We study the complexity of some computational problems in case certain stability guarantees are required. After providing suitable settings for this kind of investigations, we are able to give theoretical support to longstanding viewpoints in numerical analysis. More precisely, we focus on a set of significant examples and prove that: (1) the fastest known polynomial evaluation algorithms are not stable, (2) LUP decomposition cannot be computed stably in polylogarithmic parallel time, and (3)

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... ductions among computational problems do not in general constitute a practical tool for solving numerical problems.