Common computed Givens actions certify both pencil matrices and factor residuals

Replacing $\sqrt{2}$ by $2$ enlarges the source's raw coefficient to $v$ and its raw intended-zero bound to $h$. Apply the raw bound separately to $C,D$, square and add: the common action's pair discrepancy is at most $vP(C,D)$. The one actual target write has norm at most $h$ times its two-entry target norm, hence at most $hP(C,D)$. The triangle inequality gives $\eta P(C,D)$. This uses one generated factor for both arrays; no similarity relation between $C,D$ is assumed.

Permutations and conjugate transposes preserve Euclidean/Frobenius norms and use exact copies of stored coefficients. Write $U(rJ{)}^{\ast }=(\rho ,ϵ{)}^T$ for the normalized rounded representative, without assuming $ϵ=0$. Conjugate transposition and multiplication by $J$ give $rJ{U}^{\ast }J=(\overline{ϵ},\bar{\rho })$. The source intended-zero allowance bounds $\mathrm{\mid }ϵ\mathrm{\mid }$ by $(u+2q)\mathrm{\parallel }r{\mathrm{\parallel }}_2$; the rounded right application contributes at most $v\mathrm{\parallel }r{\mathrm{\parallel }}_2$, so its first-coordinate overwrite has the inherited bound $h\mathrm{\parallel }r{\mathrm{\parallel }}_2$ of step 1.1. For a second-coordinate right target write $U{r}^{\ast }=(\rho ,ϵ{)}^T$, giving $r{U}^{\ast }=(\bar{\rho },\overline{ϵ})$ and the same bound on that target. Thus the stated orientations select the correct coordinate without asserting exact annihilation by the normalized rounded representative. An external vector chooses a unitary but does not license any matrix zero assignment.

Write the computed pair as $Q_0^{\ast }(C,D)Z_0+F$ in post-action coordinates. Define $E=Q_{0}F{Z}_0^{\ast }$ componentwise. Unitarity gives the stated update and preserves the pair norm. Pull a separately recorded post-action write back in the same way; its saved entry and factors specify it exactly. For two consecutive actions the intermediate pair norm is at most $(1+{\mathrm{\ell }}_L)P(C,D)$, so the sum of their pulled-back errors is at most $[{\mathrm{\ell }}_L+{\mathrm{\ell }}_R(1+{\mathrm{\ell }}_L)]P(C,D)$. Pure deflation uses identity factors.

If $y$ differs from the exact current vector by at most $e$, the next raw action differs from the corresponding exact unitary action on $y$ by at most $v\mathrm{\parallel }y{\mathrm{\parallel }}_2$. That exact unitary preserves the preceding discrepancy, proving the recurrence. Rightmost factors act first in a matrix product, which gives the specified orders for $Z{e}_k$ and $Q^{\ast }$ action. Adjoints and reversals inherit the raw bound from step 2.1.

Apply [F3] to the STORED vector $\widehat{z}$, then transport $\widehat{z}-Z{e}_k$ through $A$ using $\mathrm{\parallel }A{\mathrm{\parallel }}_2\le \mathrm{\parallel }A{\mathrm{\parallel }}_F$. Step 3.2 propagates this discrepancy through $Q^{\ast }$. The final subtraction allowance in [F3] gives $e_k$. Each exact residual column is therefore bounded by $\mathrm{\parallel }{\widehat{r}}_k{\mathrm{\parallel }}_2+e_k$; the square root of the sum of their squares is at most their sum. This proves the residual certificate without using nonstored normalized factor entries as floating inputs.

For any finite complex array, its Frobenius/Euclidean norm is at most the sum of absolute real and imaginary components, by squaring and the nonnegative cross terms. These sums and the rational constants $v,h,\eta ,g_n$ use only finite field operations, absolute-value comparisons and the prescribed finite loops. Every denominator is positive under the stated hypotheses. Exact rational arithmetic realizes these instructions for rational stored data; no bit-cost bound is asserted. If an excluded rounded operation or dimension hypothesis fails, that evaluation produces no certificate, as specified.