtype in self._dtypes_to_test:
for adjoint in False, True:
operator, mat, feed_dict = self._operator_and_mat_and_feed_dict(
shape, dtype, use_placeholder=use_placeholder)
rhs = self._make_rhs(operator, adjoint=adjoint)
op_solve = operator.solve(rhs, adjoint=adjoint)
mat_solve = linalg_ops.matrix_solve(mat, rhs, adjoint=adjoint)
if not use_placeholder:
self.assertAllEqual(op_solve.get_shape(), mat_solve.get_shape())
op_solve_v, mat_solve_v = sess.run([op_solve, mat_solve],
feed_dict=feed_dict)
self.assertAC(op_solve_v, mat_solve_v)
def test_static_dims_broadcast_rhs_has_extra_dims(self):
# Since the second arg has extra dims, and the domain dim of the first arg
# is larger than the number of linear equations, code will "flip" the extra
# dims of the first arg to the far right, making extra linear equations
# (then call the matrix function, then flip back).
# We have verified that this optimization indeed happens. How? We stepped
# through with a debugger.
# batch_shape = [2]
matrix = rng.rand(3, 3)
rhs = rng.rand(2, 3, 2)
matrix_broadcast = matrix + np.zeros((2, 1, 1))
with self.cached_session():
result = linear_operator_util.matrix_solve_with_broadcast(matrix, rhs)
self.assertAllEqual((2, 3, 2), result.get_shape())
expected = linalg_ops.matrix_solve(matrix_broadcast, rhs)
self.assertAllClose(expected.eval(), self.evaluate(result))
def testSqrtSolve(self):
# Square roots are not unique, but we should still have
# S^{-T} S^{-1} x = A^{-1} x.
# In our case, we should have S = S^T, so then S^{-1} S^{-1} x = A^{-1} x.
with self.test_session():
for batch_shape in [(), (
2,
3,)]:
for k in [1, 4]:
operator, mat = self._build_operator_and_mat(batch_shape, k)
# Work with 5 simultaneous systems. 5 is arbitrary.
x = self._rng.randn(*(batch_shape + (k, 5)))
self._compare_results(
expected=linalg_ops.matrix_solve(mat, x).eval(),
actual=operator.sqrt_solve(operator.sqrt_solve(x)))
def testMultiplyInverse(self):
with ops.Graph().as_default(), self.test_session() as sess:
random_seed.set_random_seed(200)
# Create a Fisher Block.
vocab_size = 5
block = fb.EmbeddingKFACFB(lc.LayerCollection(), vocab_size)
# Add some examples.
inputs = array_ops.constant([[0, 1], [1, 2], [2, 3]])
outputs = array_ops.constant([[0.], [1.], [2.]])
block.register_additional_minibatch(inputs, outputs)
# Instantiate factor's variables. Ensure it doesn't fail.
grads = outputs**2.
damping = array_ops.constant(0.)
block.instantiate_factors(((grads,),), damping)
block._input_factor.instantiate_cov_variables()
block._output_factor.instantiate_cov_variables()
block.register_inverse()
block._input_factor.instantiate_inv_variables()
block._output_factor.instantiate_inv_variables()
# Create a sparse update.
indices = array_ops.constant([1, 3, 4])
values = array_ops.constant([[1.], [1.], [1.]])
sparse_vector = ops.IndexedSlices(
values, indices, dense_shape=[vocab_size, 1])
dense_vector = array_ops.reshape([0., 1., 0., 1., 1.], [vocab_size, 1])
# Compare Fisher-vector product against explicit result.
result = block.multiply_inverse(sparse_vector)
expected_result = linalg_ops.matrix_solve(block.full_fisher_block(),
dense_vector)
sess.run(tf_variables.global_variables_initializer())
self.assertAlmostEqual(
sess.run(expected_result[1]), sess.run(result.values[0]))
self.assertAlmostEqual(
sess.run(expected_result[3]), sess.run(result.values[1]))
self.assertAlmostEqual(
sess.run(expected_result[4]), sess.run(result.values[2]))
def _solve(self, rhs, adjoint=False):
if self.base_operator.is_non_singular is False:
raise ValueError(
"Solve not implemented unless this is a perturbation of a "
"non-singular LinearOperator.")
