SimPEG.utils.inverse_2x2_block_diagonal

SimPEG.utils.inverse_2x2_block_diagonal(a11, a12, a21, a22, return_matrix=True, **kwargs)

Invert a set of 2x2 matricies from vectors containing their elements.

Parameters

a11, a12, a21, a22(n_blocks) numpy.ndarray

All arguments a11, a12, a21, a22 are vectors which contain the corresponding element for all 2x2 matricies

return_matrixbool, optional

• True: Returns the sparse block 2x2 matrix M.

• False: Returns the vectors containing the elements of each matrix’ inverse.

Returns

(2 * n_blocks, 2 * n_blocks) scipy.sparse.coo_matrix or list of (n_blocks) numpy.ndarray

If return_matrix = False, the function will return vectors b11, b12, b21, b22. If return_matrix = True, the function will return the block matrix M

Notes

The elements of a 2x2 matrix A are given by:

$$\begin{array}{r}\begin{array}{c}A=\left[\begin{array}{cc}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}\right]\end{array}\end{array}$$

For a set of 2x2 matricies, the elements may be stored in a set of 4 distinct vectors ${\mathbf{a}}_{\mathbf{11}}$, ${\mathbf{a}}_{\mathbf{12}}$, ${\mathbf{a}}_{\mathbf{21}}$ and ${\mathbf{a}}_{\mathbf{22}}$. For each matrix, inverse_2x2_block_diagonal ouputs the vectors containing the elements of each matrix’ inverse; i.e. ${\mathbf{b}}_{\mathbf{11}}$, ${\mathbf{b}}_{\mathbf{12}}$, ${\mathbf{b}}_{\mathbf{21}}$ and ${\mathbf{b}}_{\mathbf{22}}$ where:

$$\begin{array}{r}\begin{array}{c}{A}^{-1}=B=\left[\begin{array}{cc}{b}_{11}& b_{12}\\ b_{21}& b_{22}\end{array}\right]\end{array}\end{array}$$

For special applications, we may want to output the elements of the inverses of the matricies as a 2x2 block matrix of the form:

$$\begin{array}{r}\begin{array}{c}M=\left[\begin{array}{cc}{D}_{11}& D_{12}\\ D_{21}& D_{22}\end{array}\right]\end{array}\end{array}$$

where $D_{ij}$ are diagonal matrices whose non-zero elements are defined by vector ${\mathbf{b}}_{\mathbf{ij}}$. Where n is the number of matricies, the block matrix is sparse with dimensions (2n, 2n).

Examples