Complexity Theory and Numerical Analysis ∗

نویسندگان

  • Steve Smale
  • Ya Yan Lu
چکیده

Complexity theory of numerical analysis is the study of the number of arithmetic operations required to pass from the input to the output of a numerical problem. To a large extent this requires the (global) analysis of the basic algorithms of numerical analysis. This analysis is complicated by the existence of ill-posed problems, conditioning and roundoff error. A complementary aspect (“lower bounds”) is the examination of efficiency for all algorithms solving a given problem. This study is difficult and needs a formal definition of algorithm. Highly developed complexity theory of computer science provides some inspirations to the subject at hand. Yet the nature of theoretical computer science, with its foundations in discrete Turing machines, prevents a simple transfer to a subject where real number algorithms as Newton’s method dominate. One can indeed be skeptical about a formal development of complexity into the domain of numerical analysis, where problems are solved only to a certain precision and roundoff error is central. Recall that according to computer science, an algorithm defined by a Turing machine is polynomial time if the computing time (measured by the number of Turing machine operations) T (y) on input y satisfies:

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تاریخ انتشار 1996