Safe Haskell	None

Data.Algebra

Contents

Algebraic structures
Algebraic functions
- Specialized methods on linear functions

Description

Common library for algebraic structures. Has the advantage of automatically inferring lots of useful structure, especially in the writing of linear programs. For example, here are several ways of writing 3 x - 4 y + z:

 gsum [3 *& x, (-4) *^ var y, var z]
 linCombination [(3, x), (-4, y), (1, z)]
 3 *& x ^-^ 4 *& y ^+^ var z

In addition, if we have two functions f and g, we can construct linear combinations of those functions, using exactly the same syntax. Moreover, we can multiply functions with Double coefficients by Rational values successfully. This module is intended to offer as much generality as possible without getting in your way.

Synopsis

Algebraic structures

class Group g whereSource

The algebraic structure of a group. Written additively. Required functions: zero and (^-^ or (^+^ and neg)).

Methods

zero :: gSource

(^+^) :: g -> g -> gSource

(^-^) :: g -> g -> gSource

neg :: g -> gSource

Instances

Group Bool
Group Double
Group Int
Group Integer
Integral a => Group (Ratio a)
Group g => Group (IntMap g)
Group g => Group (Poly g)
Group g => Group (a -> g)
(Group g1, Group g2) => Group (g1, g2)
(Ord k, Group g) => Group (Map k g)
(Ord v, Group c) => Group (LinExpr v c)
(Group g1, Group g2, Group g3) => Group (g1, g2, g3)
(Group g1, Group g2, Group g3, Group g4) => Group (g1, g2, g3, g4)

class Group r => Ring r whereSource

The algebraic structure of a unital ring. Assumes that the additive operation forms an abelian group, that the multiplication operation forms a group, and that multiplication distributes.

Methods

one :: rSource

(*#) :: r -> r -> rSource

Instances

Ring Bool
Ring Double
Ring Int
Ring Integer
(Group (Ratio a), Integral a) => Ring (Ratio a)
(Group (Poly r), Ring r) => Ring (Poly r)	The polynomial ring.
(Group (a -> r), Ring r) => Ring (a -> r)	The function ring.
(Group (GroupRing r g), Ord g, Group g, Ring r) => Ring (GroupRing r g)	The group ring.

class Ring f => Field f whereSource

Methods

inv :: f -> fSource

(/#) :: f -> f -> fSource

Instances

Field Double
(Ring (Ratio a), Integral a) => Field (Ratio a)

class (Ring r, Group m) => Module r m whereSource

The algebraic structure of a module. A vector space is a module with coefficients in a field.

Methods

(*^) :: r -> m -> mSource

Instances

Module Double Double
Module Int Double
Module Int Int
Module Int Integer
Module Integer Double
Module Integer Integer
(Ring Int, Group (Ratio a), Integral a) => Module Int (Ratio a)
(Ring Integer, Group (Ratio a), Integral a) => Module Integer (Ratio a)
(Ring r, Group (IntMap m), Module r m) => Module r (IntMap m)
(Ring r, Group (m1, m2), Module r m1, Module r m2) => Module r (m1, m2)
(Ring r, Group (Map k m), Ord k, Module r m) => Module r (Map k m)
(Ring r, Group (a -> m), Module r m) => Module r (a -> m)
(Ring r, Group (LinExpr v c), Ord v, Module r c) => Module r (LinExpr v c)
(Ring r, Group (m1, m2, m3), Module r m1, Module r m2, Module r m3) => Module r (m1, m2, m3)
(Ring r, Group (m1, m2, m3, m4), Module r m1, Module r m2, Module r m3, Module r m4) => Module r (m1, m2, m3, m4)
(Ring (Ratio a), Group Double, Integral a) => Module (Ratio a) Double
(Ring (Ratio a), Group (Ratio a), Integral a) => Module (Ratio a) (Ratio a)
(Ring (GroupRing r g), Group (GroupRing r g), Ord g, Group g, Ring r) => Module (GroupRing r g) (GroupRing r g)