This is a computational-related question, i couldn't find a better SE forum for this.
I have a set of equations for $ \textbf{Z}$ 3-index quantities of the form
$ S_{mi} S_{nj} Z^{k}_{ij} S_{kp} = W^{p}_{mn}$
where $\textbf{S}$ and $\textbf{W}$ are known, and the indices run from 0 to $N$ (Einstein summation convention is assumed)
The straightforward solution would be multiplying by $ \textbf{S}^{-1}$ appropiately and obtain:
$ Z^{k}_{ij} = S^{-1}_{pk} W^{p}_{mn} S^{-1}_{im} S^{-1}_{jn} $
Which is fine, unless $N$ is big, and computing the $ \textbf{S}^{-1}$ inverse complexity grows as $O(N^3)$
I was wondering if there are Gauss-Jordan or UL -like decompositions that might reduce the complexity of this problem with big $N$?