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Combines `LinearOperators`

into a blockwise lower-triangular matrix.

Inherits From: `LinearOperator`

```
tf.linalg.LinearOperatorBlockLowerTriangular(
operators, is_non_singular=None, is_self_adjoint=None,
is_positive_definite=None, is_square=None,
name='LinearOperatorBlockLowerTriangular'
)
```

This operator is initialized with a nested list of linear operators, which
are combined into a new `LinearOperator`

whose underlying matrix
representation is square and has each operator on or below the main diagonal,
and zero's elsewhere. Each element of the outer list is a list of
`LinearOperators`

corresponding to a row-partition of the blockwise structure.
The number of `LinearOperator`

s in row-partion `i`

must be equal to `i`

.

For example, a blockwise `3 x 3`

`LinearOperatorBlockLowerTriangular`

is
initialized with the list `[[op_00], [op_10, op_11], [op_20, op_21, op_22]]`

,
where the `op_ij`

, `i < 3, j <= i`

, are `LinearOperator`

instances. The
`LinearOperatorBlockLowerTriangular`

behaves as the following blockwise
matrix, where `0`

represents appropriately-sized [batch] matrices of zeros:

```
[[op_00, 0, 0],
[op_10, op_11, 0],
[op_20, op_21, op_22]]
```

Each `op_jj`

on the diagonal is required to represent a square matrix, and
hence will have shape `batch_shape_j + [M_j, M_j]`

. `LinearOperator`

s in row
`j`

of the blockwise structure must have `range_dimension`

equal to that of
`op_jj`

, and `LinearOperators`

in column `j`

must have `domain_dimension`

equal to that of `op_jj`

.

If each `op_jj`

on the diagonal has shape `batch_shape_j + [M_j, M_j]`

, then
the combined operator has shape `broadcast_batch_shape + [sum M_j, sum M_j]`

,
where `broadcast_batch_shape`

is the mutual broadcast of `batch_shape_j`

,
`j = 0, 1, ..., J`

, assuming the intermediate batch shapes broadcast.
Even if the combined shape is well defined, the combined operator's
methods may fail due to lack of broadcasting ability in the defining
operators' methods.

For example, to create a 4 x 4 linear operator combined of three 2 x 2 operators:

operator_0 = tf.linalg.LinearOperatorFullMatrix([[1., 2.], [3., 4.]]) operator_1 = tf.linalg.LinearOperatorFullMatrix([[1., 0.], [0., 1.]]) operator_2 = tf.linalg.LinearOperatorLowerTriangular([[5., 6.], [7., 8]]) operator = LinearOperatorBlockLowerTriangular( ... [[operator_0], [operator_1, operator_2]])

`operator.to_dense()`

`<tf.Tensor: shape=(4, 4), dtype=float32, numpy=`

`array([[1., 2., 0., 0.],`

`[3., 4., 0., 0.],`

`[1., 0., 5., 0.],`

`[0., 1., 7., 8.]], dtype=float32)>`

`operator.shape`

`TensorShape([4, 4])`

`operator.log_abs_determinant()`

`<tf.Tensor: shape=(), dtype=float32, numpy=4.3820267>`

`x0 = [[1., 6.], [-3., 4.]]`

`x1 = [[0., 2.], [4., 0.]]`

`x = tf.concat([x0, x1], 0) # Shape [2, 4] Tensor`

`operator.matmul(x)`

`<tf.Tensor: shape=(4, 2), dtype=float32, numpy=`

`array([[-5., 14.],`

`[-9., 34.],`

`[ 1., 16.],`

`[29., 18.]], dtype=float32)>`

The above `matmul`

is equivalent to:

tf.concat([operator_0.matmul(x0), ... operator_1.matmul(x0) + operator_2.matmul(x1)], axis=0)

