Copyright	(c) Ivan Perez and Manuel Bärenz
License	See the LICENSE file in the distribution.
Maintainer	ivan.perez@keera.co.uk
Stability	provisional
Portability	non-portable (GHC extensions)
Safe Haskell	Safe
Language	Haskell2010

Data.VectorSpace

Description

Vector space type relation and basic instances. Heavily inspired by Yampa's FRP.Yampa.VectorSpace module.

Synopsis

class Num (Groundring v) => RModule v where
- type Groundring v
class (Fractional (Groundring v), RModule v) => VectorSpace v where
type family Groundfield v :: *
class RModule v => InnerProductSpace v where
class RModule v => NormedSpace v where

Documentation

class Num (Groundring v) => RModule v where #

R-modules. A module v over a ring Groundring v is an abelian group with a linear multiplication. The hat ^ denotes the side of an operation on which the vector stands, i.e. a *^ v for v a vector.

A minimal definition should include the type Groundring and the implementations of zeroVector, ^+^, and one of *^ or ^*.

The following laws must be satisfied:

```
v1 ^+^ v2 == v2 ^+^ v1
```
```
a *^ zeroVector == zeroVector
```
@a *^ (v1 ^+^ v2) == a *^ v1 ^+^ a*^ v2
```
a *^ v == v ^* a
```
```
negateVector v == (-1) *^ v
```
```
v1 ^-^ v2 == v1 ^+^ negateVector v2
```

Minimal complete definition

zeroVector, (^+^)

