Backward stability ($\hat{f}(d)=f(d+\delta d)$)

A numerical algorithm is backward stable when its computed output is the exact output for input data perturbed by an amount comparable to the arithmetic's round-off error. The allowed input perturbations and their norm must be specified. A bound on backward error gives a forward error bound only after accounting for the problem's condition number. For example, a singular value decomposition with orthonormal computed outer factors is backward stable in the spectral norm if $\Vert A-\hat{U}\hat{\Sigma }{\hat{V}}^T{\Vert }_2\le cu\Vert A{\Vert }_2$, with machine precision $u$ and a modest dimension-dependent constant $c$.