Data.Number.ER.Real

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.

Documentation

class (Fractional rb, Ord rb) => ERRealBase rb

This class is an abstraction of a subset of real numbers with upwards rounded operations.

Instances

ERRealBase Double

ERRealBase ExtendedRational

ERRealBase ERFloat

ERRealBase b => ERRealBase (ERMachineAP b)

class (Fractional ra, Ord ra) => ERApprox ra

A type whose elements represent sets that can be used to approximate a single extended real number with arbitrary precision.

Instances

(ERRealBase b, RealFrac b) => ERApprox (ERInterval b)

class ERApprox ira => ERIntApprox ira

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.

Instances

(ERRealBase b, RealFrac b) => ERIntApprox (ERInterval b)

class ERIntApprox ra => ERApproxElementary ra

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.

Instances

(ERRealBase b, RealFrac b) => ERApproxElementary (ERInterval b)

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