Associated Types

type Groundring v #

Methods

zeroVector :: v #

(*^) :: Groundring v -> v -> v infixr 6 #

(^*) :: v -> Groundring v -> v #

negateVector :: v -> v #

(^+^) :: v -> v -> v infixl 5 #

(^-^) :: v -> v -> v infixl 5 #

Instances

RModule Double #	R-mobule instance for `Double`s.
Instance details Defined in Data.VectorSpace.Specific Associated Types type Groundring Double :: * # Methods zeroVector :: Double # (^) :: Groundring Double -> Double -> Double # (^) :: Double -> Groundring Double -> Double # negateVector :: Double -> Double # (^+^) :: Double -> Double -> Double # (^-^) :: Double -> Double -> Double #
RModule Float #	R-mobule instance for `Floating`s.
Instance details Defined in Data.VectorSpace.Specific Associated Types type Groundring Float :: * # Methods zeroVector :: Float # (^) :: Groundring Float -> Float -> Float # (^) :: Float -> Groundring Float -> Float # negateVector :: Float -> Float # (^+^) :: Float -> Float -> Float # (^-^) :: Float -> Float -> Float #
RModule Int #	R-mobule instance for `Int`s.
Instance details Defined in Data.VectorSpace.Specific Associated Types type Groundring Int :: * # Methods zeroVector :: Int # (^) :: Groundring Int -> Int -> Int # (^) :: Int -> Groundring Int -> Int # negateVector :: Int -> Int # (^+^) :: Int -> Int -> Int # (^-^) :: Int -> Int -> Int #
RModule Integer #	R-mobule instance for `Integer`s.
Instance details Defined in Data.VectorSpace.Specific Associated Types type Groundring Integer :: * # Methods zeroVector :: Integer # (^) :: Groundring Integer -> Integer -> Integer # (^) :: Integer -> Groundring Integer -> Integer # negateVector :: Integer -> Integer # (^+^) :: Integer -> Integer -> Integer # (^-^) :: Integer -> Integer -> Integer #
Num a => RModule a #	R-module instance for any number, where '^+^ is `+` and multiplication is normal multiplication.
Instance details Defined in Data.VectorSpace.Fractional Associated Types type Groundring a :: * # Methods zeroVector :: a # (^) :: Groundring a -> a -> a # (^) :: a -> Groundring a -> a # negateVector :: a -> a # (^+^) :: a -> a -> a # (^-^) :: a -> a -> a #
(Groundring a ~ Groundring b, RModule a, RModule b) => RModule (a, b) #	R-module instance for tuples.
Instance details Defined in Data.VectorSpace.Tuples Associated Types type Groundring (a, b) :: * # Methods zeroVector :: (a, b) # (^) :: Groundring (a, b) -> (a, b) -> (a, b) # (^) :: (a, b) -> Groundring (a, b) -> (a, b) # negateVector :: (a, b) -> (a, b) # (^+^) :: (a, b) -> (a, b) -> (a, b) # (^-^) :: (a, b) -> (a, b) -> (a, b) #
Num a => RModule (a, a, a) #	R-module instance for tuples with 3 elements.
Instance details Defined in Data.VectorSpace.Tuples Associated Types type Groundring (a, a, a) :: * # Methods zeroVector :: (a, a, a) # (^) :: Groundring (a, a, a) -> (a, a, a) -> (a, a, a) # (^) :: (a, a, a) -> Groundring (a, a, a) -> (a, a, a) # negateVector :: (a, a, a) -> (a, a, a) # (^+^) :: (a, a, a) -> (a, a, a) -> (a, a, a) # (^-^) :: (a, a, a) -> (a, a, a) -> (a, a, a) #
(Monad m, RModule v) => RModule (MSF m a v) #	R-module instance for MSFs.
Instance details Defined in Data.MonadicStreamFunction.Instances.VectorSpace Associated Types type Groundring (MSF m a v) :: * # Methods zeroVector :: MSF m a v # (^) :: Groundring (MSF m a v) -> MSF m a v -> MSF m a v # (^) :: MSF m a v -> Groundring (MSF m a v) -> MSF m a v # negateVector :: MSF m a v -> MSF m a v # (^+^) :: MSF m a v -> MSF m a v -> MSF m a v # (^-^) :: MSF m a v -> MSF m a v -> MSF m a v #
Num a => RModule (a, a, a, a) #	R-module instance for tuples with 4 elements.
Instance details Defined in Data.VectorSpace.Tuples Associated Types type Groundring (a, a, a, a) :: * # Methods zeroVector :: (a, a, a, a) # (^) :: Groundring (a, a, a, a) -> (a, a, a, a) -> (a, a, a, a) # (^) :: (a, a, a, a) -> Groundring (a, a, a, a) -> (a, a, a, a) # negateVector :: (a, a, a, a) -> (a, a, a, a) # (^+^) :: (a, a, a, a) -> (a, a, a, a) -> (a, a, a, a) # (^-^) :: (a, a, a, a) -> (a, a, a, a) -> (a, a, a, a) #
Num a => RModule (a, a, a, a, a) #	R-module instance for tuples with 5 elements.
Instance details Defined in Data.VectorSpace.Tuples Associated Types type Groundring (a, a, a, a, a) :: * # Methods zeroVector :: (a, a, a, a, a) # (^) :: Groundring (a, a, a, a, a) -> (a, a, a, a, a) -> (a, a, a, a, a) # (^) :: (a, a, a, a, a) -> Groundring (a, a, a, a, a) -> (a, a, a, a, a) # negateVector :: (a, a, a, a, a) -> (a, a, a, a, a) # (^+^) :: (a, a, a, a, a) -> (a, a, a, a, a) -> (a, a, a, a, a) # (^-^) :: (a, a, a, a, a) -> (a, a, a, a, a) -> (a, a, a, a, a) #

class (Fractional (Groundring v), RModule v) => VectorSpace v where #

A vector space is a module over a field, i.e. a commutative ring with inverses.

It needs to satisfy the axiom v ^ a == (1a) *^ v, which is the default implementation.

