# multi-index linear equation without computing explicit inverse

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 $N$?

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Given that there are $(N+1)^3$ components in $\mathbf{Z}$ I wouldn't worry too much about a step with $\mathcal{O}(N^3)$ complexity. –  Jyrki Lahtonen Jun 21 '11 at 16:31