Numeric.Coalgebra.Quaternion

Documentation

class Distinguished t where

Methods

e :: t

Instances

Distinguished ComplexBasis

Distinguished QuaternionBasis

Distinguished DualBasis

Distinguished DualBasis'

Distinguished QuaternionBasis'

Distinguished TrigBasis

Rig r => Distinguished (Complex r)

Rig r => Distinguished (Quaternion r)

Rig r => Distinguished (Dual r)

Rig r => Distinguished (Dual' r)

Rig r => Distinguished (Quaternion' r)

Rig r => Distinguished (Trig r)

Rig r => Distinguished (ComplexBasis -> r)

Rig r => Distinguished (QuaternionBasis -> r)

Rig r => Distinguished (DualBasis -> r)

Rig r => Distinguished (DualBasis' -> r)

Rig r => Distinguished (QuaternionBasis' -> r)

Rig r => Distinguished (TrigBasis -> r)

Rig r => Distinguished (:->: ComplexBasis r)

Rig r => Distinguished (:->: QuaternionBasis r)

Rig r => Distinguished (:->: QuaternionBasis' r)

Rig r => Distinguished (:->: TrigBasis r)

Distinguished a => Distinguished (Covector r a)

class Distinguished r => Complicated r where

Methods

i :: r

Instances

Complicated ComplexBasis

Complicated QuaternionBasis

Complicated QuaternionBasis'

Complicated TrigBasis

(Distinguished (Complex r), Rig r) => Complicated (Complex r)

(Distinguished (Quaternion r), Rig r) => Complicated (Quaternion r)

(Distinguished (Quaternion' r), Rig r) => Complicated (Quaternion' r)

(Distinguished (Trig r), Rig r) => Complicated (Trig r)

(Distinguished (ComplexBasis -> r), Rig r) => Complicated (ComplexBasis -> r)

(Distinguished (QuaternionBasis -> r), Rig r) => Complicated (QuaternionBasis -> r)

(Distinguished (QuaternionBasis' -> r), Rig r) => Complicated (QuaternionBasis' -> r)

(Distinguished (TrigBasis -> r), Rig r) => Complicated (TrigBasis -> r)

(Distinguished (:->: ComplexBasis r), Rig r) => Complicated (:->: ComplexBasis r)

(Distinguished (:->: QuaternionBasis r), Rig r) => Complicated (:->: QuaternionBasis r)

(Distinguished (:->: QuaternionBasis' r), Rig r) => Complicated (:->: QuaternionBasis' r)

(Distinguished (:->: TrigBasis r), Rig r) => Complicated (:->: TrigBasis r)

(Distinguished (Covector r a), Complicated a) => Complicated (Covector r a)

class Complicated t => Hamiltonian t where

Methods

j :: t

k :: t

Instances

Hamiltonian QuaternionBasis

Hamiltonian QuaternionBasis'

(Complicated (Quaternion r), Rig r) => Hamiltonian (Quaternion r)

(Complicated (Quaternion' r), Rig r) => Hamiltonian (Quaternion' r)

(Complicated (QuaternionBasis -> r), Rig r) => Hamiltonian (QuaternionBasis -> r)

(Complicated (QuaternionBasis' -> r), Rig r) => Hamiltonian (QuaternionBasis' -> r)

(Complicated (:->: QuaternionBasis r), Rig r) => Hamiltonian (:->: QuaternionBasis r)

(Complicated (:->: QuaternionBasis' r), Rig r) => Hamiltonian (:->: QuaternionBasis' r)

(Complicated (Covector r a), Hamiltonian a) => Hamiltonian (Covector r a)

data QuaternionBasis'

Constructors

E'

I'

J'

K'

Instances

Bounded QuaternionBasis'

Enum QuaternionBasis'

Eq QuaternionBasis'

Data QuaternionBasis'

Ord QuaternionBasis'

Read QuaternionBasis'

Show QuaternionBasis'

Ix QuaternionBasis'

Typeable QuaternionBasis'

HasTrie QuaternionBasis'

Distinguished QuaternionBasis'

Complicated QuaternionBasis'

Hamiltonian QuaternionBasis'

MonadReader QuaternionBasis' Quaternion'

(Semiring r, TriviallyInvolutive r, Rng r) => Coalgebra r QuaternionBasis'

dual quaternion comultiplication

(TriviallyInvolutive r, Semiring r) => Algebra r QuaternionBasis'

the trivial diagonal algebra

(UnitalAlgebra r QuaternionBasis', CounitalCoalgebra r QuaternionBasis', TriviallyInvolutive r, Rng r) => Bialgebra r QuaternionBasis'

(Coalgebra r QuaternionBasis', TriviallyInvolutive r, Rng r) => CounitalCoalgebra r QuaternionBasis'

(Algebra r QuaternionBasis', TriviallyInvolutive r, Semiring r) => UnitalAlgebra r QuaternionBasis'

