Demmel, James Steve Smale and the geometry of ill-conditioning. (English) Zbl 0791.65036 Hirsch, M. W. (ed.) et al., From topology to computation: Proceedings of the Smalefest. Papers presented at the conference “From topology to computation: Unity and diversity in the mathematical sciences” held at the University of California at Berkeley, USA, August 5-9, 1990 in honor of Stephen Smale’s 60th birthday. New York: Springer-Verlag. 305-316 (1993). The work of S. Smale et al. on average case analysis of algorithms and modeling real computations, has stimulated much further work. The author surveys some known results and open problems in (i) the probability that a random numerical problem is difficult, (ii) the complexity of condition estimation, and (iii) the use of regularization to solve ill-posed or ill-conditioned problems.For the entire collection see [Zbl 0779.00016]. Reviewer: W.C.Rheinboldt (Pittsburgh) Cited in 1 Document MSC: 65J05 General theory of numerical analysis in abstract spaces 65J20 Numerical solutions of ill-posed problems in abstract spaces; regularization 65Y20 Complexity and performance of numerical algorithms 65F35 Numerical computation of matrix norms, conditioning, scaling 68Q15 Complexity classes (hierarchies, relations among complexity classes, etc.) 68Q25 Analysis of algorithms and problem complexity 65-03 History of numerical analysis 68-03 History of computer science 01A65 Development of contemporary mathematics Keywords:Steve Smale; geometry of ill-conditioning; complexity of algorithms; condition estimation; average case analysis of algorithms; modeling real computations; regularization; ill-conditioned problems Biographic References: Smale, S. PDFBibTeX XMLCite \textit{J. Demmel}, in: From topology to computation: Proceedings of the Smalefest. Papers presented at the conference ''From topology to computation: Unity and diversity in the mathematical sciences'' held at the University of California at Berkeley, USA, August 5-9, 1990 in honor of Stephen Smale's 60th birthday. New York: Springer-Verlag. 305--316 (1993; Zbl 0791.65036)