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.

Basic constructions

Computable real numbers

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

Main article: Computable real function

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

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.

References

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

Computability and Complexity in Analysis Network