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
| Year | \( n \) | |
| 1950 | \( n=20 \) | (Wilkinson) |
| 1965 | \( n=200 \) | (Forsythe et al.) |
| 1980 | \( n=2000 \) | Linpack |
| 1995 | \( n=20000 \) | Lapack |
| 2017 | \( n\sim 10^5 \) | Lapack |
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.