The mathematics behind the inverse of large sparse matrices is very interesting and widely used in several fields. Sometimes, It is required to find the inverse of these kinds of matrices. However, finding the same is computationally costly. I want to know, the related research, what happens when a single entry (or a few entries) are perturbed in the original matrix then how much it will affect the entries of inverse of the matrix.
This is a very interesting and intriguing Q. Maybe the idea below can be used.
Long ago , Prof.W.A.Lewis (IIT Chicago) showed a method of removing a transmission line impedance in a network, by using a negative impedance in parallel to the line to be removed ... this kept all the the remaining network impedances unchanged. Current flow in a network is related to matrix inversion of sorts. Conjecture that this idea could be used more formally.
Good Luck
Cheers
A standard trick in these cases is to use the Sherman Morrison formula
https://en.wikipedia.org/wiki/Sherman%E2%80%93Morrison_formula
However, the inverse of a sparse matrix does not have to be sparse and in particular one does not want to store inverses for sparse large matrices, so that the formula would rather have to be applied to the action A^-1b of the inverse on the right hand side of the linear system, so as to correct the solution to the original linear system A^-1b with a hopefully limited amount of operations.
Please notice that this is a very generic comment, I am sure somebody in the sparse solver community will have studied the problem in much greater depth.
Dear Bharat Soni
In addition to the helpful answers, I would like to add:
Perturbing some entries of the symmetric sparse matrix will cause the loss of the matrix's symmetric property. Consequently, you need to turn into the general form of the sparse matrix.
Regards
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