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Calculate sparse inverse matrix in MATLAB

I have a large square n x n matrix where n = 1,000,000. I need to calculate the inverse of this matrix. This would take a long time to do directly. However, the inverse matrix is very sparse. I know which elements of the inverse matrix are non-zero, and each row (and column) only has 100 non-zero entries.
So I thought I could directly calculate each of these 100,000,000 elements of the inverse matrix. Therefore I need MATLAB code to calculate individual elements of an inverse matrix in a much faster way than simply taking the inverse and checking the individual elements.

That is, given a matrix M, find element (i,j) of inv(M) quickly and directly using MATLAB. The answer must be much faster to calculate than simply taking the inverse and checking the appropriate elements.

tadsyoung

ASKER

C or C++ code I could compile into a MEX file would also be okay.

d-glitch

What is the context of this question?
What is the data and where are you getting it?
What sort of job/project/assignment is this?

Isn't this functionality built into MATLAB already?

http://www.mathworks.com/access/helpdesk/help/techdoc/math/f6-19352.html#f6-41044

tadsyoung

ASKER

This is an attempt to make a research project work faster. The input matrix is a covariance matrix, and I am trying to obtain the inverse covariance matrix. Matlab can, of course, take the inverse of a matrix. But because my matrix is so large, and has special properties (i.e. the output matrix is sparse) I thought there must surely be a better way to do it.

This is how the input matrix (cov) is created, and how it is used (out):

for j=1:size(x,1);
            covpart(:,:,j)=gaussians(j)*(distance_vector(:,j)*distance_vector(:,j)');      
end
cov=sum(covpart,3); 

out=exp(-.5*(distance_vector'/cov)*distance_vector);

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tadsyoung

ASKER

"distance_vector" in line 6 is a different variable than any of the distance vectors in line 2. Sorry for the confusion there.