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

#### For example:

Create a [2, 3] batch of 4 x 4 linear operators:

matrix_44 = tf.random.normal(shape=[2, 3, 4, 4]) operator_44 = tf.linalg.LinearOperatorFullMatrix(matrix_44)

Create a [1, 3] batch of 5 x 4 linear operators:

matrix_54 = tf.random.normal(shape=[1, 3, 5, 4]) operator_54 = tf.linalg.LinearOperatorFullMatrix(matrix_54)

Create a [1, 3] batch of 5 x 5 linear operators:

matrix_55 = tf.random.normal(shape=[1, 3, 5, 5]) operator_55 = tf.linalg.LinearOperatorFullMatrix(matrix_55)

Combine to create a [2, 3] batch of 9 x 9 operators:

operator_99 = LinearOperatorBlockLowerTriangular( ... [[operator_44], [operator_54, operator_55]]) operator_99.shape TensorShape([2, 3, 9, 9])

Create a shape [2, 1, 9] batch of vectors and apply the operator to it.

x = tf.random.normal(shape=[2, 1, 9]) y = operator_99.matvec(x) y.shape TensorShape([2, 3, 9])

Create a blockwise list of vectors and apply the operator to it. A blockwise list is returned.

x4 = tf.random.normal(shape=[2, 1, 4]) x5 = tf.random.normal(shape=[2, 3, 5]) y_blockwise = operator_99.matvec([x4, x5]) y_blockwise[0].shape TensorShape([2, 3, 4]) y_blockwise[1].shape TensorShape([2, 3, 5])

#### Performance

Suppose `operator`

is a `LinearOperatorBlockLowerTriangular`

consisting of `D`

row-partitions and `D`

column-partitions, such that the total number of
operators is `N = D * (D + 1) // 2`

.

`operator.matmul`

has complexity equal to the sum of the`matmul`

complexities of the individual operators.`operator.solve`

has complexity equal to the sum of the`solve`

complexities of the operators on the diagonal and the`matmul`

complexities of the operators off the diagonal.`operator.determinant`

has complexity equal to the sum of the`determinant`

complexities of the operators on the diagonal.

#### 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:

: Iterable of iterables of`operators`

`LinearOperator`

objects, each with the same`dtype`

. Each element of`operators`

corresponds to a row- partition, in top-to-bottom order. The operators in each row-partition are filled in left-to-right. For example,`operators = [[op_0], [op_1, op_2], [op_3, op_4, op_5]]`

creates a`LinearOperatorBlockLowerTriangular`

with full block structure`[[op_0, 0, 0], [op_1, op_2, 0], [op_3, op_4, op_5]]`

. The number of operators in the`i`

th row must be equal to`i`

, such that each operator falls on or below the diagonal of the blockwise structure.`LinearOperator`

s that fall on the diagonal (the last elements of each row) must be square. The other`LinearOperator`

s must have domain dimension equal to the domain dimension of the`LinearOperator`

s in the same column-partition, and range dimension equal to the range dimension of the`LinearOperator`

s in the same row-partition.: Expect that this operator is non-singular.`is_non_singular`

: Expect that this operator is equal to its hermitian transpose.`is_self_adjoint`

: Expect that this operator is positive definite, meaning the quadratic form`is_positive_definite`

`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: Expect that this operator acts like square [batch] matrices. This will raise a`is_square`

`ValueError`

if set to`False`

.: A name for this`name`

`LinearOperator`

.

#### Attributes:

: Returns the adjoint of the current`H`

`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]`

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

If this operator acts like the batch matrix

`A`

with`A.shape = [B1,...,Bb, M, N]`

, then this returns`N`

.: The`dtype`

`DType`

of`Tensor`

s handled by this`LinearOperator`

.: List of graph dependencies of this`graph_parents`

`LinearOperator`

. (deprecated)`is_non_singular`

`is_positive_definite`

`is_self_adjoint`

: Return`is_square`

`True/False`

depending on if this operator is square.`operators`

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

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`

.: Rank (in the sense of tensors) of matrix corresponding to this operator.`tensor_rank`

If this operator acts like the batch matrix

`A`

with`A.shape = [B1,...,Bb, M, N]`

, then this returns`b + 2`

.

