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  "body": "\n## Summary\n\nAn algorithm is numerically stable when the errors it makes, such as [[round-off errors|CPN6SD3BTV]], are not amplified beyond what the problem itself forces.\nThe strongest common form is backward stability: the computed answer is the exact answer for a slightly perturbed input, as in [[backward error analysis|CMXHRNWY35]].\nA stable algorithm applied to a well-conditioned problem gives an accurate answer; for an ill-conditioned problem even a stable algorithm can give large errors.\nFor methods for differential equations, stability means instead that errors in the steps stay bounded, as checked by [[von Neumann stability analysis|2Z7NASDQB6]].\n\n## Explanations\n\n### L1 human\n\nTwo formulas can be equal in exact arithmetic but behave very differently on a computer.\nFor example, the roots of $x^2 - 10^8 x + 1 = 0$ are about $10^8$ and $10^{-8}$.\nThe textbook formula computes the small root as $(10^8 - \\sqrt{10^{16} - 4})/2$, a difference of two almost equal numbers, and loses most of its digits.\nThe equivalent formula $2/(10^8 + \\sqrt{10^{16} - 4})$ has no such subtraction and is accurate.\nThe second formula is a stable way to compute the small root and the first is not.\n",
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