SimPEG.utils.inverse_3x3_block_diagonal

SimPEG.utils.inverse_3x3_block_diagonal(a11, a12, a13, a21, a22, a23, a31, a32, a33, return_matrix=True, **kwargs)

Invert a set of 3x3 matricies from vectors containing their elements.

Parameters

a11, a12, …, a33(n_blocks) numpy.ndarray

Vectors which contain the corresponding element for all 3x3 matricies

return_matrixbool, optional

• True: Returns the sparse block 3x3 matrix M (default).

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

Returns

(3 * n_blocks, 3 * n_blocks) scipy.sparse.coo_matrix or list of (n_blocks)

numpy.ndarray. If return_matrix = False, the function will return vectors b11, b12, b13, b21, b22, b23, b31, b32, b33. If return_matrix = True, the function will return the block matrix M.

Notes

The elements of a 3x3 matrix A are given by:

$$\begin{array}{r}\begin{array}{c}A=\left[\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33}\end{array}\right]\end{array}\end{array}$$

For a set of 3x3 matricies, the elements may be stored in a set of 9 distinct vectors ${\mathbf{a}}_{\mathbf{11}}$, ${\mathbf{a}}_{\mathbf{12}}$, …, ${\mathbf{a}}_{\mathbf{33}}$. For each matrix, inverse_3x3_block_diagonal ouputs the vectors containing the elements of each matrix’ inverse; i.e. ${\mathbf{b}}_{\mathbf{11}}$, ${\mathbf{b}}_{\mathbf{12}}$, …, ${\mathbf{b}}_{\mathbf{33}}$ where:

$$\begin{array}{r}\begin{array}{c}{A}^{-1}=B=\left[\begin{array}{ccc}{b}_{11}& b_{12}& b_{13}\\ b_{21}& b_{22}& b_{23}\\ b_{31}& b_{32}& b_{33}\end{array}\right]\end{array}\end{array}$$

For special applications, we may want to output the elements of the inverses of the matricies as a 3x3 block matrix of the form:

$$\begin{array}{r}\begin{array}{c}M=\left[\begin{array}{ccc}{D}_{11}& D_{12}& D_{13}\\ D_{21}& D_{22}& D_{23}\\ D_{31}& D_{32}& D_{33}\end{array}\right]\end{array}\end{array}$$

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 (3n, 3n).

Examples