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  "body": "\n## Summary\n\nBackward error analysis studies a computed answer by asking for which nearby input it is the exact answer.\nThe size of that input change is the backward error.\nThe error of the answer, the forward error, is then bounded by about the [[condition number|AB3S6MHJ8X]] times the backward error.\nThis splits the analysis into a property of the algorithm (small backward error, that is, [[numerical stability|1AKB4XHMTH]]) and a property of the problem (its conditioning).\nIt was developed and made systematic by James H. Wilkinson in his work on numerical linear algebra.\n\n## Explanations\n\n### L1 human\n\nInstead of asking \"how wrong is my answer?\", backward error analysis asks \"for which slightly different question is my answer exactly right?\".\nIf the question had to change only by an amount comparable to the rounding errors, the algorithm did as well as one can hope.\nFor example, a computed root $\\hat r$ of a polynomial may be an exact root of a polynomial whose coefficients differ from the given ones only in the last digit.\nWhether the answer is then close to the true root depends on how sensitive the problem is, not on the algorithm.\n",
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