Name: numeric-prelude
Version: 0.0.2
License: GPL
License-File: LICENSE
Author: Dylan Thurston , Henning Thielemann , Mikael Johansson
Maintainer: Henning Thielemann
Homepage: http://www.haskell.org/haskellwiki/Mathematical_prelude_discussion
Package-URL: http://darcs.haskell.org/numericprelude/
Category: Math
Stability: Experimental
Synopsis: An experimental alternative hierarchy of numeric type classes
Description:
Revisiting the Numeric Classes
.
The Prelude for Haskell 98 offers a well-considered set of numeric
classes which cover the standard numeric types
('Integer', 'Int', 'Rational', 'Float', 'Double', 'Complex') quite well.
But they offer limited extensibility and have a few other flaws.
In this proposal we will revisit these classes, addressing the following concerns:
.
[1] The current Prelude defines no semantics for the fundamental operations.
For instance, presumably addition should be associative
(or come as close as feasible),
but this is not mentioned anywhere.
.
[2] There are some superfluous superclasses.
For instance, 'Eq' and 'Show' are superclasses of 'Num'.
Consider the data type
@ data IntegerFunction a = IF (a -> Integer) @
One can reasonably define all the methods of 'Algebra.Ring.C' for
@IntegerFunction a@ (satisfying good semantics),
but it is impossible to define non-bottom instances of 'Eq' and 'Show'.
In general, superclass relationship should indicate
some semantic connection between the two classes.
.
[3] In a few cases, there is a mix of semantic operations and
representation-specific operations.
'toInteger', 'toRational',
and the various operations in 'RealFloating' ('decodeFloat', ...)
are the main examples.
.
[4] In some cases, the hierarchy is not finely-grained enough:
Operations that are often defined independently are lumped together.
For instance, in a financial application one might want a type \"Dollar\",
or in a graphics application one might want a type \"Vector\".
It is reasonable to add two Vectors or Dollars,
but not, in general, reasonable to multiply them.
But the programmer is currently forced to define a method for '(*)'
when she defines a method for '(+)'.
.
In specifying the semantics of type classes,
I will state laws as follows:
.
> (a + b) + c === a + (b + c)
.
The intended meaning is extensional equality:
The rest of the program should behave in the same way
if one side is replaced with the other.
Unfortunately, the laws are frequently violated by standard instances;
the law above, for instance, fails for 'Float':
.
> (1e20 + (-1e20)) + 1.0 = 1.0
> 1e20 + ((-1e20) + 1.0) = 0.0
.
For inexact number types like floating point types,
thus these laws should be interpreted as guidelines rather than absolute rules.
In particular, the compiler is not allowed to use them.
Unless stated otherwise, default definitions should also be taken as laws.
.
Thanks to Brian Boutel, Joe English, William Lee Irwin II, Marcin
Kowalczyk, Ketil Malde, Tom Schrijvers, Ken Shan, and Henning
Thielemann for helpful comments.
.
Scope & Limitations\/TODO:
* It might be desireable to split Ord up into Poset and Ord
(a total ordering).
This is not addressed here.
.
* In some cases, this hierarchy may not be fine-grained enough.
For instance, time spans (\"5 minutes\") can be added to times (\"12:34\"),
but two times are not addable. (\"12:34 + 8:23\")
As it stands,
users have to use a different operator for adding time spans to times
than for adding two time spans.
Similar issues arise for vector space et al.
This is a consciously-made tradeoff, but might be changed.
This becomes most serious when dealing with quantities with units
like @length\/distance^2@, for which @(*)@ as defined here is useless.
(One way to see the issue: should
@ f x y = iterate (x *) y @
have principal type
@ (Ring.C a) => a -> a -> [a] @
or something like
@ (Ring.C a, Module a b) => a -> b -> [b] @
?)
.
* I stuck with the Haskell 98 names.
In some cases I find them lacking.
