# tf.contrib.linalg.LinearOperatorTriL

### class tf.contrib.linalg.LinearOperatorTriL

See the guide: Linear Algebra (contrib) > LinearOperator

LinearOperator acting like a [batch] square lower triangular matrix.

This operator acts like a [batch] lower triangular 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 N matrix.

LinearOperatorTriL is initialized with a Tensor having dimensions [B1,...,Bb, N, N]. The upper triangle of the last two dimensions is ignored.

# Create a 2 x 2 lower-triangular linear operator.
tril = [[1., 2.], [3., 4.]]
operator = LinearOperatorTriL(tril)

# The upper triangle is ignored.
operator.to_dense()
==> [[1., 0.]
[3., 4.]]

operator.shape
==> [2, 2]

operator.log_determinant()
==> scalar Tensor

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

# Create a [2, 3] batch of 4 x 4 linear operators.
tril = tf.random_normal(shape=[2, 3, 4, 4])
operator = LinearOperatorTriL(tril)


#### Shape compatibility

This operator acts on [batch] matrix with compatible shape. x is a batch matrix with compatible shape for apply and solve if

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


#### Performance

Suppose operator is a LinearOperatorTriL of shape [N, N], and x.shape = [N, R]. Then

• operator.apply(x) involves N^2 * R multiplications.
• operator.solve(x) involves N * R size N back-substitutions.
• operator.determinant() involves a size N reduce_prod.

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. 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.

### 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__(tril, is_non_singular=None, is_self_adjoint=None, is_positive_definite=None, name='LinearOperatorTriL')

Initialize a LinearOperatorTriL.

#### Args:

• tril: Shape [B1,...,Bb, N, N] with b >= 0, N >= 0. The lower triangular part of tril defines this operator. The strictly upper triangle is ignored. Allowed dtypes: float32, float64.
• is_non_singular: Expect that this operator is non-singular. This operator is non-singular if and only if its diagonal elements are all non-zero.
• is_self_adjoint: Expect that this operator is equal to its hermitian transpose. This operator is self-adjoint only if it is diagonal with real-valued diagonal entries. In this case it is advised to use LinearOperatorDiag.
• is_positive_definite: Expect that this operator is positive definite, meaning the real part of all eigenvalues is positive. 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
• name: A name for this LinearOperator.

#### Raises:

• TypeError: If diag.dtype is not an allowed type.

### 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.

### apply(x, adjoint=False, name='apply')

Transform x with left multiplication: x --> Ax.

#### 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.
• name: A name for this Op.

#### Returns:

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

### assert_non_singular(name='assert_non_singular')

Returns an Op that asserts this operator is non singular.

### assert_positive_definite(name='assert_positive_definite')

Returns an Op that asserts this operator is positive definite.

Here, positive definite means the real part of all eigenvalues is positive. We do not require the operator to be self-adjoint.

#### Args:

• name: A name to give this Op.

#### Returns:

An Op that asserts this operator is positive definite.

### assert_self_adjoint(name='assert_self_adjoint')

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

### batch_shape_dynamic(name='batch_shape_dynamic')

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(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.

### domain_dimension_dynamic(name='domain_dimension_dynamic')

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(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.

### range_dimension_dynamic(name='range_dimension_dynamic')

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_dynamic(name='shape_dynamic')

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(rhs, adjoint=False, name='solve')

Solve R (batch) systems of equations exactly: A X = rhs.

Examples:

# Create an operator acting like a 10 x 2 x 2 matrix.
operator = LinearOperator(...)
operator.shape # = 10 x 2 x 2

# Solve one linear system (R = 1) for every member of the length 10 batch.
RHS = ... # shape 10 x 2 x 1
X = operator.solve(RHS)  # shape 10 x 2 x 1

# Solve five linear systems (R = 5) for every member of the length 10 batch.
RHS = ... # shape 10 x 2 x 5
X = operator.solve(RHS)
X[3, :, 2]  # Solution to the linear system A[3, :, :] X = RHS[3, :, 2]


#### Args:

• rhs: Tensor with same dtype as this operator and compatible shape. See class docstring for definition of compatibility.
• adjoint: Python bool. If True, solve the system involving the adjoint of this LinearOperator.
• name: A name scope to use for ops added by this method.

#### Returns:

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

#### Raises:

• ValueError: If self.is_non_singular is False.

### tensor_rank_dynamic(name='tensor_rank_dynamic')

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(name='to_dense')`

Return a dense (batch) matrix representing this operator.