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	None
Language	Haskell2010

Data.VectorSpace

Contents

Vector space classes
- Utilities to work with n-tuples for n = 3, 4, 5
Vector spaces from arbitrary Fractionals

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
- zeroVector :: v
- (*^) :: Groundring v -> v -> v
- (^*) :: v -> Groundring v -> v
- negateVector :: v -> v
- (^+^) :: v -> v -> v
- (^-^) :: v -> v -> v
class (Fractional (Groundring v), RModule v) => VectorSpace v where
- (^/) :: v -> Groundfield v -> v
type Groundfield v = Groundring v
class RModule v => InnerProductSpace v where
- dot :: v -> v -> Groundfield v
class (Floating (Groundfield v), InnerProductSpace v, VectorSpace v) => NormedSpace v where
- norm :: v -> Groundfield v
normalize :: (Eq (Groundfield v), NormedSpace v) => v -> v
break3Tuple :: (a, b, c) -> ((a, b), c)
join3Tuple :: ((a, b), c) -> (a, b, c)
break4Tuple :: (a, b, c, d) -> ((a, b), (c, d))
join4Tuple :: ((a, b), (c, d)) -> (a, b, c, d)
break5Tuple :: (a, b, c, d, e) -> ((a, b), (c, d, e))
join5Tuple :: ((a, b), (c, d, e)) -> (a, b, c, d, e)
newtype FractionalVectorSpace a = FractionalVectorSpace {
- getFractional :: a
}

Vector space classes

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 #
Instance details Defined in Data.VectorSpace Associated Types type Groundring Double :: Type # Methods zeroVector :: Double # (^) :: Groundring Double -> Double -> Double # (^) :: Double -> Groundring Double -> Double # negateVector :: Double -> Double # (^+^) :: Double -> Double -> Double # (^-^) :: Double -> Double -> Double #
RModule Float #
Instance details Defined in Data.VectorSpace Associated Types type Groundring Float :: Type # Methods zeroVector :: Float # (^) :: Groundring Float -> Float -> Float # (^) :: Float -> Groundring Float -> Float # negateVector :: Float -> Float # (^+^) :: Float -> Float -> Float # (^-^) :: Float -> Float -> Float #
RModule Int #
Instance details Defined in Data.VectorSpace Associated Types type Groundring Int :: Type # Methods zeroVector :: Int # (^) :: Groundring Int -> Int -> Int # (^) :: Int -> Groundring Int -> Int # negateVector :: Int -> Int # (^+^) :: Int -> Int -> Int # (^-^) :: Int -> Int -> Int #
RModule Integer #
Instance details Defined in Data.VectorSpace Associated Types type Groundring Integer :: Type # Methods zeroVector :: Integer # (^) :: Groundring Integer -> Integer -> Integer # (^) :: Integer -> Groundring Integer -> Integer # negateVector :: Integer -> Integer # (^+^) :: Integer -> Integer -> Integer # (^-^) :: Integer -> Integer -> Integer #
Num a => RModule (FractionalVectorSpace a) #
Instance details Defined in Data.VectorSpace Associated Types type Groundring (FractionalVectorSpace a) :: Type # Methods zeroVector :: FractionalVectorSpace a # (^) :: Groundring (FractionalVectorSpace a) -> FractionalVectorSpace a -> FractionalVectorSpace a # (^) :: FractionalVectorSpace a -> Groundring (FractionalVectorSpace a) -> FractionalVectorSpace a # negateVector :: FractionalVectorSpace a -> FractionalVectorSpace a # (^+^) :: FractionalVectorSpace a -> FractionalVectorSpace a -> FractionalVectorSpace a # (^-^) :: FractionalVectorSpace a -> FractionalVectorSpace a -> FractionalVectorSpace a #
(Groundring a ~ Groundring b, RModule a, RModule b) => RModule (a, b) #
Instance details Defined in Data.VectorSpace Associated Types type Groundring (a, b) :: Type # 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) #
(Groundring a ~ Groundring b, Groundring a ~ Groundring c, RModule a, RModule b, RModule c) => RModule (a, b, c) #
Instance details Defined in Data.VectorSpace Associated Types type Groundring (a, b, c) :: Type # Methods zeroVector :: (a, b, c) # (^) :: Groundring (a, b, c) -> (a, b, c) -> (a, b, c) # (^) :: (a, b, c) -> Groundring (a, b, c) -> (a, b, c) # negateVector :: (a, b, c) -> (a, b, c) # (^+^) :: (a, b, c) -> (a, b, c) -> (a, b, c) # (^-^) :: (a, b, c) -> (a, b, c) -> (a, b, c) #
(Monad m, RModule v) => RModule (MSF m a v) #	R-module instance for `MSF`s.
Instance details Defined in Data.MonadicStreamFunction.Instances.VectorSpace Associated Types type Groundring (MSF m a v) :: Type # 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 #
(Groundring a ~ Groundring b, Groundring a ~ Groundring c, Groundring a ~ Groundring d, RModule a, RModule b, RModule c, RModule d) => RModule (a, b, c, d) #
Instance details Defined in Data.VectorSpace Associated Types type Groundring (a, b, c, d) :: Type # Methods zeroVector :: (a, b, c, d) # (^) :: Groundring (a, b, c, d) -> (a, b, c, d) -> (a, b, c, d) # (^) :: (a, b, c, d) -> Groundring (a, b, c, d) -> (a, b, c, d) # negateVector :: (a, b, c, d) -> (a, b, c, d) # (^+^) :: (a, b, c, d) -> (a, b, c, d) -> (a, b, c, d) # (^-^) :: (a, b, c, d) -> (a, b, c, d) -> (a, b, c, d) #
(Groundring a ~ Groundring b, Groundring a ~ Groundring c, Groundring a ~ Groundring d, Groundring a ~ Groundring e, RModule a, RModule b, RModule c, RModule d, RModule e) => RModule (a, b, c, d, e) #
Instance details Defined in Data.VectorSpace Associated Types type Groundring (a, b, c, d, e) :: Type # Methods zeroVector :: (a, b, c, d, e) # (^) :: Groundring (a, b, c, d, e) -> (a, b, c, d, e) -> (a, b, c, d, e) # (^) :: (a, b, c, d, e) -> Groundring (a, b, c, d, e) -> (a, b, c, d, e) # negateVector :: (a, b, c, d, e) -> (a, b, c, d, e) # (^+^) :: (a, b, c, d, e) -> (a, b, c, d, e) -> (a, b, c, d, e) # (^-^) :: (a, b, c, d, e) -> (a, b, c, d, e) -> (a, b, c, d, e) #

