= Backward stability
{title2=$\widehat f(d)=f(d+\delta d)$}
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 $\|A-\widehat U\widehat\Sigma\widehat V^T\|_2\leq c u\|A\|_2$, with machine precision $u$ and a modest dimension-dependent constant $c$.
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