# tf.linalg.LinearOperator

## Class LinearOperator

### Aliases:

• Class tf.contrib.linalg.LinearOperator
• Class tf.linalg.LinearOperator

See the guide: Linear Algebra (contrib) > LinearOperator

Base class defining a [batch of] linear operator[s].

Subclasses of LinearOperator provide a access to common methods on a (batch) matrix, without the need to materialize the matrix. This allows:

• Matrix free computations
• Operators that take advantage of special structure, while providing a consistent API to users.

#### Subclassing

To enable a public method, subclasses should implement the leading-underscore version of the method. The argument signature should be identical except for the omission of name="...". For example, to enable matmul(x, adjoint=False, name="matmul") a subclass should implement _matmul(x, adjoint=False).

#### Performance contract

Subclasses should only implement the assert methods (e.g. assert_non_singular) if they can be done in less than O(N^3) time.

Class docstrings should contain an explanation of computational complexity. Since this is a high-performance library, attention should be paid to detail, and explanations can include constants as well as Big-O notation.

#### Shape compatibility

LinearOperator sub classes should operate on a [batch] matrix with compatible shape. Class docstrings should define what is meant by compatible shape. Some sub-classes may not support batching.

An example is:

x is a batch matrix with compatible shape for matmul if

operator.shape = [B1,...,Bb] + [M, N],  b >= 0,
x.shape =   [B1,...,Bb] + [N, R]


rhs is a batch matrix with compatible shape for solve if

operator.shape = [B1,...,Bb] + [M, N],  b >= 0,
rhs.shape =   [B1,...,Bb] + [M, R]


#### Example docstring for subclasses.

This operator acts like a (batch) matrix A with shape [B1,...,Bb, M, N] for some b >= 0. The first b indices index a batch member. For every batch index (i1,...,ib), A[i1,...,ib, : :] is an m x n matrix. Again, this matrix A may not be materialized, but for purposes of identifying and working with compatible arguments the shape is relevant.

Examples:

some_tensor = ... shape = ????
operator = MyLinOp(some_tensor)

operator.shape()
==> [2, 4, 4]

operator.log_abs_determinant()
==> Shape [2] Tensor

x = ... Shape [2, 4, 5] Tensor

operator.matmul(x)
==> Shape [2, 4, 5] Tensor


#### Shape compatibility

This operator acts on batch matrices with compatible shape. FILL IN WHAT IS MEANT BY COMPATIBLE SHAPE

FILL THIS IN

#### 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.

## Properties

### 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.get_shape()[:-2]

#### Returns:

TensorShape, statically determined, may be undefined.

### 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.

#### Returns:

Dimension object.

### dtype

The DType of Tensors handled by this LinearOperator.

### graph_parents

List of graph dependencies of this LinearOperator.

### is_square

Return True/False depending on if this operator is square.

### name

Name prepended to all ops created by this LinearOperator.

### 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.

#### Returns:

Dimension object.

### 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.get_shape().

#### Returns:

TensorShape, statically determined, may be undefined.

### 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.

#### Args:

• name: A name for this Op.

#### Returns:

Python integer, or None if the tensor rank is undefined.

## Methods

### __init__

__init__(
dtype,
graph_parents=None,
is_non_singular=None,
is_positive_definite=None,
is_square=None,
name=None
)


Initialize the LinearOperator.

This is a private method for subclass use. Subclasses should copy-paste this __init__ documentation.

#### Args:

• dtype: The type of the this LinearOperator. Arguments to matmul and solve will have to be this type.
• graph_parents: Python list of graph prerequisites of this LinearOperator Typically tensors that are passed during initialization.
• is_non_singular: Expect that this operator is non-singular.
• is_self_adjoint: Expect that this operator is equal to its hermitian transpose. If dtype is real, this is equivalent to being symmetric.
• 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:

• ValueError: If any member of graph_parents is None or not a Tensor.
• ValueError: If hints are set incorrectly.

