Updated 31 May 2018
Matlab has very good built-in support for fast matrix inversion exploiting the structure of a matrix. See the algorithms section of the documentation on `mldivide` for more information: https://www.mathworks.com/help/matlab/ref/mldivide.html#bt4jslc-6
The functions provided here were initially written to support a latent Gaussian Process inference implementation, where we frequently encounter large matrices which have sub-matrices with "nice" structure, but the full matrix does not. These functions implement matrix inversion (`blockinv`) and division (`blockmldivide` and `blockmrdivide`) by extracting sub-matrices of a user-defined size and calling the matlab built-ins on them. In certain cases, this means that the built-ins are able to exploit structure in the sub-matrices for very fast inversion and quickly combine the results together.
In general, expect these functions to be slower than simply using built-ins unless you are sure that your sub-matrices (but not the full matrix) have the kind of structure exploited by mldivide.
Richard Lange (2020). Block Matrix Inverse tools (https://www.github.com/wrongu/block-matrix-inverse-tools), GitHub. Retrieved .