(Bialgebra r QuaternionBasis', TriviallyInvolutive r, InvolutiveSemiring r, Rng r) => HopfAlgebra r QuaternionBasis'

(Coalgebra r QuaternionBasis', TriviallyInvolutive r, InvolutiveSemiring r, Rng r) => InvolutiveCoalgebra r QuaternionBasis'

(Algebra r QuaternionBasis', TriviallyInvolutive r, InvolutiveSemiring r, Rng r) => InvolutiveAlgebra r QuaternionBasis'

Rig r => Distinguished (QuaternionBasis' -> r)

Rig r => Distinguished (:->: QuaternionBasis' r)

(Distinguished (QuaternionBasis' -> r), Rig r) => Complicated (QuaternionBasis' -> r)

(Distinguished (:->: QuaternionBasis' r), Rig r) => Complicated (:->: QuaternionBasis' r)

(Complicated (QuaternionBasis' -> r), Rig r) => Hamiltonian (QuaternionBasis' -> r)

(Complicated (:->: QuaternionBasis' r), Rig r) => Hamiltonian (:->: QuaternionBasis' r)

data Quaternion' a

Constructors

Quaternion' a a a a

Instances

Monad Quaternion'

Functor Quaternion'

Typeable1 Quaternion'

Applicative Quaternion'

Foldable Quaternion'

Traversable Quaternion'

Distributive Quaternion'

Keyed Quaternion'

Zip Quaternion'

ZipWithKey Quaternion'

Indexable Quaternion'

Lookup Quaternion'

Adjustable Quaternion'

FoldableWithKey Quaternion'

FoldableWithKey1 Quaternion'

TraversableWithKey Quaternion'

TraversableWithKey1 Quaternion'

Representable Quaternion'

Traversable1 Quaternion'

Foldable1 Quaternion'

Apply Quaternion'

Bind Quaternion'

MonadReader QuaternionBasis' Quaternion'

(Semiring r, Additive (Quaternion' s), RightModule r s) => RightModule r (Quaternion' s)

(Semiring r, Additive (Quaternion' s), LeftModule r s) => LeftModule r (Quaternion' s)

(Additive r, TriviallyInvolutive r, Rng r) => Quadrance r (Quaternion' r)

Eq a => Eq (Quaternion' a)

(Typeable (Quaternion' a), Data a) => Data (Quaternion' a)

Read a => Read (Quaternion' a)

Show a => Show (Quaternion' a)

(Additive (Quaternion' r), Idempotent r) => Idempotent (Quaternion' r)

(Additive (Quaternion' r), Abelian r) => Abelian (Quaternion' r)

(Additive (Quaternion' r), Partitionable r) => Partitionable (Quaternion' r)

Additive r => Additive (Quaternion' r)

(LeftModule Natural (Quaternion' r), RightModule Natural (Quaternion' r), Monoidal r) => Monoidal (Quaternion' r)

(Additive (Quaternion' r), Abelian (Quaternion' r), Multiplicative (Quaternion' r), TriviallyInvolutive r, Semiring r) => Semiring (Quaternion' r)

(TriviallyInvolutive r, Semiring r) => Multiplicative (Quaternion' r)

(LeftModule Integer (Quaternion' r), RightModule Integer (Quaternion' r), Monoidal (Quaternion' r), Group r) => Group (Quaternion' r)

(Multiplicative (Quaternion' r), TriviallyInvolutive r, Ring r) => Unital (Quaternion' r)

(Unital (Quaternion' r), TriviallyInvolutive r, Ring r, Division r) => Division (Quaternion' r)

(Semiring (Quaternion' r), Unital (Quaternion' r), Monoidal (Quaternion' r), TriviallyInvolutive r, Ring r) => Rig (Quaternion' r)

(Rig (Quaternion' r), Rng (Quaternion' r), TriviallyInvolutive r, Ring r) => Ring (Quaternion' r)

(Multiplicative (Quaternion' r), TriviallyInvolutive r, Rng r) => InvolutiveMultiplication (Quaternion' r)

Rig r => Distinguished (Quaternion' r)

(Distinguished (Quaternion' r), Rig r) => Complicated (Quaternion' r)

(Complicated (Quaternion' r), Rig r) => Hamiltonian (Quaternion' r)

(Semiring (Quaternion' r), Additive (Quaternion' r), TriviallyInvolutive r, Rng r) => RightModule (Quaternion' r) (Quaternion' r)

(Semiring (Quaternion' r), Additive (Quaternion' r), TriviallyInvolutive r, Rng r) => LeftModule (Quaternion' r) (Quaternion' r)

complicate' :: Complicated c => QuaternionBasis' -> (c, c)

Cayley-Dickson quaternion isomorphism (one way)

vectorPart' :: (Representable f, Key f ~ QuaternionBasis') => f r -> (r, r, r)

scalarPart' :: (Representable f, Key f ~ QuaternionBasis') => f r -> r