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LinearOperator acting like a [batch] of toeplitz matrices.

Inherits From: LinearOperator, Module

This operator acts like a [batch] Toeplitz 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. This matrix A is not materialized, but for purposes of broadcasting this shape will be relevant.

Description in terms of toeplitz matrices

Toeplitz means that A has constant diagonals. Hence, A can be generated with two vectors. One represents the first column of the matrix, and the other represents the first row.

Below is a 4 x 4 example:

A = |a b c d|
    |e a b c|
    |f e a b|
    |g f e a|

Example of a Toeplitz operator.

# Create a 3 x 3 Toeplitz operator.
col = [1., 2., 3.]
row = [1., 4., -9.]
operator = LinearOperatorToeplitz(col, row)

==> [[1., 4., -9.],
     [2., 1., 4.],
     [3., 2., 1.]]

==> [3, 3]

==> scalar Tensor

x = ... Shape [3, 4] Tensor
==> Shape [3, 4] Tensor

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] to [D1,...,Dd]

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.

col Shape [B1,...,Bb, N] Tensor with b >= 0 N >= 0. The first column of the operator. Allowed dtypes: float16, float