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.
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.
The testBlockFunctions.m script generates random matrices and asserts that blockinv, blockmldivide, and blockmrdivide are within a reasonable tolerance of their built-in counterparts.
The profileBlockInv.m script generates random structured matrices of increasing size and profiles the performance of built-in versus block functions.