Comparison between Gauss and Cramer

Why is Gauss preferred in practice? The reason is the computational cost, which grows much more slowly as the number of unknowns increases.

Observation — Comparison with Cramer

Cramer requires $4$ determinants of size $3\times 3$ ($\Delta,\Delta_x,\Delta_y,\Delta_z$). Gauss requires $\approx n^3/3$ multiplications: for $n=3$ it is $9$ against $\approx 24$, but for $n=10$ it is $\approx 330$ against $\sim 10^7$. Gauss always wins for large $n$. It is the standard algorithm in linear algebra packages (NumPy, Matlab, R).

Links

Topics: Linear systems
Concepts: Computational complexity · Gauss’s method · Cramer’s rule
Skills: Estimate
People: Gabriel Cramer · Carl Friedrich Gauss