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I want to know that what is the efficient way to inverse a sparse matrix.

I want to implement this type of equation using sparse matrix.

$ \Sigma\ = \Omega^{-1}$ Here $ \Omega $ is a Sparse Matrix or it is efficient to say that $ \Omega $ is a band diagonal matrix whose diagonal and off diagonal element have some values and rest of the entries are zero.

My equation is

$ \mu = \Omega^{-1} \ Xi $

Now what is the efficient way to compute Matrix inverse?

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    $\begingroup$ It's rarely a good idea to form the inverse of a matrix $A$ explicitly. Usually you are solving $Ax = b$, and you want $x$ only. The best method for solving $Ax = b$ depends on what kind of sparsity pattern $A$ has. Special methods have been developed for banded matrices. This is discussed in Golub and Van Loan, for example. There is probably a library you can use that will do it for you. $\endgroup$ – littleO Sep 5 '18 at 1:03
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In general there is no efficient way to invert a sparse matrix, since the inverse of a sparse matrix need not be sparse. But you have much more structure because your matrix is band diagonal. Moreover, based on my reading of your question, you only have a single off-diagonal band. In this case you don't need to actually an inverse of $\Omega$. You can just do backward substitution on $\Omega$ and it will be efficient.

See examples in https://www.mathworks.com/content/dam/mathworks/mathworks-dot-com/moler/lu.pdf if you need more help.

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