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.

Minimal complete definition

Nothing

Methods

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

Instances

VectorSpace Double #
Instance details Defined in Data.VectorSpace Methods (^/) :: Double -> Groundfield Double -> Double #
VectorSpace Float #
Instance details Defined in Data.VectorSpace Methods (^/) :: Float -> Groundfield Float -> Float #
Fractional a => VectorSpace (FractionalVectorSpace a) #
Instance details Defined in Data.VectorSpace Methods (^/) :: FractionalVectorSpace a -> Groundfield (FractionalVectorSpace a) -> FractionalVectorSpace a #
(Groundfield a ~ Groundfield b, VectorSpace a, VectorSpace b) => VectorSpace (a, b) #
Instance details Defined in Data.VectorSpace Methods (^/) :: (a, b) -> Groundfield (a, b) -> (a, b) #
(Groundring a ~ Groundring b, Groundring a ~ Groundring c, VectorSpace a, VectorSpace b, VectorSpace c) => VectorSpace (a, b, c) #
Instance details Defined in Data.VectorSpace Methods (^/) :: (a, b, c) -> Groundfield (a, b, c) -> (a, b, c) #
(Monad m, VectorSpace v) => VectorSpace (MSF m a v) #	Vector-space instance for `MSF`s.
Instance details Defined in Data.MonadicStreamFunction.Instances.VectorSpace Methods (^/) :: MSF m a v -> Groundfield (MSF m a v) -> MSF m a v #
(Groundring a ~ Groundring b, Groundring a ~ Groundring c, Groundring a ~ Groundring d, VectorSpace a, VectorSpace b, VectorSpace c, VectorSpace d) => VectorSpace (a, b, c, d) #
Instance details Defined in Data.VectorSpace Methods (^/) :: (a, b, c, d) -> Groundfield (a, b, c, d) -> (a, b, c, d) #
(Groundring a ~ Groundring b, Groundring a ~ Groundring c, Groundring a ~ Groundring d, Groundring a ~ Groundring e, VectorSpace a, VectorSpace b, VectorSpace c, VectorSpace d, VectorSpace e) => VectorSpace (a, b, c, d, e) #
Instance details Defined in Data.VectorSpace Methods (^/) :: (a, b, c, d, e) -> Groundfield (a, b, c, d, e) -> (a, b, c, d, e) #