Methods

(^/) :: v -> Groundfield v -> v infixl 6 #

Instances

VectorSpace Double #	Vector-space instance for `Double`.
Instance details Defined in Data.VectorSpace.Specific Methods (^/) :: Double -> Groundfield Double -> Double #
VectorSpace Float #	Vector-space instance for `Floating`s.
Instance details Defined in Data.VectorSpace.Specific Methods (^/) :: Float -> Groundfield Float -> Float #
Fractional a => VectorSpace a #	Vector-space instance for any fractional, where vectorial division is normal number division.
Instance details Defined in Data.VectorSpace.Fractional Methods (^/) :: a -> Groundfield a -> a #
(Groundfield a ~ Groundfield b, VectorSpace a, VectorSpace b) => VectorSpace (a, b) #	Vector-space instance for tuples.
Instance details Defined in Data.VectorSpace.Tuples Methods (^/) :: (a, b) -> Groundfield (a, b) -> (a, b) #
Fractional a => VectorSpace (a, a, a) #	Vector-space instance for tuples with 3 elements.
Instance details Defined in Data.VectorSpace.Tuples Methods (^/) :: (a, a, a) -> Groundfield (a, a, a) -> (a, a, a) #
(Monad m, VectorSpace v) => VectorSpace (MSF m a v) #	Vector-space instance for MSFs.
Instance details Defined in Data.MonadicStreamFunction.Instances.VectorSpace Methods (^/) :: MSF m a v -> Groundfield (MSF m a v) -> MSF m a v #
Fractional a => VectorSpace (a, a, a, a) #	Vector-space instance for tuples with 4 elements.
Instance details Defined in Data.VectorSpace.Tuples Methods (^/) :: (a, a, a, a) -> Groundfield (a, a, a, a) -> (a, a, a, a) #
Fractional a => VectorSpace (a, a, a, a, a) #	Vector-space instance for tuples with 5 elements.
Instance details Defined in Data.VectorSpace.Tuples Methods (^/) :: (a, a, a, a, a) -> Groundfield (a, a, a, a, a) -> (a, a, a, a, a) #

type family Groundfield v :: * #

The ground ring of a vector space is required to be commutative and to possess inverses. It is then called the "ground field". Commutativity amounts to the law a * b = b * a, and the existence of inverses is given by the requirement of the Fractional type class.

Instances

type Groundfield v #
Instance details Defined in Data.VectorSpace type Groundfield v = Groundring v

class RModule v => InnerProductSpace v where #

An inner product space is a module with an inner product, i.e. a map dot satisfying

```
v1 dot v2 == v2 dot v1
```

(v1 ^+^ v2) dot v3 == v1 dot v3 ^+^ v2 dot v3

```
(a *^ v1) dot v2 == a *^ v1 dot v2
```

Minimal complete definition

dot

Methods

dot :: v -> v -> Groundfield v infix 6 #

Instances

Num a => InnerProductSpace a #	Inner-product instance for any number.
Instance details Defined in Data.VectorSpace.Fractional Methods dot :: a -> a -> Groundfield a #
(Groundfield a ~ Groundfield b, InnerProductSpace a, InnerProductSpace b) => InnerProductSpace (a, b) #	Inner Product Space instance for tuples.
Instance details Defined in Data.VectorSpace.Tuples Methods dot :: (a, b) -> (a, b) -> Groundfield (a, b) #
Num a => InnerProductSpace (a, a, a) #	Inner Product Space instance for tuples with 3 elements.
Instance details Defined in Data.VectorSpace.Tuples Methods dot :: (a, a, a) -> (a, a, a) -> Groundfield (a, a, a) #
Num a => InnerProductSpace (a, a, a, a) #	Inner Product Space instance for tuples with 4 elements.
Instance details Defined in Data.VectorSpace.Tuples Methods dot :: (a, a, a, a) -> (a, a, a, a) -> Groundfield (a, a, a, a) #
Num a => InnerProductSpace (a, a, a, a, a) #	Inner Product Space instance for tuples with 5 elements.
Instance details Defined in Data.VectorSpace.Tuples Methods dot :: (a, a, a, a, a) -> (a, a, a, a, a) -> Groundfield (a, a, a, a, a) #

class RModule v => NormedSpace v where #

A normed space is a module with a norm, i.e. a function norm satisfying

```
norm (a ^* v) = a ^* norm v
```
norm (v1 ^+^ v2) <= norm v1 ^+^ norm v2 (the "triangle inequality")

A typical example is sqrt (v dot v), for an inner product space.

Minimal complete definition

norm

Methods

norm :: v -> Groundfield v #