Direct or non-iterative methods require for matrices of dimensionality $n\times n$ typically $O(n^3)$ operations. These methods are normally called standard methods and are used for dimensionalities $n \sim 10^5$ or smaller. A brief historical overview

shows that in the course of 60 years the dimension that direct diagonalization methods can handle has increased by almost a factor of $10^4$. However, it pales beside the progress achieved by computer hardware, from flops to petaflops, a factor of almost $10^{15}$. We see clearly played out in history the $O(n^3)$ bottleneck of direct matrix algorithms.

Sloppily speaking, when $n\sim 10^4$ is cubed we have $O(10^{12})$ operations, which is smaller than the $10^{15}$ increase in flops.