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LinearOperator
acting like a [batch] square tridiagonal matrix.
Inherits From: LinearOperator
tf.linalg.LinearOperatorTridiag(
diagonals, diagonals_format=_COMPACT, is_non_singular=None,
is_self_adjoint=None, is_positive_definite=None, is_square=None,
name='LinearOperatorTridiag'
)
This operator acts like a [batch] square tridiagonal matrix A
with shape
[B1,...,Bb, N, N]
for some b >= 0
. The first b
indices index a
batch member. For every batch index (i1,...,ib)
, A[i1,...,ib, : :]
is
an N x M
matrix. This matrix A
is not materialized, but for
purposes of broadcasting this shape will be relevant.
Example usage:
Create a 3 x 3 tridiagonal linear operator.
superdiag = [3., 4., 5.]
diag = [1., -1., 2.]
subdiag = [6., 7., 8]
operator = tf.linalg.LinearOperatorTridiag(
[superdiag, diag, subdiag],
diagonals_format='sequence')
operator.to_dense()
<tf.Tensor: shape=(3, 3), dtype=float32, numpy=
array([[ 1., 3., 0.],
[ 7., -1., 4.],
[ 0., 8., 2.]], dtype=float32)>
operator.shape
TensorShape([3, 3])
Scalar Tensor output.
operator.log_abs_determinant()
<tf.Tensor: shape=(), dtype=float32, numpy=4.3307333>
Create a [2, 3] batch of 4 x 4 linear operators.
diagonals = tf.random.normal(shape=[2, 3, 3, 4])
operator = tf.linalg.LinearOperatorTridiag(
diagonals,
diagonals_format='compact')
Create a shape [2, 1, 4, 2] vector. Note that this shape is compatible since the batch dimensions, [2, 1], are broadcast to operator.batch_shape = [2, 3].
y = tf.random.normal(shape=[2, 1, 4, 2])
x = operator.solve(y)
x
<tf.Tensor: shape=(2, 3, 4, 2), dtype=float32, numpy=...,
dtype=float32)>
Shape compatibility
This operator acts on [batch] matrix with compatible shape.
x
is a batch matrix with compatible shape for matmul
and solve
if
operator.shape = [B1,...,Bb] + [N, N], with b >= 0
x.shape = [C1,...,Cc] + [N, R],
and [C1,...,Cc] broadcasts with [B1,...,Bb].
Performance
Suppose operator
is a LinearOperatorTridiag
of shape [N, N]
,
and x.shape = [N, R]
. Then
operator.matmul(x)
will take O(N * R) time.operator.solve(x)
will take O(N * R) time.
If instead operator
and x
have shape [B1,...,Bb, N, N]
and
[B1,...,Bb, N, R]
, every operation increases in complexity by B1*...*Bb
.
Matrix property hints
This LinearOperator
is initialized with boolean flags of the form is_X
,
for X = non_singular, self_adjoint, positive_definite, square
.
These have the following meaning:
- If
is_X == True
, callers should expect the operator to have the propertyX
. This is a promise that should be fulfilled, but is not a runtime assert. For example, finite floating point precision may result in these promises being violated. - If
is_X == False
, callers should expect the operator to not haveX
. - If
is_X == None
(the default), callers should have no expectation either way.
Args | |
---|---|
diagonals
|
Tensor or list of Tensor s depending on diagonals_format .
If If If In every case, these |
diagonals_format
|
one of matrix , sequence , or compact . Default is
compact .
|
is_non_singular
|
Expect that this operator is non-singular. |
is_self_adjoint
|
Expect that this operator is equal to its hermitian
transpose. If diag.dtype is real, this is auto-set to True .
|
is_positive_definite
|
Expect that this operator is positive definite,
meaning the quadratic form x^H A x has positive real part for all
nonzero x . Note that we do not require the operator to be
self-adjoint to be positive-definite. See:
https://en.wikipedia.org/wiki/Positive-definite_matrix#Extension_for_non-symmetric_matrices
|
is_square
|
Expect that this operator acts like square [batch] matrices. |
name
|
A name for this LinearOperator .
|
Raises | |
---|---|
TypeError
|
If diag.dtype is not an allowed type.
|
ValueError
|
If diag.dtype is real, and is_self_adjoint is not True .
|
Attributes | |
---|---|
H
|
Returns the adjoint of the current LinearOperator .
Given |
batch_shape
|
TensorShape of batch dimensions of this LinearOperator .
If this operator acts like the batch matrix |
diagonals
|
|
diagonals_format
|
|
domain_dimension
|
Dimension (in the sense of vector spaces) of the domain of this operator.
If this operator acts like the batch matrix |
dtype
|
The DType of Tensor s handled by this LinearOperator .
|
graph_parents
|
List of graph dependencies of this LinearOperator . (deprecated)
|
is_non_singular
|
|
is_positive_definite
|
|
is_self_adjoint
|
|
is_square
|
Return True/False depending on if this operator is square.
|
range_dimension
|
Dimension (in the sense of vector spaces) of the range of this operator.
If this operator acts like the batch matrix |
shape
|
TensorShape of this LinearOperator .
If this operator acts like the batch matrix |
tensor_rank
|
Rank (in the sense of tensors) of matrix corresponding to this operator.
If this operator acts like the batch matrix |
Methods
add_to_tensor
add_to_tensor(
x, name='add_to_tensor'
)
Add matrix represented by this operator to x
. Equivalent to A + x
.
Args | |
---|---|
x
|
Tensor with same dtype and shape broadcastable to self.shape .
|
name
|
A name to give this Op .
|
Returns | |
---|---|
A Tensor with broadcast shape and same dtype as self .
|
adjoint
adjoint(
name='adjoint'
)
Returns the adjoint of the current LinearOperator
.
Given A
representing this LinearOperator
, return A*
.
Note that calling self.adjoint()
and self.H
are equivalent.
Args | |
---|---|
name
|
A name for this Op .
|
Returns | |
---|---|
LinearOperator which represents the adjoint of this LinearOperator .
|
assert_non_singular
assert_non_singular(
name='assert_non_singular'
)
Returns an Op
that asserts this operator is non singular.
This operator is considered non-singular if
ConditionNumber < max{100, range_dimension, domain_dimension} * eps,
eps := np.finfo(self.dtype.as_numpy_dtype).eps
Args | |
---|---|
name
|
A string name to prepend to created ops. |
Returns | |
---|---|
An Assert Op , that, when run, will raise an InvalidArgumentError if
the operator is singular.
|
assert_positive_definite
assert_positive_definite(
name='assert_positive_definite'
)
Returns an Op
that asserts this operator is positive definite.
Here, positive definite means that the quadratic form x^H A x
has positive
real part for all nonzero x
. Note that we do not require the operator to
be self-adjoint to be posi