  Traditional linear stability analysis based on matrix diagonalization is a
computationally intensive $O(n^3)$ process for $n$-dimensional systems of
differential equations, posing substantial limitations for the exploration of
Turing systems of pattern formation where an additional wave-number parameter
needs to be investigated. In this study, we introduce an efficient $O(n)$
technique that leverages Gershgorin's theorem to determine upper limits on
regions of parameter space and the wave number beyond which Turing
instabilities cannot occur. This method offers a streamlined avenue for
exploring the phase diagrams of other complex multiparametric models, such as
those found in systems biology.