type Groundfield v = Groundring 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.

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
```

Methods

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

Instances

Num a => InnerProductSpace (FractionalVectorSpace a) #
Instance details Defined in Data.VectorSpace Methods dot :: FractionalVectorSpace a -> FractionalVectorSpace a -> Groundfield (FractionalVectorSpace a) #
(Groundfield a ~ Groundfield b, InnerProductSpace a, InnerProductSpace b) => InnerProductSpace (a, b) #
Instance details Defined in Data.VectorSpace Methods dot :: (a, b) -> (a, b) -> Groundfield (a, b) #
(Groundring a ~ Groundring b, Groundring a ~ Groundring c, InnerProductSpace a, InnerProductSpace b, InnerProductSpace c) => InnerProductSpace (a, b, c) #
Instance details Defined in Data.VectorSpace Methods dot :: (a, b, c) -> (a, b, c) -> Groundfield (a, b, c) #
(Groundring a ~ Groundring b, Groundring a ~ Groundring c, Groundring a ~ Groundring d, InnerProductSpace a, InnerProductSpace b, InnerProductSpace c, InnerProductSpace d) => InnerProductSpace (a, b, c, d) #
Instance details Defined in Data.VectorSpace Methods dot :: (a, b, c, d) -> (a, b, c, d) -> Groundfield (a, b, c, d) #
(Groundring a ~ Groundring b, Groundring a ~ Groundring c, Groundring a ~ Groundring d, Groundring a ~ Groundring e, InnerProductSpace a, InnerProductSpace b, InnerProductSpace c, InnerProductSpace d, InnerProductSpace e) => InnerProductSpace (a, b, c, d, e) #
Instance details Defined in Data.VectorSpace Methods dot :: (a, b, c, d, e) -> (a, b, c, d, e) -> Groundfield (a, b, c, d, e) #

class (Floating (Groundfield v), InnerProductSpace v, VectorSpace 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

Nothing

Methods

norm :: v -> Groundfield v #

Instances

Floating a => NormedSpace (FractionalVectorSpace a) #
Instance details Defined in Data.VectorSpace Methods norm :: FractionalVectorSpace a -> Groundfield (FractionalVectorSpace a) #
(Groundfield a ~ Groundfield b, NormedSpace a, NormedSpace b) => NormedSpace (a, b) #
Instance details Defined in Data.VectorSpace Methods norm :: (a, b) -> Groundfield (a, b) #
(Groundring a ~ Groundring b, Groundring a ~ Groundring c, NormedSpace a, NormedSpace b, NormedSpace c) => NormedSpace (a, b, c) #
Instance details Defined in Data.VectorSpace Methods norm :: (a, b, c) -> Groundfield (a, b, c) #
(Groundring a ~ Groundring b, Groundring a ~ Groundring c, Groundring a ~ Groundring d, NormedSpace a, NormedSpace b, NormedSpace c, NormedSpace d) => NormedSpace (a, b, c, d) #
Instance details Defined in Data.VectorSpace Methods norm :: (a, b, c, d) -> Groundfield (a, b, c, d) #
(Groundring a ~ Groundring b, Groundring a ~ Groundring c, Groundring a ~ Groundring d, Groundring a ~ Groundring e, NormedSpace a, NormedSpace b, NormedSpace c, NormedSpace d, NormedSpace e) => NormedSpace (a, b, c, d, e) #
Instance details Defined in Data.VectorSpace Methods norm :: (a, b, c, d, e) -> Groundfield (a, b, c, d, e) #

normalize :: (Eq (Groundfield v), NormedSpace v) => v -> v #

