Is it possible to compute individual elements of a matrix inverse?

I have a large sparse regular matrix. Instead of computing the full inverse of it, I just want to determine individual elements of it (mainly the diagonal elements of the inverse). Is there any way to do that in Matlab?

[Stephen23]

Some light reading to get you started:

https://math.stackexchange.com/questions/978051/find-diagonal-of-inverse-matrix

http://www-users.cs.umn.edu/~saad/PDF/Sparse_days_06_15_2010.pdf: "Computing the diagonal of the inverse of a sparse matrix"

http://www-users.cs.umn.edu/~saad/PDF/umsi-2010-42.pdf: "A Probing Method for Computing the Diagonal of the Matrix Inverse"

https://web.stanford.edu/~lexing/publication/diagonal.pdf: "FAST ALGORITHM FOR EXTRACTING THE DIAGONAL OF THE INVERSE MATRIX WITH APPLICATION TO THE ELECTRONIC STRUCTURE ANALYSIS OF METALLIC SYSTEMS"

Question: how big is the matrix?