Scaled Householder reflector formation and application

Let $\mathbb{F}\in \{\mathbb{R},\mathbb{C}\}$. Input a real active column $x\in {\mathbb{R}}^r\subseteq {\mathbb{F}}^r$ and trailing block $B\in {\mathbb{F}}^{r\times q}$ with entries in the floating-point system, $r\ge 1$, $q\ge 0$. If every component of $x$ is zero, return the identity action and a copy of $B$, with the active output column set to exact zeros, so that $C_{+}=[0e_{1}B]$; return $(v=0,\widehat{\tau }=0,\widehat{\alpha }=0,C_{+})$ marked identity. Otherwise compute $s={\max }_{1\le i\le r}\mathrm{\mid }{x}_i\mathrm{\mid }>0$, and then, in increasing $i$, compute $z_i=\mathrm{fl}(x_i\mathrm{/}s)$. Initialize $t=0$ and, in increasing $i$, compute $a_i=\mathrm{fl}(z_i{z}_i)$ followed by $t=\mathrm{fl}(t+a_i)$. Compute $h=\mathrm{fl}(\sqrt{t})$ and then $\widehat{r}=\mathrm{fl}(sh)$. Set $\sigma =1$ if $x_1=0$ and $\sigma =\mathrm{sign}(x_1)$ otherwise, and set $\widehat{\alpha }=-\sigma \widehat{r}$ using the exact sign change. Compute $v_1=\mathrm{fl}(x_1-\widehat{\alpha })$, then copy $v_i=x_i$ exactly for $i=2,\dots ,r$.

Initialize $w=0$ and, in increasing $i$, compute $a_i=\mathrm{fl}(v_i{v}_i)$ followed by $w=\mathrm{fl}(w+a_i)$. Then compute $\widehat{\tau }=\mathrm{fl}(2\mathrm{/}w)$. For each column $j=1,\dots ,q$ in increasing order, initialize $d_j=0$ and, in increasing $i$, compute $a_{ij}=\mathrm{fl}(v_i{B}_{ij})$ followed by $d_j=\mathrm{fl}(d_j+a_{ij})$. After all $d_j$ have been computed, in increasing $i$ compute $p_i=\mathrm{fl}(\widehat{\tau }{v}_i)$. Then, in increasing $j$ and within each column in increasing $i$, compute $U_{ij}=\mathrm{fl}(p_i{d}_j)$ and ${\widehat{B}}_{ij}=\mathrm{fl}(B_{ij}-U_{ij})$ in that order. The complete active output is $C_{+}=[\widehat{\alpha }{e}_1\widehat{B}]$, with exact zeros written below its first diagonal entry. If any elementary operation fails, return the first range failure with its stage and indices in the order above. Also return failure if $t$, $\widehat{r}$, or $w$ is not positive normal, or any intermediate is nonfinite. Otherwise return $(v,\widehat{\tau },\widehat{\alpha },C_{+})$ marked reflector. For a separate block $Y\in {\mathbb{F}}^{r\times q^{\mathrm{\prime }}}$, $q^{\mathrm{\prime }}\ge 0$, apply the stored reflector by computing its sequential column dots in increasing column order, then the vector $p_i=\mathrm{fl}(\widehat{\tau }{v}_i)$ in increasing $i$, and then each rounded product and subtraction by increasing column and row; omit the target-column overwrite. Store reflector factors with their identity/reflector marker to represent the exact orthogonal product; a rounded explicit dense product is a separate output with its own arithmetic error. For a nonzero stored $v$, the associated exact reflector is $H(v)=I-2v{v}^T\mathrm{/}(v^{T}v)$, using Householder reflectors in real or complex inner-product spaces and the QR convention of Full, reduced, and column-pivoted computational QR factorisations. Since $v$ is real, this matrix acts on ${\mathbb{F}}^r$ as an orthogonal reflector over $\mathbb{R}$ and a unitary reflector over $\mathbb{C}$. This definition does not assert a local backward-error bound.