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The book I am reading by Trefethen and Bau defines backward statbility in the following way...

A problem is backward stable if for each x, $$ \tilde{f}(x) = f(x) $$ for some $\tilde{x}$ with $\frac{||\tilde{x} - x||}{||x||} = O(\epsilon)$.

It also explains the statement by saying "A backward stable algorithm gives exactly the right answer to nearly the right question. "

The part I dont understand is according to the definition, which part is the question and nearly the right question?

user1559897
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  • Without more context, it is hard to say, though I think the statement "a backward stable algorithm gives exactly the right answer to nearly the right question" might be referring to the fact that when you are solving problems numerically, usually you are either trying to solve an exact problem approximately, or you are trying to solve an approximate problem exactly. – Matthew Cassell Oct 04 '19 at 18:44
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    The right question is "Find the value of $f(x)$". Nearly the right question is "Find the value of $f(\tilde x)$". A backward stable algorithm gives $\tilde f(x)$, which as the equation states is exactly the right answer to the latter question. (You have either quoted the book incorrectly, or there is a misprint in your copy; the right-hand side should be $f(\tilde x)$.) –  Oct 04 '19 at 19:31

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