Sparse matrix inversion

I have the impedance matrix $$Y$$, formulated from an electrical network by augmented nodal analysis. The matrix $$Y$$ is shown as an image to illustrate its feature visually, where all the white blocks are zero.

What could be the best possible way to inverse such matrix which is required to solve the linear equation in the form $$Yx=b$$? Furthermore, in each iteration of the network solution, only the first $$8\times 8$$ elements get new values. All other elements remain as is. Any suggestion will be highly appreciated.

• It sounds like you just need to solve la linear equation, not invert a matrix, in which case, don;t invert the matrix. Jul 18 '19 at 1:06
• Can you please suggest me a method of solving the linear equation that will be computationally efficient in this case. Note that the matrix is not a "positive definite" as it fails Cholesky decomposition. Thanks Jul 18 '19 at 1:17
• how large is your matrix $Y$? depending on the overall size of $Y$ wrt to the $8\times 8$ changing block, there might be several alternatives to consider. Jul 18 '19 at 2:10
• As shown in figure Y is a 19x19 matrix where the first 8X8 is changing in every iteration. Jul 18 '19 at 2:37