### add_to_tensor

add_to_tensor(
x,
)


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.

### 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 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

assert_self_adjoint(name='assert_self_adjoint')


Returns an Op that asserts this operator is self-adjoint.

Here we check that this operator is exactly equal to its hermitian transpose.

#### 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 not self-adjoint.

### batch_shape_tensor

batch_shape_tensor(name='batch_shape_tensor')


Shape of batch dimensions of this operator, determined at runtime.

If this operator acts like the batch matrix A with A.shape = [B1,...,Bb, M, N], then this returns a Tensor holding [B1,...,Bb].

#### Args:

• name: A name for this Op.

#### Returns:

int32 Tensor

### determinant

determinant(name='det')


Determinant for every batch member.

#### Args:

• name: A name for this Op.

#### Returns:

Tensor with shape self.batch_shape and same dtype as self.

#### Raises:

• NotImplementedError: If self.is_square is False.

### diag_part

diag_part(name='diag_part')


Efficiently get the [batch] diagonal part of this operator.

If this operator has shape [B1,...,Bb, M, N], this returns a Tensor diagonal, of shape [B1,...,Bb, min(M, N)], where diagonal[b1,...,bb, i] = self.to_dense()[b1,...,bb, i, i].

my_operator = LinearOperatorDiag([1., 2.])

# Efficiently get the diagonal
my_operator.diag_part()
==> [1., 2.]

# Equivalent, but inefficient method
tf.matrix_diag_part(my_operator.to_dense())
==> [1., 2.]


#### Args:

• name: A name for this Op.

#### Returns:

• diag_part: A Tensor of same dtype as self.

### domain_dimension_tensor

domain_dimension_tensor(name='domain_dimension_tensor')


Dimension (in the sense of vector spaces) of the domain of this operator.

Determined at runtime.

If this operator acts like the batch matrix A with A.shape = [B1,...,Bb, M, N], then this returns N.

#### Args:

• name: A name for this Op.

#### Returns:

int32 Tensor

### log_abs_determinant

log_abs_determinant(name='log_abs_det')


Log absolute value of determinant for every batch member.

#### Args:

• name: A name for this Op.

#### Returns:

Tensor with shape self.batch_shape and same dtype as self.

#### Raises:

• NotImplementedError: If self.is_square is False.

### matmul

matmul(
x,
name='matmul'
)


Transform [batch] matrix x with left multiplication: x --> Ax.

# Make an operator acting like batch matrix A.  Assume A.shape = [..., M, N]
operator = LinearOperator(...)
operator.shape = [..., M, N]

X = ... # shape [..., N, R], batch matrix, R > 0.

Y = operator.matmul(X)
Y.shape
==> [..., M, R]

Y[..., :, r] = sum_j A[..., :, j] X[j, r]


#### Args:

• x: Tensor with compatible shape and same dtype as self. See class docstring for definition of compatibility.
• adjoint: Python bool. If True, left multiply by the adjoint: A^H x.
• adjoint_arg: Python bool. If True, compute A x^H where x^H is the hermitian transpose (transposition and complex conjugation).
• name: A name for this Op.

#### Returns:

A Tensor with shape [..., M, R] and same dtype as self.

### matvec

matvec(
x,
name='matvec'
)


Transform [batch] vector x with left multiplication: x --> Ax.

# Make an operator acting like batch matric A.  Assume A.shape = [..., M, N]
operator = LinearOperator(...)

X = ... # shape [..., N], batch vector

Y = operator.matvec(X)
Y.shape
==> [..., M]

Y[..., :] = sum_j A[..., :, j] X[..., j]


#### Args:

• x: Tensor with compatible shape and same dtype as self. x is treated as a [batch] vector meaning for every set of leading dimensions, the last dimension defines a vector. See class docstring for definition of compatibility.
• adjoint: Python bool. If True, left multiply by the adjoint: A^H x.
• name: A name for this Op.

#### Returns:

A Tensor with shape [..., M] and same dtype as self.

