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This paper shows for a large class of numerical methods that such a <jats:italic>Trade-off Principle<\/jats:italic> between error and evaluation stability is unavoidable. It is an instance of a <jats:italic>no free lunch theorem<\/jats:italic>. Here, <jats:italic>evaluation<\/jats:italic> is the mathematical map that takes input data to output data. This is independent from the numerical routine that calculates the output. Therefore, <jats:italic>evaluation<\/jats:italic> stability is different from <jats:italic>computational<\/jats:italic> stability. The setting is confined to recovery of functions from data, but it includes solving differential equations by writing such methods as a recovery of functions under constraints imposed by differential operators and boundary values. The trade-off principle bounds the product of two terms from below. The first is related to errors, and the second turns out to be related to evaluation instability. Under certain conditions satisfied for splines and kernel-based interpolation, both can be minimized. Then the lower bound is attained, and the error term is the inverse of the instability term. As a byproduct, it is shown that Kansa\u2019s Unsymmetric Collocation Method sacrifices accuracy for improved evaluation stability, when compared to symmetric collocation.<\/jats:p>","DOI":"10.1007\/s10444-023-10026-2","type":"journal-article","created":{"date-parts":[[2023,3,29]],"date-time":"2023-03-29T04:02:36Z","timestamp":1680062556000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":13,"title":["Small errors imply large evaluation instabilities"],"prefix":"10.1007","volume":"49","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7828-4355","authenticated-orcid":false,"given":"Robert","family":"Schaback","sequence":"first","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2023,3,29]]},"reference":[{"key":"10026_CR1","doi-asserted-by":"publisher","unstructured":"Schaback, R.: Error estimates and condition numbers for radial basis function interpolation. Adv. Comput. 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