Numeric.Coalgebra.Quaternion

Documentation

class Distinguished t where

Minimal complete definition

e

Methods

e :: t

Instances

Distinguished TrigBasis

Methods

e :: TrigBasis

Distinguished QuaternionBasis'

Methods

e :: QuaternionBasis'

Distinguished DualBasis'

Methods

e :: DualBasis'

Distinguished QuaternionBasis

Methods

e :: QuaternionBasis

Distinguished DualBasis

Methods

e :: DualBasis

Distinguished ComplexBasis

Methods

e :: ComplexBasis

Rig r => Distinguished (Trig r)

Methods

e :: Trig r

Rig r => Distinguished (Quaternion' r)

Methods

e :: Quaternion' r

Rig r => Distinguished (Dual' r)

Methods

e :: Dual' r

Rig r => Distinguished (Quaternion r)

Methods

e :: Quaternion r

Rig r => Distinguished (Dual r)

Methods

e :: Dual r

Rig r => Distinguished (Complex r)

Methods

e :: Complex r

Rig r => Distinguished (TrigBasis -> r)

Methods

e :: TrigBasis -> r

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

Methods

e :: QuaternionBasis' -> r

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

Methods

e :: DualBasis' -> r

Rig r => Distinguished (QuaternionBasis -> r)

Methods

e :: QuaternionBasis -> r

Rig r => Distinguished (DualBasis -> r)

Methods

e :: DualBasis -> r

Rig r => Distinguished (ComplexBasis -> r)

Methods

e :: ComplexBasis -> r

Distinguished a => Distinguished (Covector r a)

Methods

e :: Covector r a

class Distinguished r => Complicated r where

Minimal complete definition

i

Methods

i :: r

Instances

Complicated TrigBasis

Methods

i :: TrigBasis

Complicated QuaternionBasis'

Methods

i :: QuaternionBasis'

Complicated QuaternionBasis

Methods

i :: QuaternionBasis

Complicated ComplexBasis

Methods

i :: ComplexBasis

Rig r => Complicated (Trig r)

Methods

i :: Trig r

Rig r => Complicated (Quaternion' r)

Methods

i :: Quaternion' r

Rig r => Complicated (Quaternion r)

Methods

i :: Quaternion r

Rig r => Complicated (Complex r)

Methods

i :: Complex r

Rig r => Complicated (TrigBasis -> r)

Methods

i :: TrigBasis -> r

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

Methods

i :: QuaternionBasis' -> r

Rig r => Complicated (QuaternionBasis -> r)

Methods

i :: QuaternionBasis -> r

Rig r => Complicated (ComplexBasis -> r)

Methods

i :: ComplexBasis -> r

Complicated a => Complicated (Covector r a)

Methods

i :: Covector r a

class Complicated t => Hamiltonian t where

Minimal complete definition

j, k

Methods

j :: t

k :: t

Instances

Hamiltonian QuaternionBasis'

Methods

j :: QuaternionBasis'

k :: QuaternionBasis'

Hamiltonian QuaternionBasis

Methods

j :: QuaternionBasis

k :: QuaternionBasis

Rig r => Hamiltonian (Quaternion' r)

Methods

j :: Quaternion' r

k :: Quaternion' r

Rig r => Hamiltonian (Quaternion r)

Methods

j :: Quaternion r

k :: Quaternion r

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

Methods

j :: QuaternionBasis' -> r

k :: QuaternionBasis' -> r

Rig r => Hamiltonian (QuaternionBasis -> r)

Methods

j :: QuaternionBasis -> r

k :: QuaternionBasis -> r

Hamiltonian a => Hamiltonian (Covector r a)

Methods

j :: Covector r a

k :: Covector r a

data QuaternionBasis'

Constructors

E'

I'

J'

K'

Instances

Bounded QuaternionBasis'

Methods

minBound :: QuaternionBasis'

maxBound :: QuaternionBasis'

Enum QuaternionBasis'

Methods

succ :: QuaternionBasis' -> QuaternionBasis'

pred :: QuaternionBasis' -> QuaternionBasis'

toEnum :: Int -> QuaternionBasis'

fromEnum :: QuaternionBasis' -> Int

enumFrom :: QuaternionBasis' -> [QuaternionBasis']

enumFromThen :: QuaternionBasis' -> QuaternionBasis' -> [QuaternionBasis']

enumFromTo :: QuaternionBasis' -> QuaternionBasis' -> [QuaternionBasis']

enumFromThenTo :: QuaternionBasis' -> QuaternionBasis' -> QuaternionBasis' -> [QuaternionBasis']

