mixed forward-backward stability

Pure backward stability is a demanding promise: the computed answer is the EXACT answer to a nearby problem, not just a close answer. Some perfectly good algorithms cannot quite make that promise, yet they are still trustworthy. Mixed forward-backward stability is the slightly relaxed standard that covers them: the computed answer is NEARLY the exact answer to a NEARBY problem — a small wobble allowed at both ends, on the output and on the input.

Stated carefully: an algorithm for y = f(x) is mixed forward-backward stable if the computed y-hat satisfies y-hat + delta-y = f(x + delta-x), where BOTH the output perturbation ||delta-y|| / ||y|| and the input perturbation ||delta-x|| / ||x|| are of order the unit roundoff. Contrast the two purer notions: backward stability forces delta-y = 0 (exactly right for a perturbed input); forward stability would force delta-x = 0 (nearly right for the exact input). Mixed stability allows a small amount of each, which is why it is the natural and most common stability guarantee in real numerical analysis — many algorithms achieve it even when full backward stability is unattainable.

The reassuring fact is that mixed stability gives the same practical accuracy bound as backward stability: the forward error is still of order (condition number times unit roundoff). So for the purpose of trusting an answer, a mixed-stable algorithm is essentially as good as a backward-stable one — the relaxation buys analytical breathing room without costing accuracy. As always, the result is only as accurate as the problem's conditioning permits; mixed stability, like its purer cousins, certifies the algorithm, not the problem.