LU vs. QR Decomposition

The main difference is that LU decomposition is a specialized tool for solving square linear systems, while QR decomposition is a more robust and general method primarily used for solving overdetermined least-squares problems.

Key Differences

Feature	LU Decomposition	QR Decomposition
Primary Use Case	Solving square systems: \(Ax=b\)	Solving least-squares problems: \(Ax \approx b\)
Applicability	Requires a square, invertible matrix \(A\).	Works for any \(m \times n\) matrix \(A\) with full column rank.
Numerical Stability	Can be unstable without pivoting. LU with partial pivoting is generally stable in practice but can fail in rare cases.	Inherently more stable because it uses orthogonal matrices, which do not amplify rounding errors. This is its biggest advantage.
Computational Cost	Faster for square systems. Costs \(\approx \frac{2}{3}n^3\) flops.	Slower for square systems. Costs \(\approx \frac{4}{3}n^3\) flops (about twice as much as LU).

Which Method Should You Choose? 🤔

Your choice depends entirely on the problem you are trying to solve.

• Choose LU decomposition (with partial pivoting) when: You need to solve a square linear system \(Ax=b\) and speed is a priority. It is the faster, standard method for this specific task and is stable enough for most applications.

• Choose QR decomposition when: You are solving a least-squares problem (e.g., from a regression or data-fitting task where you have more equations than unknowns). QR is the standard, most numerically reliable method for this. You should also prefer QR if you have a square system but suspect it is very ill-conditioned and you need maximum numerical stability, even at the cost of speed.