The normal equations square the condition number
Because κ(XᵀX) = κ(X)², QR’s error scales with κ(X)·ε while the normal equations’ scales with κ(X)²·ε.
Same data, same objective, error squared. The cost is one line of code: writing X.T @ X at all.
On the real degree-10 airline Vandermonde:
κ(X) = 2.16e7κ(XᵀX) = 4.65e14— the square, to one digit- coefficient error by the normal equations: 1.95e-03
- coefficient error by QR: 2.04e-14
Eleven orders of magnitude apart.