AERN-Real- arbitrary precision interval arithmetic for approximating exact real numbers

Portabilitynon-portable (requires fenv.h)



This module bundles some of the most important functionality of the AERN-Real package. It is intended to be imported *qualified*.

AERN-Real provides datatypes and abstractions for approximating exact real numbers and a basic arithmetic over such approximations. The approach is inspired to some degree by Mueller's iRRAM and Lambov's RealLib (both are C++ libraries for exact real arithmetic).

Abstractions are provided via 4 type classes:

  • ERRealBase: generalises floating point numbers (not exported here, used only internally)
  • ERApprox: generalises measurable subsets of real numbers
  • ERIntApprox: generalises interval neighbourhoods of real numbers
  • ERApproxElementary: generalises real number approximations that support elementary operations

For ERRealBase we give several implementations. The default is an arbitrary precision floating point type that uses Double for lower precisions and an Integer-based simulation for higher precisions. Rational numbers can be used as one of the alternatives. Augustsson's Data.Number.BigFloat can be easily wrapped as an instance of ERRealBase except that it uses a different method to control precision. Optionally, one can also have MPFR floating point numbers via package hmpfr if compiled with USE_MPFR.

ERIntApprox is implemented via outwards-rounded arbitrary precision interval arithmetic. Any instance of ERRealBase can be used for the endpoints of the intervals.

ERApproxElementary is implemented generically for any implementation of ERIntApprox. This way some of the most common elementary operations are provided, notably: sqrt, exp, log, sin, cos, atan. These operations converge to an arbitrary precision and also work well over larger intervals without excessive wrapping.

There is also some support for generic Taylor series, interval Newton method and simple numerical integration.