Certified QZ residuals give a backward error bound for both pencil matrices

Under [A1], apply [F1] and [F2] to the finite list of left/right QZ updates, keeping each explicit deflation as its own local perturbation. This gives $Q^{\ast }(A+{\mathrm{\Delta }}_A)Z=S$ and $Q^{\ast }(B+{\mathrm{\Delta }}_B)Z=T$ with the asserted sum of local pair error bounds. This step claims no size bound beyond the supplied certificates.

From the definitions of $R_A,R_B$, set ${\mathrm{\Delta }}_A=-Q{R}_A{Z}^{\ast }$ and ${\mathrm{\Delta }}_B=-Q{R}_B{Z}^{\ast }$. Exact unitarity gives $Q^{\ast }{\mathrm{\Delta }}_{A}Z=-R_A$ and $Q^{\ast }{\mathrm{\Delta }}_{B}Z=-R_B$, so adding the residuals proves both displayed backward equations. By [F3], $\mathrm{\parallel }{\mathrm{\Delta }}_A{\mathrm{\parallel }}_F=\mathrm{\parallel }{R}_A{\mathrm{\parallel }}_F$ and $\mathrm{\parallel }{\mathrm{\Delta }}_B{\mathrm{\parallel }}_F=\mathrm{\parallel }{R}_B{\mathrm{\parallel }}_F$; squaring and adding proves the exact pair equality.

For the specified implementation, [F4] gives $P(E_j^A,E_j^B)\le {\mathrm{\ell }}_{j}P(A_{j-1},B_{j-1})$. Put each pure tolerance deflation at its own identity stage, and pull back any post-action write before using [F2]. Structural writes are already counted in $E_j$, so summing proves the concrete displayed bound without double counting.

Under the independent residual route's successful-operation hypotheses, [F4] bounds each exact residual column by its computed diagnostic norm plus the propagated factor, stored-input matvec and subtraction allowances. Summing these column bounds gives certified residual upper bounds $b_A,b_B$, hence step 1.2 gives pair error at most $\sqrt{b_A^2+b_B^2}$ (or conservatively $b_A+b_B$). If that route fails, no certificate is claimed from its diagnostic; the local-accumulation route remains available on its own hypotheses.