Past exam of the options II course of the University of Oxford / 2011 / P / P4 / d

Multiply $Ax=b$ by $Q^T$ to get the equivalent triangular systemFor nonsingular $A$, compute $c=Q^{T}b$ and use back substitution:Orthogonal transformations preserve the Euclidean norm, making this a numerically stable reduction. The factors can be applied as stored Householder transformations without explicitly forming $Q$.

Since $Q$ is invertible, $A$ is singular exactly when $R$ is singular. For an upper triangular matrix,Thus a singular $A$ forces at least one zero diagonal entry in $R$. In that case, ordinary division-based back substitution must be replaced by a consistency/rank analysis: the system may have no solution or infinitely many solutions, and a rank-revealing factorization can identify the free parameters.