Excerpt from On Condition Numbers and the Distance to the Nearest Ill-Posed Problem

The condition number of a problem measures the sensitivity of the answer to small changes in the input. We call the problem ill-posed if its condition number is infinite. The ill-posed problems typically form a lower dimensional surface in the space of problems. It turns out that for many problems of numerical analysis, there is a simple relationship between the condition number of a problem and the shortest distance from that problem to the surface of ill-posed ones: the shortest distance is proportional to the reciprocal of the condition number, or bounded by the reciprocal of the condition number. Sometimes, the distance is bounded below by the reciprocal of the condition number squared. This is true for matrix inversion, computing eigenvalues and eigenvectors, finding zeros of polynomials.

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