|Portability||non-portable (requires fenv.h)|
Datatypes and abstractions for approximating exact real numbers and a basic arithmetic over such approximations. The design 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: abstracts floating point numbers
ERApprox: abstracts neighbourhoods of real numbers
ERIntApprox: abstracts neighbourhoods of real numbers that are known to be intervals
ERApproxElementary: abstracts 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.
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.
- class (Fractional rb, Ord rb) => ERRealBase rb
- class (Fractional ra, Ord ra) => ERApprox ra
- class ERApprox ira => ERIntApprox ira
- class ERIntApprox ra => ERApproxElementary ra
- module Data.Number.ER.Real.DefaultRepr
- module Data.Number.ER.Real.Approx.Sequence
- module Data.Number.ER.Real.Arithmetic.Taylor
- module Data.Number.ER.Real.Arithmetic.Newton
- module Data.Number.ER.Real.Arithmetic.Integration
- module Data.Number.ER.BasicTypes
This class is an abstraction of a subset of real numbers with upwards rounded operations.
A type whose elements represent sets that can be used to approximate a single extended real number with arbitrary precision.
A type whose elements represent sets that can be used to approximate a recursive set of closed extended real number intervals with arbitrary precision.
A type whose elements represent real *intervals* that can be used to approximate a single extended real number with arbitrary precision.
Sometimes, these types can be used to approximate a closed extended real number interval with arbitrary precision. Nevetheless, this is not guaranteed.
A class defining various common real number operations in a approximation-aware fashion, ie introducing effort indices.
All operations here have default implementations based on Data.Number.ER.Real.Arithmetic.Elementary.