Quick way to Invert the matrix (A+D) where the inverse of matrix A is known and the matrix D is diagonal.

I have a known matrix A, of which I have computed the inverse of already. I need to add a diagonal matrix to A and invert the sum quickly as A is a large matrix and I need to do this in an optimization loop. The matrix A is nxn and full rank and is invertable, and the matrix D is diagonal, nxn, and also of full rank. Any help would be appreciated. The idea is that inverting (A+D) takes too long as the matrix is large, and the hope is that by knowing the matrix A and its inverse ahead of time, that information can be used to invert the matrix (A+D) quickly.

[Steven Lord]

Are you assuming that (A+D) is invertible?

A = eye(4);
D = -A;

The inverse of A is known (it is A itself.) The inverse of D is also known (it is D itself.) A+D does not have an inverse.

Answer 1

[James Tursa]

Are the values in D small compared to A^-1 ? Maybe you can make use of one of the forms listed here:

https://en.wikipedia.org/wiki/Woodbury_matrix_identity

E.g., if D is small compared to A^-1 maybe you can use a couple of terms of the "perturbative expansions" series.

See also this discussion:

https://math.stackexchange.com/questions/792477/inverse-of-symmetric-matrix-plus-diagonal-matrix-if-square-matrixs-inverse-is-k