Neglecting backwards compatibility, we have renamed classes as follows:
Num --> Ring
Fractional --> Field
Floating --> Algebraic, Transcendental
RealFloat --> RealTranscental
.
* It's slightly unfortunate that 'abs' can no longer be used for complex numbers,
since it is standard mathematically.
'magnitude' or more generally 'Algebra.NormedSpace.Euclidean.norm' can be used.
But it had the wrong type before,
and I couldn't see how to fit it in without complicating the hierarchy.
.
.
Additional standard libraries might include Enum, IEEEFloat (including
the bulk of the functions in Haskell 98's RealFloat class),
VectorSpace, Ratio.T, and Lattice.
Tested-With: GHC==6.4.1, GHC==6.8.2
Cabal-Version: >=1.2
Build-Type: Simple
Extra-Source-Files:
Makefile
docs/NOTES
docs/README
Flag splitBase
description: Choose the new smaller, split-up base package.
Library
Build-Depends: parsec >= 1, HUnit >=1 && <2, QuickCheck >=1 && <2, non-negative >=0.0.1 && <0.1
If flag(splitBase)
Build-Depends: base >= 2, array, containers, random
Else
Build-Depends: base >= 1.0 && < 2
GHC-Options: -Wall
Hs-source-dirs: src
Exposed-modules:
Algebra.Additive
Algebra.Algebraic
Algebra.Differential
Algebra.DimensionTerm
Algebra.DivisibleSpace
Algebra.Field
Algebra.Indexable
Algebra.IntegralDomain
Algebra.NonNegative
Algebra.Lattice
Algebra.Laws
Algebra.Module
Algebra.ModuleBasis
Algebra.Monoid
Algebra.NormedSpace.Euclidean
Algebra.NormedSpace.Maximum
Algebra.NormedSpace.Sum
Algebra.OccasionallyScalar
Algebra.PrincipalIdealDomain
Algebra.Real
Algebra.RealField
Algebra.RealIntegral
Algebra.RealTranscendental
Algebra.RightModule
Algebra.Ring
Algebra.ToInteger
Algebra.ToRational
Algebra.Transcendental
Algebra.Units
Algebra.Vector
Algebra.VectorSpace
Algebra.ZeroTestable
MathObj.Algebra
MathObj.DiscreteMap
MathObj.LaurentPolynomial
MathObj.Matrix
MathObj.PartialFraction
MathObj.Permutation
MathObj.Permutation.CycleList
MathObj.Permutation.CycleList.Check
MathObj.Permutation.Table
MathObj.Polynomial
MathObj.PowerSeries
MathObj.PowerSeries.DifferentialEquation
MathObj.PowerSeries.Example
MathObj.PowerSeries.Mean
MathObj.PowerSeries2
MathObj.PowerSum
MathObj.RootSet
MyPrelude
Number.Complex
Number.DimensionTerm
Number.DimensionTerm.SI
Number.FixedPoint
Number.FixedPoint.Check
Number.NonNegative
Number.PartiallyTranscendental
Number.Peano
Number.Positional
Number.Positional.Check
Number.Quaternion
Number.Ratio
Number.ResidueClass
Number.ResidueClass.Check
Number.ResidueClass.Maybe
Number.ResidueClass.Func
Number.ResidueClass.Reader
Number.OccasionallyScalarExpression
Number.SI.Unit
Number.SI
Number.Physical.Unit
Number.Physical.UnitDatabase
Number.Physical
Number.Physical.Read
Number.Physical.Show
NumericPrelude
NumericPrelude.Condition
NumericPrelude.List
NumericPrelude.Monad
NumericPrelude.Text
PreludeBase
Executable test
Hs-Source-Dirs: src, test
Main-Is: Test.hs
Executable testsuite
Hs-Source-Dirs: src, test
GHC-Options: -Wall
Other-modules:
Test.NumericPrelude.Utility
Test.NumericPrelude.List
Test.MathObj.PartialFraction
Test.MathObj.Polynomial
Test.MathObj.PowerSeries
Main-Is: Test/Run.hs