tf.linalg.LinearOperatorBlockLowerTriangular Combines LinearOperators into a blockwise lower-triangular matrix. Inherits From: LinearOperator, Module  View aliases  Compat aliases for migration 
See Migration guide for more details. tf.compat.v1.linalg.LinearOperatorBlockLowerTriangular  
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 LinearOperators 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]. LinearOperators 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)
<tf.Tensor: shape=(4, 2), dtype=float32, numpy=
array([[-5., 14.],
       [-9., 34.],
       [ 1., 16.],
       [29., 18.]], 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] + [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
  operators   Iterable of iterables of 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 ith row must be equal to i, such that each operator falls on or below the diagonal of the blockwise structure. LinearOperators that fall on the diagonal (the last elements of each row) must be square. The other LinearOperators must have domain dimension equal to the domain dimension of the LinearOperators in the same column-partition, and range dimension equal to the range dimension of the LinearOperators in the same row-partition.  
  is_non_singular   Expect that this operator is non-singular.  
  is_self_adjoint   Expect that this operator is equal to its hermitian transpose.  
  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. This will raise a ValueError if set to False.  
  name   A name for this LinearOperator.   
 


 Raises
  TypeError   If all operators do not have the same dtype.  
  ValueError   If operators is empty, contains an erroneous number of elements, or contains operators with incompatible shapes.   
 


 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] 
 
  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.  
  operators  
 
  parameters   Dictionary of parameters used to instantiate 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. 
 
  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 View source 
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 View source 
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  
 cholesky View source 
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
  name   A name for this Op.   
 


 Returns   LinearOperator which represents the lower triangular matrix in the Cholesky decomposition.  

 


 Raises
  ValueError   When the LinearOperator is not hinted to be positive definite and self adjoint.    cond View source 
cond(
    name='cond'
)
 Returns the condition number of this linear operator.
 


 Args
  name   A name for this Op.   
 


 Returns   Shape [B1,...,Bb] Tensor of same dtype as self.  
 determinant View source 
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 View source 
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
  name   A name for this Op.   
 


 Returns
  diag_part   A Tensor of same dtype as self.    domain_dimension_tensor View source 
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  
 eigvals View source 
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. 
Note: This currently only supports self-adjoint operators.

 


 Args
  name   A name for this Op.   
 


 Returns   Shape [B1,...,Bb, N] Tensor of same dtype as self.  
 inverse View source 
inverse(
    name='inverse'
)
 Returns the Inverse of this LinearOperator. Given A representing this LinearOperator, return a LinearOperator representing A^-1.
 


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


 Returns   LinearOperator representing inverse of this matrix.  

 


 Raises
  ValueError   When the LinearOperator is not hinted to be non_singular.    log_abs_determinant View source 
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 View source 
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 LinearOperators or Tensors. See class docstring for definition of shape 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 LinearOperator or Tensor with shape [..., M, R] and same dtype as self, or if x is blockwise, a list of Tensors with shapes that concatenate to [..., M, R].  
 matvec View source 
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 Tensors. Tensors 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.  
  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 View source 
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 View source 
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 View source 
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 kth 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 Tensors. Tensors 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.  
  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 View source 
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 Tensors (for blockwise operators). Tensors 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.  
  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 View source 
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 View source 
to_dense(
    name='to_dense'
)
 Return a dense (batch) matrix representing this operator. trace View source 
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.  
 __matmul__ View source 
__matmul__(
    other
)