Computable analysis

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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.

Basic constructions[edit]

Computable real numbers[edit]

Main article: Computable number

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 functions[edit]

A function f \colon \mathbb{R} \to \mathbb{R} is sequentially computable if, for every computable sequence \{x_i\}_{i=1}^\infty of real numbers, the sequence \{f(x_i) \}_{i=1}^\infty is also computable.

Basic results[edit]

The computable real numbers form a real closed field. 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.

See also[edit]

References[edit]

  • Oliver Aberth (1980), Computable analysis, McGraw-Hill, 1980.
  • 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, 2000.

External links[edit]