Optimization of the FIND Algorithm to Compute the Inverse of a Sparse Matrix

The FIND algorithm is a fast algorithm designed to calculate entries of the inverse of a sparse matrix. Such calculation is critical in many applications, e.g., quantum transport in nano-devices. For a 2D device discretized as N × N mesh, the best known algorithms have a running time of O(N
⁴
), whereas FIND only requires O(N
³
), although with a larger constant factor. By exploiting the extra sparsity and symmetry, the size of the problem where FIND becomes faster than others may decrease from a 130 × 130 mesh down to a 40 × 40 mesh. This improvement will make the optimized FIND algorithm appealing to small problems as well, thus becoming competitive for most real applications.