### range_dimension_tensor

range_dimension_tensor(name='range_dimension_tensor')


Dimension (in the sense of vector spaces) of the range of this operator.

Determined at runtime.

If this operator acts like the batch matrix A with A.shape = [B1,...,Bb, M, N], then this returns M.

#### Args:

• name: A name for this Op.

#### Returns:

int32 Tensor

### shape_tensor

shape_tensor(name='shape_tensor')


Shape of this LinearOperator, determined at runtime.

If this operator acts like the batch matrix A with A.shape = [B1,...,Bb, M, N], then this returns a Tensor holding [B1,...,Bb, M, N], equivalent to tf.shape(A).

#### Args:

• name: A name for this Op.

#### Returns:

int32 Tensor

### solve

solve(
rhs,
name='solve'
)


Solve (exact or approx) R (batch) systems of equations: A X = rhs.

The returned Tensor will be close to an exact solution if A is well conditioned. Otherwise closeness will vary. See class docstring for details.

Examples:

# Make an operator acting like batch matrix A.  Assume A.shape = [..., M, N]
operator = LinearOperator(...)
operator.shape = [..., M, N]

# Solve R > 0 linear systems for every member of the batch.
RHS = ... # shape [..., M, R]

X = operator.solve(RHS)
# X[..., :, r] is the solution to the r'th linear system
# sum_j A[..., :, j] X[..., j, r] = RHS[..., :, r]

operator.matmul(X)
==> RHS


#### Args:

• rhs: Tensor with same dtype as this operator and compatible shape. rhs is treated like a [batch] matrix meaning for every set of leading dimensions, the last two dimensions defines a matrix. See class docstring for definition of compatibility.
• adjoint: Python bool. If True, solve the system involving the adjoint of this LinearOperator: A^H X = rhs.
• adjoint_arg: Python bool. If True, solve A X = rhs^H where rhs^H is the hermitian transpose (transposition and complex conjugation).
• name: A name scope to use for ops added by this method.

#### Returns:

Tensor with shape [...,N, R] and same dtype as rhs.

#### Raises:

• NotImplementedError: If self.is_non_singular or is_square is False.

### solvevec

solvevec(
rhs,
name='solve'
)


Solve single equation with best effort: A X = rhs.

The returned Tensor will be close to an exact solution if A is well conditioned. Otherwise closeness will vary. See class docstring for details.

Examples:

# Make an operator acting like batch matrix A.  Assume A.shape = [..., M, N]
operator = LinearOperator(...)
operator.shape = [..., M, N]

# Solve one linear system for every member of the batch.
RHS = ... # shape [..., M]

X = operator.solvevec(RHS)
# X is the solution to the linear system
# sum_j A[..., :, j] X[..., j] = RHS[..., :]

operator.matvec(X)
==> RHS


#### Args:

• rhs: Tensor with same dtype as this operator. rhs is treated like a [batch] vector meaning for every set of leading dimensions, the last dimension defines a vector. See class docstring for definition of compatibility regarding batch dimensions.
• adjoint: Python bool. If True, solve the system involving the adjoint of this LinearOperator: A^H X = rhs.
• name: A name scope to use for ops added by this method.

#### Returns:

Tensor with shape [...,N] and same dtype as rhs.

#### Raises:

• NotImplementedError: If self.is_non_singular or is_square is False.

### tensor_rank_tensor

tensor_rank_tensor(name='tensor_rank_tensor')


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.

#### Args:

• name: A name for this Op.

#### Returns:

int32 Tensor, determined at runtime.

### to_dense

to_dense(name='to_dense')


Return a dense (batch) matrix representing this operator.

### trace

trace(name='trace')


Trace of the linear operator, equal to sum of self.diag_part().

If the operator is square, this is also the sum of the eigenvalues.

#### Args:

• name: A name for this Op.

#### Returns:

Shape [B1,...,Bb] Tensor of same dtype as self.