#### Raises:

: If all operators do not have the same`TypeError`

`dtype`

.: If`ValueError`

`operators`

is empty, contains an erroneous number of elements, or contains operators with incompatible shapes.

## Methods

`__matmul__`

```
__matmul__(
other
)
```

`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`

.: A name to give this`name`

`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:

: A name for this`name`

`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:

: A string name to prepend to created ops.`name`

#### 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:

: A name to give this`name`

`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:

: A string name to prepend to created ops.`name`

#### 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:

: A name for this`name`

`Op`

.

#### Returns:

`int32`

`Tensor`

`cholesky`

```
cholesky(
name='cholesky'
)
```

Returns a Cholesky factor as a `LinearOperator`

.

Given `A`

representing this `LinearOperator`

, if `A`

is positive definite
self-adjoint, return `L`

, where `A = L L^T`

, i.e. the cholesky
decomposition.

#### Args:

: A name for this`name`

`Op`

.

#### Returns:

`LinearOperator`

which represents the lower triangular matrix
in the Cholesky decomposition.

#### Raises:

: When the`ValueError`

`LinearOperator`

is not hinted to be positive definite and self adjoint.

`cond`

```
cond(
name='cond'
)
```

Returns the condition number of this linear operator.

#### Args:

: A name for this`name`

`Op`

.

#### Returns:

Shape `[B1,...,Bb]`

`Tensor`

of same `dtype`

as `self`

.

`determinant`

```
determinant(
name='det'
)
```

Determinant for every batch member.

#### Args:

: A name for this`name`

`Op`

.

#### Returns:

`Tensor`

with shape `self.batch_shape`

and same `dtype`

as `self`

.

#### Raises:

: If`NotImplementedError`

`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.linalg.diag_part(my_operator.to_dense())
==> [1., 2.]
```

#### Args:

: A name for this`name`

`Op`

.

#### Returns:

: A`diag_part`

`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:

: A name for this`name`

`Op`

.

#### Returns:

`int32`

`Tensor`

`eigvals`

```
eigvals(
name='eigvals'
)
```

Returns the eigenvalues of this linear operator.

If the operator is marked as self-adjoint (via `is_self_adjoint`

)
this computation can be more efficient.

#### Args:

: A name for this`name`

`Op`

.

#### Returns:

Shape `[B1,...,Bb, N]`

`Tensor`

of same `dtype`

as `self`

.

`inverse`

```
inverse(
name='inverse'
)
```

Returns the Inverse of this `LinearOperator`

.

Given `A`

representing this `LinearOperator`

, return a `LinearOperator`

representing `A^-1`

.

#### Args:

: A name scope to use for ops added by this method.`name`

#### Returns:

`LinearOperator`

representing inverse of this matrix.

#### Raises:

: When the`ValueError`

`LinearOperator`

is not hinted to be`non_singular`

.

`log_abs_determinant`

```
log_abs_determinant(
name='log_abs_det'
)
```

Log absolute value of determinant for every batch member.

#### Args:

: A name for this`name`

`Op`

.

#### Returns:

`Tensor`

with shape `self.batch_shape`

and same `dtype`

as `self`

.

#### Raises:

: If`NotImplementedError`

`self.is_square`

is`False`

.

`matmul`

```
matmul(
x, adjoint=False, adjoint_arg=False, 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`

`LinearOperator`

,`Tensor`

with compatible shape and same`dtype`

as`self`

, or a blockwise iterable of`LinearOperator`

s or`Tensor`

s. See class docstring for definition of shape compatibility.: Python`adjoint`

`bool`

. If`True`

, left multiply by the adjoint:`A^H x`

.: Python`adjoint_arg`

`bool`

. If`True`

, compute`A x^H`

where`x^H`

is the hermitian transpose (transposition and complex conjugation).: A name for this`name`

`Op`

.

