Steve Smale and the Geometry of Ill-Conditioning

The work of Steve Smale and his colleagues on average case analysis of algorithms [30, 31] and modeling real computations [3, 32] introduced methods and models not previously used in numerical analysis and complexity theory. In particular, the use of integral geometry to bound the sizes of sets of problems where algorithms “go bad” and the introduction of a model of real computation to clearly formulate complexity questions have inspired a great deal of other work. In this chapter, we will survey past results and open problems in three areas: the probability that a random numerical problem is difficult, the complexity of condition estimation, and the use of regularization to solve ill-posed or ill-conditioned problems; we will define all these terms below.

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