Local stable reflectors and rotations give an aggregate SVD backward error

Let $C_0\in {\mathbb{F}}^{M\times p}$, $M\ge p$, be the tall orientation used by Householder bidiagonalization and its finite precision status, and let $B_J\in {\mathbb{F}}^{p\times p}$ be the final bidiagonal-QR work matrix returned by Finite budget implicit shifted QR for bidiagonal singular values. Set $C_J=\left[\begin{array}{c}{B}_J\\ 0_{M-p,p}\end{array}\right]\in {\mathbb{F}}^{M\times p}$. Suppose the computed bidiagonal reduction and QR sweeps perform $J$ stages, each either an elementary left/right unitary update or an identity update for a deflation/cleanup write. At stage $j$, assume the actual active matrix obeys $C_j=L_j^{\ast }(C_{j-1}+E_j)R_j+D_j,$ where $L_j,R_j$ are unitary and ${\eta }_j\ge 0$, $\mathrm{\parallel }{E}_j{\mathrm{\parallel }}_F\le {\eta }_j\mathrm{\parallel }{C}_{j-1}{\mathrm{\parallel }}_F$, and $D_j$ is the additive representation of entries discarded by deflation or cleanup at that stage, with ${\delta }_j=\mathrm{\parallel }{D}_j{\mathrm{\parallel }}_F$. A deflation/cleanup write is represented by $L_j=R_j=I$, $E_j=0$, and the recorded additive change $D_j$. The deflation record returned by the algorithm identifies each such entry and its old value, so it determines $D_j$. Then the returned tall transformed matrix $C_J$ satisfies $C_0+E=L{C}_J{R}^{\ast }$ for unitary accumulated factors $L,R$ and some $E$ with $\mathrm{\parallel }E{\mathrm{\parallel }}_F\le ({\prod }_{j=1}^J(1+{\eta }_j)-1)\mathrm{\parallel }{C}_0{\mathrm{\parallel }}_F+{\sum }_{j=1}^J{\delta }_j{\prod }_{\mathrm{\ell }=j+1}^J(1+{\eta }_{\mathrm{\ell }})\mathrm{.}$ If no deflation or other cleanup is performed, every ${\delta }_j=0$. For the successful scaled arithmetic realization Scaled real and complex Householder and Givens update arithmetic, assume also $0<u<1$ and $(r_j-1)u<1$ for every Householder dimension $r_j$. Under these arithmetic and dimension conditions, Scaled complex Householder and Givens steps have explicit local backward errors establishes these hypotheses: choose ${\eta }_j={\eta }_H(r_j)$ for Householder updates, ${\eta }_j={\eta }_G$ for Givens updates with an actual target zero write, and its smaller separate-block coefficient when no matrix entry is overwritten. This covers real/complex left and right updates. Every additional discarded entry is included in $D_j$. Rounded dense accumulators remain separate from these exact represented products.