Divides a vector by its norm, resulting in a vector of norm 1. Throws an error on vectors with norm 0.

Utilities to work with n-tuples for n = 3, 4, 5

break3Tuple :: (a, b, c) -> ((a, b), c) #

join3Tuple :: ((a, b), c) -> (a, b, c) #

break4Tuple :: (a, b, c, d) -> ((a, b), (c, d)) #

join4Tuple :: ((a, b), (c, d)) -> (a, b, c, d) #

break5Tuple :: (a, b, c, d, e) -> ((a, b), (c, d, e)) #

join5Tuple :: ((a, b), (c, d, e)) -> (a, b, c, d, e) #

Vector spaces from arbitrary `Fractional`s

newtype FractionalVectorSpace a #

Wrap an arbitrary Fractional in this newtype in order to get VectorSpace, and related instances.

Constructors

FractionalVectorSpace
Fields getFractional :: a

Instances

Fractional a => Fractional (FractionalVectorSpace a) #
Instance details Defined in Data.VectorSpace Methods (/) :: FractionalVectorSpace a -> FractionalVectorSpace a -> FractionalVectorSpace a # recip :: FractionalVectorSpace a -> FractionalVectorSpace a # fromRational :: Rational -> FractionalVectorSpace a #
Num a => Num (FractionalVectorSpace a) #
Instance details Defined in Data.VectorSpace Methods (+) :: FractionalVectorSpace a -> FractionalVectorSpace a -> FractionalVectorSpace a # (-) :: FractionalVectorSpace a -> FractionalVectorSpace a -> FractionalVectorSpace a # (*) :: FractionalVectorSpace a -> FractionalVectorSpace a -> FractionalVectorSpace a # negate :: FractionalVectorSpace a -> FractionalVectorSpace a # abs :: FractionalVectorSpace a -> FractionalVectorSpace a # signum :: FractionalVectorSpace a -> FractionalVectorSpace a # fromInteger :: Integer -> FractionalVectorSpace a #
Floating a => NormedSpace (FractionalVectorSpace a) #
Instance details Defined in Data.VectorSpace Methods norm :: FractionalVectorSpace a -> Groundfield (FractionalVectorSpace a) #
Num a => InnerProductSpace (FractionalVectorSpace a) #
Instance details Defined in Data.VectorSpace Methods dot :: FractionalVectorSpace a -> FractionalVectorSpace a -> Groundfield (FractionalVectorSpace a) #
Fractional a => VectorSpace (FractionalVectorSpace a) #
Instance details Defined in Data.VectorSpace Methods (^/) :: FractionalVectorSpace a -> Groundfield (FractionalVectorSpace a) -> FractionalVectorSpace a #
Num a => RModule (FractionalVectorSpace a) #
Instance details Defined in Data.VectorSpace Associated Types type Groundring (FractionalVectorSpace a) :: Type # Methods zeroVector :: FractionalVectorSpace a # (^) :: Groundring (FractionalVectorSpace a) -> FractionalVectorSpace a -> FractionalVectorSpace a # (^) :: FractionalVectorSpace a -> Groundring (FractionalVectorSpace a) -> FractionalVectorSpace a # negateVector :: FractionalVectorSpace a -> FractionalVectorSpace a # (^+^) :: FractionalVectorSpace a -> FractionalVectorSpace a -> FractionalVectorSpace a # (^-^) :: FractionalVectorSpace a -> FractionalVectorSpace a -> FractionalVectorSpace a #
type Groundring (FractionalVectorSpace a) #
Instance details Defined in Data.VectorSpace type Groundring (FractionalVectorSpace a) = a

Vector space classes

Utilities to work with n-tuples for n = 3, 4, 5

Vector spaces from arbitrary Fractionals

Vector spaces from arbitrary `Fractional`s