The condition number note measured how sensitive a problem is, independent of any algorithm. This note supplies the matching concept for the algorithm itself: backward stability, which asks not "how close is the computed answer to the true answer?" but the more subtle and more useful question, "is the computed answer the exact answer to a slightly different problem?" This reframing is the single most important idea in numerical analysis, and it is the concept that finally makes precise every "numerically preferred" claim made informally earlier in the unit.
Learning objectives. By the end of this note you should be able to:
- state the definition of backward stability precisely;
- explain why backward error, not forward error, is the natural quantity an algorithm controls;
- combine backward stability with the condition number to bound the forward error;
- explain why a stable algorithm can still give an inaccurate answer on an ill-conditioned problem.
In this lesson
- The definition
- Combining stability and conditioning
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