# How can I approximate this large matrix?

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Jerry Guern on 4 Nov 2022
Commented: Jerry Guern on 4 Nov 2022
I have a sparse symmetric matrix M, roughly 10^6 x 10^6, but only about 100 values in each column (or row) are non-zero, so they'll all fit memory / disk space. Values are complex, and all magnitudes are < 10^-1 I need an approximation of
M * inv(eye() - M) = M + M^2 + M^3 + M^4 + ...
The inverse in there won't be sparse, but the produc hopefully will be.
So, if I have the nonzero values and their locations tabled up, what's the best way to go about this calculation in MATLAB?
Are there library tools avaiable for large sparse matrices?
And should I try to approximate the LHS or the RHS? I know the RHS can be approximated by calculating M^2, M^4, M^8,... and then multiplying M * (I+M) * (I+M^2) * (I+M^4) * (I+M^8) = M + M^2 + ... + M^16. But I'm hoping there might be some iterative way to approximate the LHS. And in either case I'd still need to know how to set M up as a sparse matrix and take advantage of M's sparseness.

Bruno Luong on 4 Nov 2022
Edited: Bruno Luong on 4 Nov 2022
T = M;
niter = 5:
for k=1:niter
T = M*(speye(size(M)) + T);
end
This returns
T = M + M^2 + .... + M^6
Bruno Luong on 4 Nov 2022
Edited: Bruno Luong on 4 Nov 2022
But MATLAB sparse stores only non-zeros elements.
If your matrix are big and not sparse enough to be store as sparse in the memory of your PC, then you are simply stuck.
I don't know what you meant by tool, MATLAB matrix library comprises a comprehensive array of functions that can deal with sparse matrix.
Jerry Guern on 4 Nov 2022
Ah, thank you, this was helpful on the sparse matrix handling part. I realize now I probably should have posted this as two separate questions, one on the sparse matrix tools and one on the approximation.

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