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:

\[ \begin{align}\begin{aligned}A = \begin{bmatrix} a_{11} & a_{12} & a_{13} \\a_{21} & a_{22} & a_{23} \\a_{31} & a_{32} & a_{33} \end{bmatrix}\end{aligned}\end{align} \]

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

\[ \begin{align}\begin{aligned}A^{-1} = B = \begin{bmatrix} b_{11} & b_{12} & b_{13} \\b_{21} & b_{22} & b_{23} \\b_{31} & b_{32} & b_{33} \end{bmatrix}\end{aligned}\end{align} \]

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

\[ \begin{align}\begin{aligned}M = \begin{bmatrix} D_{11} & D_{12} & D_{13} \\D_{21} & D_{22} & D_{23} \\D_{31} & D_{32} & D_{33} \end{bmatrix}\end{aligned}\end{align} \]

where \(D_{ij}\) are diagonal matrices whose non-zero elements are defined by vector \(\mathbf{b_{ij}}\). Where n is the number of matricies, the block matrix is sparse with dimensions (3n, 3n).

Examples