# pylops.Block¶

pylops.Block(ops, nproc=1, dtype=None)[source]

Block operator.

Create a block operator from N lists of M linear operators each.

Parameters: ops : list List of lists of operators to be combined in block fashion. Alternatively, numpy.ndarray or scipy.sparse matrices can be passed in place of one or more operators. nproc : int, optional Number of processes used to evaluate the N operators in parallel using multiprocessing. If nproc=1, work in serial mode. dtype : str, optional Type of elements in input array.

Notes

In mathematics, a block or a partitioned matrix is a matrix that is interpreted as being broken into sections called blocks or submatrices. Similarly a block operator is composed of N sets of M linear operators each such that its application in forward mode leads to

$\begin{split}\begin{bmatrix} \mathbf{L}_{1,1} & \mathbf{L}_{1,2} & \ldots & \mathbf{L}_{1,M} \\ \mathbf{L}_{2,1} & \mathbf{L}_{2,2} & \ldots & \mathbf{L}_{2,M} \\ \vdots & \vdots & \ddots & \vdots \\ \mathbf{L}_{N,1} & \mathbf{L}_{N,2} & \ldots & \mathbf{L}_{N,M} \end{bmatrix} \begin{bmatrix} \mathbf{x}_{1} \\ \mathbf{x}_{2} \\ \vdots \\ \mathbf{x}_{M} \end{bmatrix} = \begin{bmatrix} \mathbf{L}_{1,1} \mathbf{x}_{1} + \mathbf{L}_{1,2} \mathbf{x}_{2} + \mathbf{L}_{1,M} \mathbf{x}_{M} \\ \mathbf{L}_{2,1} \mathbf{x}_{1} + \mathbf{L}_{2,2} \mathbf{x}_{2} + \mathbf{L}_{2,M} \mathbf{x}_{M} \\ \vdots \\ \mathbf{L}_{N,1} \mathbf{x}_{1} + \mathbf{L}_{N,2} \mathbf{x}_{2} + \mathbf{L}_{N,M} \mathbf{x}_{M} \end{bmatrix}\end{split}$

$\begin{split}\begin{bmatrix} \mathbf{L}_{1,1}^H & \mathbf{L}_{2,1}^H & \ldots & \mathbf{L}_{N,1}^H \\ \mathbf{L}_{1,2}^H & \mathbf{L}_{2,2}^H & \ldots & \mathbf{L}_{N,2}^H \\ \vdots & \vdots & \ddots & \vdots \\ \mathbf{L}_{1,M}^H & \mathbf{L}_{2,M}^H & \ldots & \mathbf{L}_{N,M}^H \end{bmatrix} \begin{bmatrix} \mathbf{y}_{1} \\ \mathbf{y}_{2} \\ \vdots \\ \mathbf{y}_{N} \end{bmatrix} = \begin{bmatrix} \mathbf{L}_{1,1}^H \mathbf{y}_{1} + \mathbf{L}_{2,1}^H \mathbf{y}_{2} + \mathbf{L}_{N,1}^H \mathbf{y}_{N} \\ \mathbf{L}_{1,2}^H \mathbf{y}_{1} + \mathbf{L}_{2,2}^H \mathbf{y}_{2} + \mathbf{L}_{N,2}^H \mathbf{y}_{N} \\ \vdots \\ \mathbf{L}_{1,M}^H \mathbf{y}_{1} + \mathbf{L}_{2,M}^H \mathbf{y}_{2} + \mathbf{L}_{N,M}^H \mathbf{y}_{N} \end{bmatrix}\end{split}$
Attributes: shape : tuple Operator shape explicit : bool Operator contains a matrix that can be solved explicitly (True) or not (False)