#### Returns:

A `LinearOperator`

or `Tensor`

with shape `[..., M, R]`

and same `dtype`

as `self`

, or if `x`

is blockwise, a list of `Tensor`

s with shapes that
concatenate to `[..., M, R]`

.

`matvec`

```
matvec(
x, adjoint=False, name='matvec'
)
```

Transform [batch] vector `x`

with left multiplication: `x --> Ax`

.

```
# Make an operator acting like batch matrix 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`

, or an iterable of`Tensor`

s.`Tensor`

s are treated a [batch] vectors, meaning for every set of leading dimensions, the last dimension defines a vector. See class docstring for definition of compatibility.: Python`adjoint`

`bool`

. If`True`

, left multiply by the adjoint:`A^H x`

.: A name for this`name`

`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:

: A name for this`name`

`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:

: A name for this`name`

`Op`

.

#### Returns:

`int32`

`Tensor`

`solve`

```
solve(
rhs, adjoint=False, adjoint_arg=False, 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.

Given the blockwise `n + 1`

-by-`n + 1`

linear operator:

op = [[A_00 0 ... 0 ... 0], [A_10 A_11 ... 0 ... 0], ... [A_k0 A_k1 ... A_kk ... 0], ... [A_n0 A_n1 ... A_nk ... A_nn]]

we find `x = op.solve(y)`

by observing that

`y_k = A_k0.matmul(x_0) + A_k1.matmul(x_1) + ... + A_kk.matmul(x_k)`

and therefore

```
x_k = A_kk.solve(y_k -
A_k0.matmul(x_0) - ... - A_k(k-1).matmul(x_(k-1)))
```

where `x_k`

and `y_k`

are the `k`

th blocks obtained by decomposing `x`

and `y`

along their appropriate axes.

We first solve `x_0 = A_00.solve(y_0)`

. Proceeding inductively, we solve
for `x_k`

, `k = 1..n`

, given `x_0..x_(k-1)`

.

The adjoint case is solved similarly, beginning with
`x_n = A_nn.solve(y_n, adjoint=True)`

and proceeding backwards.

#### 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, or a list of`Tensor`

s.`Tensor`

s are treated like a [batch] matrices meaning for every set of leading dimensions, the last two dimensions defines a matrix. See class docstring for definition of compatibility.: Python`adjoint`

`bool`

. If`True`

, solve the system involving the adjoint of this`LinearOperator`

:`A^H X = rhs`

.: Python`adjoint_arg`

`bool`

. If`True`

, solve`A X = rhs^H`

where`rhs^H`

is the hermitian transpose (transposition and complex conjugation).: A name scope to use for ops added by this method.`name`

#### Returns:

`Tensor`

with shape `[...,N, R]`

and same `dtype`

as `rhs`

.

#### Raises:

: If`NotImplementedError`

`self.is_non_singular`

or`is_square`

is False.

`solvevec`

```
solvevec(
rhs, adjoint=False, 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, or list of`Tensor`

s (for blockwise operators).`Tensor`

s are treated as [batch] vectors, meaning for every set of leading dimensions, the last dimension defines a vector. See class docstring for definition of compatibility regarding batch dimensions.: Python`adjoint`

`bool`

. If`True`

, solve the system involving the adjoint of this`LinearOperator`

:`A^H X = rhs`

.: A name scope to use for ops added by this method.`name`

#### Returns:

`Tensor`

with shape `[...,N]`

and same `dtype`

as `rhs`

.

#### Raises:

: If`NotImplementedError`

`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:

: A name for this`name`

`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:

: A name for this`name`

`Op`

.

#### Returns:

Shape `[B1,...,Bb]`

`Tensor`

of same `dtype`

as `self`

.