For this routine two examples are presented. Both illustrate how the reduction to bidiagonal form of a matrix $A$ may be preceded by a $QR$ or $LQ$ factorization of $A$.

In the first example, $m>n$, and The routine first performs a $QR$ factorization of $A$ as $A=Q_{a}R$ and then reduces the factor $R$ to bidiagonal form $B$: $R=Q_{b}B{P}^{\mathrm{T}}$. Finally it forms $Q_a$ and calls dormbr to form $Q=Q_a{Q}_b$.

In the second example, $m<n$, and The routine first performs an $LQ$ factorization of $A$ as $A=L{P}_a^{\mathrm{T}}$ and then reduces the factor $L$ to bidiagonal form $B$: $L=QB{P}_b^{\mathrm{T}}$. Finally it forms $P_b^{\mathrm{T}}$ and calls dormbr to form $P^{\mathrm{T}}=P_b^{\mathrm{T}}{P}_a^{\mathrm{T}}$.