Find the best approximate solution to \( A \mathbf{x} \approx \mathbf{b} \) using QR Decomposition.
Minimizes the Euclidean norm \( \| A \mathbf{x} - \mathbf{b} \|_2 \).
When the system $A\mathbf{x} = \mathbf{b}$ is overdetermined (more equations than unknowns), we find the solution that minimizes the residual:
$ \min_{\mathbf{x}} \| A\mathbf{x} - \mathbf{b} \|_2 $
QR Method:
Factor $A = QR$ where $Q$ is orthogonal and $R$ is upper triangular. Then solve:
$ R\mathbf{x} = Q^T\mathbf{b} $
Advantages: