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A function is well-conditioned roughly if it's computable with small forwards error. Often, there are problems like computing the Singular Value Decomposition, which are fundamentally ill-conditioned, but which may nevertheless admit good backwards stable algorithms. I'm interested in whether there's a general term for something being computable in a backwards stable manner. Such a term would be useful because it appears to be a more common property than well-conditionedness.

Alternatively, I may be looking at this topic in the wrong way.

wlad
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The conditioning of a problem and the stability of a specific algorithm for solving said problem are two important issues that are unrelated. The conditioning of a function measures the sensitivity of the output to small changes in the input. The stability of an algorithm measures the sensitivity of the output to rounding errors during the calculation. If the problem is well-conditioned and the algorithm is backwards stable, then the output will be computed with a small forward error, see this answer. How we define small depends on the context, see this answer for a few examples. We can do nothing to change the conditioning of a problem, but we can easily degrade the stability of an algorithm if we ignore the differences between exact and finite precision arithmetic. Finally, before we classify a problem as well-conditioned or ill-conditioned we have to consider the context as explained in this answer.

To the best of my knowledge there is no specialized term that signals that a problem has a backwards stable algorithm.

Carl Christian
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  • "The conditioning of a problem and the stability of a specific algorithm for solving said problem are two important issues that are unrelated." - Carl, where did you get the mistaken impression that I thought differently? – wlad Jan 17 '23 at 01:44
  • In fact, this doesn't answer my question. Your last paragraph. – wlad Jan 17 '23 at 01:44
  • @wlad "... where did you get the impression that I thought differently?". There exists precise definitions of condition numbers and the classification of well-conditioned or ill-conditioned depends on the application and the hardware.Therefore, your statement "A function is well-conditioned roughly if it's computable with small forwards error." suggested to me an incomplete view of the topic. To me, the word "nevertheless" has the connotation of surprise. There are papers that prove that algorithms are backward stable without introducing a specific term. Your last line suggested doubt. – Carl Christian Jan 17 '23 at 09:00