tf.linalg.LinearOperatorTridiag

LinearOperator acting like a [batch] square tridiagonal matrix.

Inherits From: LinearOperator

View aliases

Compat aliases for migration

See Migration guide for more details.

tf.compat.v1.linalg.LinearOperatorTridiag

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 property X. 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 have X.
• If is_X == None (the default), callers should have no expectation either way.

Args
diagonals	Tensor or list of Tensors depending on diagonals_format. If diagonals_format=sequence, this is a list of three Tensor's each with shape [B1, ..., Bb, N], b >= 0, N >= 0, representing the superdiagonal, diagonal and subdiagonal in that order. Note the superdiagonal is padded with an element in the last position, and the subdiagonal is padded with an element in the front. If diagonals_format=matrix this is a [B1, ... Bb, N, N] shaped Tensor representing the full tridiagonal matrix. If diagonals_format=compact this is a [B1, ... Bb, 3, N] shaped Tensor with the second to last dimension indexing the superdiagonal, diagonal and subdiagonal in that order. Note the superdiagonal is padded with an element in the last position, and the subdiagonal is padded with an element in the front. In every case, these Tensors are all floating dtype.
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 A representing this LinearOperator, return A*. Note that calling self.adjoint() and self.H are equivalent.
batch_shape	TensorShape of batch dimensions of this LinearOperator. If this operator acts like the batch matrix A with A.shape = [B1,...,Bb, M, N], then this returns TensorShape([B1,...,Bb]), equivalent to A.shape[:-2]
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 A with A.shape = [B1,...,Bb, M, N], then this returns N.
dtype	The DType of Tensors handled by this LinearOperator.
graph_parents	List of graph dependencies of this LinearOperator. (deprecated) Warning: THIS FUNCTION IS DEPRECATED. It will be removed in a future version. Instructions for updating: Do not call graph_parents.
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 A with A.shape = [B1,...,Bb, M, N], then this returns M.
shape	TensorShape of this LinearOperator. If this operator acts like the batch matrix A with A.shape = [B1,...,Bb, M, N], then this returns TensorShape([B1,...,Bb, M, N]), equivalent to A.shape.
tensor_rank	Rank (in the sense of tensors) of matrix corresponding to this operator. If this operator acts like the batch matrix A with A.shape = [B1,...,Bb, M, N], then this returns b + 2.

Methods

add_to_tensor

View source

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

View source

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

View source

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

View source

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 positive definite.

Args
name	A name to give this Op.

Returns
An Assert Op, that, when run, will raise an InvalidArgumentError if the operator is not positive definite.

assert_self_adjoint