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Computable analysis

In mathematics and computer science, computable analysis is the study of mathematical analysis from the perspective of computability theory. It is concerned with the parts of real analysis and functional analysis that can be carried out in a computable manner. The field is closely related to constructive analysis and numerical analysis.

Contents

Basic constructionsEdit

Computable real numbersEdit

Computable numbers are the real numbers that can be computed to within any desired precision by a finite, terminating algorithm. They are also known as the recursive numbers or the computable reals.

Computable real functionsEdit

A function   is sequentially computable if, for every computable sequence   of real numbers, the sequence   is also computable.

Basic resultsEdit

The computable real numbers form a real closed field (Weihrauch 2000, p. 180). The equality relation on computable real numbers is not computable, but for unequal computable real numbers the order relation is computable.

Computable real functions map computable real numbers to computable real numbers. The composition of computable real functions is again computable. Every computable real function is continuous (Weihrauch 2000, p. 6).

See alsoEdit

ReferencesEdit

  • Oliver Aberth (1980), Computable analysis, McGraw-Hill, ISBN 0-0700-0079-4.
  • Marian Pour-El and Ian Richards, Computability in Analysis and Physics, Springer-Verlag, 1989.
  • Stephen G. Simpson (1999), Subsystems of second-order arithmetic.
  • Klaus Weihrauch (2000), Computable analysis, Springer, ISBN 3-540-66817-9.

External linksEdit