# The Woodbury formula gives:
# https://en.wikipedia.org/wiki/Woodbury_matrix_identity
# (L + UDV^H)^{-1}
# = L^{-1} - L^{-1} U (D^{-1} + V^H L^{-1} U)^{-1} V^H L^{-1}
# = L^{-1} - L^{-1} U C^{-1} V^H L^{-1}
# where C is the capacitance matrix, C := D^{-1} + V^H L^{-1} U
# Note also that, with ^{-H} being the inverse of the adjoint,
# (L + UDV^H)^{-H}
# = L^{-H} - L^{-H} V C^{-H} U^H L^{-H}
l = self.base_operator
if adjoint:
v = self.u
u = self.v
else:
v = self.v
u = self.u
# L^{-1} rhs
linv_rhs = l.solve(rhs, adjoint=adjoint)
# V^H L^{-1} rhs
vh_linv_rhs = math_ops.matmul(v, linv_rhs, adjoint_a=True)
# C^{-1} V^H L^{-1} rhs
if self._use_cholesky:
capinv_vh_linv_rhs = linalg_ops.cholesky_solve(
self._chol_capacitance, vh_linv_rhs)
else:
capinv_vh_linv_rhs = linalg_ops.matrix_solve(
self._capacitance, vh_linv_rhs, adjoint=adjoint)
# U C^{-1} V^H M^{-1} rhs
u_capinv_vh_linv_rhs = math_ops.matmul(u, capinv_vh_linv_rhs)
# L^{-1} U C^{-1} V^H L^{-1} rhs
linv_u_capinv_vh_linv_rhs = l.solve(u_capinv_vh_linv_rhs, adjoint=adjoint)
# L^{-1} - L^{-1} U C^{-1} V^H L^{-1}
return linv_rhs - linv_u_capinv_vh_linv_rhs
def test_dynamic_dims_broadcast_64bit(self):
# batch_shape = [2, 2]
matrix = rng.rand(2, 3, 3)
rhs = rng.rand(2, 1, 3, 7)
matrix_broadcast = matrix + np.zeros((2, 2, 1, 1))
rhs_broadcast = rhs + np.zeros((2, 2, 1, 1))
matrix_ph = array_ops.placeholder(dtypes.float64)
rhs_ph = array_ops.placeholder(dtypes.float64)
with self.cached_session() as sess:
result, expected = sess.run(
[
linear_operator_util.matrix_solve_with_broadcast(
matrix_ph, rhs_ph),
linalg_ops.matrix_solve(matrix_broadcast, rhs_broadcast)
],
feed_dict={
matrix_ph: matrix,
rhs_ph: rhs,
})
self.assertAllEqual(expected, result)
def _solve(self, rhs, adjoint=False):
if self._is_spd:
return linalg_ops.cholesky_solve(self._chol, rhs)
return linalg_ops.matrix_solve(self._matrix, rhs, adjoint=adjoint)
def _solve(self, rhs, adjoint=False, adjoint_arg=False): rhs = ops.convert_to_tensor(rhs, name="rhs") assert not adjoint_arg, "Not implemented for this test class." return linalg_ops.matrix_solve(self._matrix, rhs, adjoint=adjoint)
def matrix_exponential(input, name=None): # pylint: disable=redefined-builtin
r"""Computes the matrix exponential of one or more square matrices.
exp(A) = \sum_{n=0}^\infty A^n/n!
The exponential is computed using a combination of the scaling and squaring
method and the Pade approximation. Details can be found in:
Nicholas J. Higham, "The scaling and squaring method for the matrix
exponential revisited," SIAM J. Matrix Anal. Applic., 26:1179-1193, 2005.
The input is a tensor of shape `[..., M, M]` whose inner-most 2 dimensions
form square matrices. The output is a tensor of the same shape as the input
containing the exponential for all input submatrices `[..., :, :]`.
Args:
input: A `Tensor`. Must be `float16`, `float32`, `float64`, `complex64`,
or `complex128` with shape `[..., M, M]`.
name: A name to give this `Op` (optional).
Returns:
the matrix exponential of the input.
Raises:
ValueError: An unsupported type is provided as input.
