Left and right scaled Householder errors give a concrete similarity bound

Use a successful real or complex scaled Householder formation on active dimension $m\ge 1$, with $0<u<1$ and $mu<1$. Let $P=P^{\ast }$ be its stored vector's exact unitary reflector, embedded in an $n\times n$ identity. Apply it on the left to the active rows of $H$, using the specified rounded rank-one order, and on the right to the active columns: compute each row dot $d_i=\mathrm{fl}({\sum }_a{H}_{i,s_a}^L{v}_a)$, $c_a=\mathrm{fl}(\widehat{\tau }{\bar{v}}_a)$, the product $\mathrm{fl}(d_i{c}_a)$, and subtraction in that order. Every nonzero exact intermediate is normal; otherwise return the specified failure.

For real arithmetic put $b_m=(1+u)c_m+3u$, with $c_m,{\kappa }_m$ from The scaled computed Householder step satisfies the local backward error premise. With an actual matrix-target column overwrite use $a_m={\kappa }_m$; without such an overwrite use $a_m=b_m$. For complex arithmetic put $b_m=(1+u)c_U+3u$, with $c_U,{\eta }_H(m)$ from Scaled complex Householder and Givens steps have explicit local backward errors, and use $a_m={\eta }_H(m)$ or $a_m=b_m$, respectively. Set ${\alpha }_m=a_m+b_m+a_m{b}_m\mathrm{.}$ Then the complete computed similarity has a perturbation $E$ satisfying $H_{+}=P^{\ast }(H+E)P,\mathrm{\parallel }E{\mathrm{\parallel }}_F\le {\alpha }_m\mathrm{\parallel }H{\mathrm{\parallel }}_F\mathrm{.}$ Identity branches use $E=0$. If a separate deflation first zeros entries of $H$ by $D$, the same assertion holds with $H+D$ inside the similarity and the bound still uses $\mathrm{\parallel }H{\mathrm{\parallel }}_F$, since deleting entries cannot increase Frobenius norm. All explicit target zero writes are included in $a_m$; no unrelated matrix entry is overwritten when the reflector is formed from an external polynomial vector.