# Gustafson's law

**[Gustafson's law](https://en.wikipedia.org/wiki/Gustafson%27s_law)** models speedup when problem size grows with hardware (**weak scaling**), yielding near-linear speedup: $S(n) = n - \alpha(n-1)$ where $\alpha$ is the sequential fraction. Unlike [[amdahls-law|Amdahl's law]], speedup grows with processor count, making it applicable to most HPC workloads that scale problem size with available hardware.

At 1% sequential overhead and 256 cores, expected speedup is ~250×. This reflects real practice: larger machines solve larger problems, not the same problem faster.

## Example

This table shows speedup scaling with problem size (Gustafson regime).

```
Sequential fraction (α)  S(n=16)  S(n=64)   S(n=256)
1%                      ≈15.8×   ≈63.4×    ≈253.5×
5%                      ≈15.2×   ≈60.8×    ≈243.2×
10%                     ≈14.5×   ≈58.5×    ≈230.4×
```