@compatibility(scipy)
Equivalent to scipy.linalg.expm
@end_compatibility
"""
with ops.name_scope(name, 'matrix_exponential', [input]):
matrix = ops.convert_to_tensor(input, name='input')
if matrix.shape[-2:] == [0, 0]:
return matrix
batch_shape = matrix.shape[:-2]
if not batch_shape.is_fully_defined():
batch_shape = array_ops.shape(matrix)[:-2]
# reshaping the batch makes the where statements work better
matrix = array_ops.reshape(
matrix, array_ops.concat(([-1], array_ops.shape(matrix)[-2:]), axis=0))
l1_norm = math_ops.reduce_max(
math_ops.reduce_sum(math_ops.abs(matrix),
axis=array_ops.size(array_ops.shape(matrix)) - 2),
axis=-1)
const = lambda x: constant_op.constant(x, l1_norm.dtype)
def _nest_where(vals, cases):
assert len(vals) == len(cases) - 1
if len(vals) == 1:
return array_ops.where(
math_ops.less(l1_norm, const(vals[0])), cases[0], cases[1])
else:
return array_ops.where(
math_ops.less(l1_norm, const(vals[0])), cases[0],
_nest_where(vals[1:], cases[1:]))
if matrix.dtype in [dtypes.float16, dtypes.float32, dtypes.complex64]:
maxnorm = const(3.925724783138660)
squarings = math_ops.maximum(
math_ops.floor(
math_ops.log(l1_norm / maxnorm) / math_ops.log(const(2.0))), 0)
u3, v3 = _matrix_exp_pade3(matrix)
u5, v5 = _matrix_exp_pade5(matrix)
u7, v7 = _matrix_exp_pade7(
matrix / math_ops.pow(
constant_op.constant(2.0, dtype=matrix.dtype),
math_ops.cast(squarings, matrix.dtype))[...,
array_ops.newaxis,
array_ops.newaxis])
conds = (4.258730016922831e-001, 1.880152677804762e+000)
u = _nest_where(conds, (u3, u5, u7))
v = _nest_where(conds, (v3, v5, v7))
elif matrix.dtype in [dtypes.float64, dtypes.complex128]:
maxnorm = const(5.371920351148152)
squarings = math_ops.maximum(
math_ops.floor(
math_ops.log(l1_norm / maxnorm) / math_ops.log(const(2.0))), 0)
u3, v3 = _matrix_exp_pade3(matrix)
u5, v5 = _matrix_exp_pade5(matrix)
u7, v7 = _matrix_exp_pade7(matrix)
u9, v9 = _matrix_exp_pade9(matrix)
u13, v13 = _matrix_exp_pade13(
matrix / math_ops.pow(
constant_op.constant(2.0, dtype=matrix.dtype),
math_ops.cast(squarings, matrix.dtype))[...,
array_ops.newaxis,
array_ops.newaxis])
conds = (1.495585217958292e-002,
2.539398330063230e-001,
9.504178996162932e-001,
2.097847961257068e+000)
u = _nest_where(conds, (u3, u5, u7, u9, u13))
v = _nest_where(conds, (v3, v5, v7, v9, v13))
else:
raise ValueError(
'tf.linalg.expm does not support matrices of type %s' % matrix.dtype)
numer = u + v
denom = -u + v
result = linalg_ops.matrix_solve(denom, numer)
max_squarings = math_ops.reduce_max(squarings)
i = const(0.0)
c = lambda i, r: math_ops.less(i, max_squarings)
def b(i, r):
return i+1, array_ops.where(math_ops.less(i, squarings),
math_ops.matmul(r, r), r)
_, result = control_flow_ops.while_loop(c, b, [i, result])
if not matrix.shape.is_fully_defined():
return array_ops.reshape(
result,
array_ops.concat((batch_shape, array_ops.shape(result)[-2:]), axis=0))
return array_ops.reshape(result, batch_shape.concatenate(result.shape[-2:]))
def _solve(self, rhs, adjoint=False, adjoint_arg=False):
rhs = linear_operator_util.matrix_adjoint(rhs) if adjoint_arg else rhs
if self._is_spd:
return linalg_ops.cholesky_solve(self._chol, rhs)
return linalg_ops.matrix_solve(self._matrix, rhs, adjoint=adjoint)