The scaled computed Householder step satisfies the local backward error premise

If $x=0$, the algorithm copies $B$ and the zero column; take $H=I$ and $E=0$. Otherwise put $N=\mathrm{\parallel }x{\mathrm{\parallel }}_2$, ${\alpha }_0=-\sigma N$, $v_0=x-{\alpha }_0{e}_1$ and $V_0=\mathrm{\parallel }{v}_0{\mathrm{\parallel }}_2\ge N$. The nonnegative scaled squared-norm sum has each exact squared input multiplied by at most $r+2$ factors $1+\delta$, $\mathrm{\mid }\delta \mathrm{\mid }\le u$: two from scaling and at most $r$ from squaring and subsequent additions. Thus the computed norm divided by $N$ lies between $(1-u{)}^{r\mathrm{/}2+3}$ and $(1+u{)}^{r\mathrm{/}2+3}$ after square root and rescaling. For exponent $p\ge 1$, convexity of $t^p$ on $t>0$ gives $1-(1-u{)}^p\le (1+u{)}^p-1$. Hence $\mathrm{\mid }\widehat{\alpha }-{\alpha }_0\mathrm{\mid }\le e_{r}N$. Only the first stored coordinate can differ from $v_0$, and subtraction gives $\mathrm{\parallel }v-v_0{\mathrm{\parallel }}_2\le (e_r+u(2+e_r))N=d_{r}N$. Success gives nonzero stored $v$.

Put $V=\mathrm{\parallel }v{\mathrm{\parallel }}_2$. The rounded positive dot product satisfies $(1-u{)}^r{V}^2\le w\le (1+u{)}^r{V}^2$ by expansion of its products and additions, including exact zeros. The division defining $\widehat{\tau }$ therefore gives $\mathrm{\mid }\widehat{\tau }{V}^2\mathrm{/}2-1\mathrm{\mid }\le a_r$; the upper deviation dominates the lower by multiplying the positive denominators. A sequential signed dot product differs from its exact value by at most $g_r{\sum }_i\mathrm{\mid }{v}_i{B}_{ij}\mathrm{\mid }\le g_{r}V\mathrm{\parallel }{B}_{:j}{\mathrm{\parallel }}_2$, by the same factor expansion, [F4], and Cauchy-Schwarz [F5].

For nonzero vectors $a,b$, adding and subtracting $a\mathrm{/}\mathrm{\parallel }b\mathrm{\parallel }$ and using the reverse triangle inequality from [F5] gives $\mathrm{\parallel }a\mathrm{/}\mathrm{\parallel }a\mathrm{\parallel }-b\mathrm{/}\mathrm{\parallel }b\mathrm{\parallel }\mathrm{\parallel }\le 2\mathrm{\parallel }a-b\mathrm{\parallel }\mathrm{/}\mathrm{\parallel }b\mathrm{\parallel }$. For unit $z,w$, expanding $z{z}^T-w{w}^T=(z-w)z^T+w(z-w{)}^T$ and applying [F5] and [F7] to the rank-one terms gives operator norm at most $2\mathrm{\parallel }z-w\mathrm{\parallel }$. Consequently $\mathrm{\parallel }H(v)-H(v_0){\mathrm{\parallel }}_2\le 8\mathrm{\parallel }v-v_0\mathrm{\parallel }\mathrm{/}{V}_0\le 8d_r$. Since $H(v_0)x={\alpha }_0{e}_1$, the overwritten column differs from $H(v)x$ by at most $(e_r+8d_r)N$.

The exact rank-one update has norm at most $2\mathrm{\parallel }B{\mathrm{\parallel }}_F$: for $z=v\mathrm{/}V$, each column satisfies $\mathrm{\parallel }z{z}^T{B}_{:j}{\mathrm{\parallel }}_2=\mathrm{\mid }{z}^T{B}_{:j}\mathrm{\mid }\le \mathrm{\parallel }{B}_{:j}{\mathrm{\parallel }}_2$ by [F5], and summing the squared column bounds gives the Frobenius estimate by [F6]. Replacing its coefficient by $\widehat{\tau }$, its dot by the computed dot, and its two multiplications by rounded multiplications bounds the update error by $c_r\mathrm{\parallel }B{\mathrm{\parallel }}_F$: successively the nonnegative majorants multiply by $1+a_r$, $1+g_r$, and $(1+u{)}^2$, then subtract the exact majorant $2$. Final subtraction adds at most $u(\mathrm{\parallel }B{\mathrm{\parallel }}_F+\mathrm{\parallel }U{\mathrm{\parallel }}_F)\le u(3+c_r)\mathrm{\parallel }B{\mathrm{\parallel }}_F$. Thus $\mathrm{\parallel }\widehat{B}-H(v)B{\mathrm{\parallel }}_F\le ((1+u)c_r+3u)\mathrm{\parallel }B{\mathrm{\parallel }}_F$. The same computation applies to any separate block $Y$.

The first-column and remaining-column errors occupy disjoint columns, so their squared Frobenius norms add by [F6]; the same entrywise formula gives $\mathrm{\parallel }C{\mathrm{\parallel }}_F^2=\mathrm{\parallel }x{\mathrm{\parallel }}_2^2+\mathrm{\parallel }B{\mathrm{\parallel }}_F^2$. Steps 2.1 and 2.2 give $\mathrm{\parallel }{C}_{+}-H(v)C{\mathrm{\parallel }}_F\le {\kappa }_r\mathrm{\parallel }C{\mathrm{\parallel }}_F$. Choose $E=H(v)C_{+}-C$; orthogonality gives $H(C+E)=C_{+}$, and Frobenius norm invariance follows because $H$ preserves the Euclidean norm of each column and [F6] sums their squares. Each displayed coefficient increases with $r$ and has zero constant term and finite first derivative at $u=0$, proving the stated asymptotics.