Eq QuaternionBasis'

Methods

(==) :: QuaternionBasis' -> QuaternionBasis' -> Bool

(/=) :: QuaternionBasis' -> QuaternionBasis' -> Bool

Data QuaternionBasis'

Methods

gfoldl :: (forall d b. Data d => c (d -> b) -> d -> c b) -> (forall g. g -> c g) -> QuaternionBasis' -> c QuaternionBasis'

gunfold :: (forall b r. Data b => c (b -> r) -> c r) -> (forall r. r -> c r) -> Constr -> c QuaternionBasis'

toConstr :: QuaternionBasis' -> Constr

dataTypeOf :: QuaternionBasis' -> DataType

dataCast1 :: Typeable (* -> *) t => (forall d. Data d => c (t d)) -> Maybe (c QuaternionBasis')

dataCast2 :: Typeable (* -> * -> *) t => (forall d e. (Data d, Data e) => c (t d e)) -> Maybe (c QuaternionBasis')

gmapT :: (forall b. Data b => b -> b) -> QuaternionBasis' -> QuaternionBasis'

gmapQl :: (r -> r' -> r) -> r -> (forall d. Data d => d -> r') -> QuaternionBasis' -> r

gmapQr :: (r' -> r -> r) -> r -> (forall d. Data d => d -> r') -> QuaternionBasis' -> r

gmapQ :: (forall d. Data d => d -> u) -> QuaternionBasis' -> [u]

gmapQi :: Int -> (forall d. Data d => d -> u) -> QuaternionBasis' -> u

gmapM :: Monad m => (forall d. Data d => d -> m d) -> QuaternionBasis' -> m QuaternionBasis'

gmapMp :: MonadPlus m => (forall d. Data d => d -> m d) -> QuaternionBasis' -> m QuaternionBasis'

gmapMo :: MonadPlus m => (forall d. Data d => d -> m d) -> QuaternionBasis' -> m QuaternionBasis'

Ord QuaternionBasis'

Methods

compare :: QuaternionBasis' -> QuaternionBasis' -> Ordering

(<) :: QuaternionBasis' -> QuaternionBasis' -> Bool

(<=) :: QuaternionBasis' -> QuaternionBasis' -> Bool

(>) :: QuaternionBasis' -> QuaternionBasis' -> Bool

(>=) :: QuaternionBasis' -> QuaternionBasis' -> Bool

max :: QuaternionBasis' -> QuaternionBasis' -> QuaternionBasis'

min :: QuaternionBasis' -> QuaternionBasis' -> QuaternionBasis'

Read QuaternionBasis'

Methods

readsPrec :: Int -> ReadS QuaternionBasis'

readList :: ReadS [QuaternionBasis']

readPrec :: ReadPrec QuaternionBasis'

readListPrec :: ReadPrec [QuaternionBasis']

Show QuaternionBasis'

Methods

showsPrec :: Int -> QuaternionBasis' -> ShowS

show :: QuaternionBasis' -> String

showList :: [QuaternionBasis'] -> ShowS

Ix QuaternionBasis'

Methods

range :: (QuaternionBasis', QuaternionBasis') -> [QuaternionBasis']

index :: (QuaternionBasis', QuaternionBasis') -> QuaternionBasis' -> Int

unsafeIndex :: (QuaternionBasis', QuaternionBasis') -> QuaternionBasis' -> Int

inRange :: (QuaternionBasis', QuaternionBasis') -> QuaternionBasis' -> Bool

rangeSize :: (QuaternionBasis', QuaternionBasis') -> Int

unsafeRangeSize :: (QuaternionBasis', QuaternionBasis') -> Int

Distinguished QuaternionBasis'

Methods

e :: QuaternionBasis'

Complicated QuaternionBasis'

Methods

i :: QuaternionBasis'

Hamiltonian QuaternionBasis'

Methods

j :: QuaternionBasis'

k :: QuaternionBasis'

MonadReader QuaternionBasis' Quaternion'

Methods

ask :: Quaternion' QuaternionBasis'

local :: (QuaternionBasis' -> QuaternionBasis') -> Quaternion' a -> Quaternion' a

reader :: (QuaternionBasis' -> a) -> Quaternion' a

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

dual quaternion comultiplication

Methods

comult :: (QuaternionBasis' -> r) -> QuaternionBasis' -> QuaternionBasis' -> r

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

the trivial diagonal algebra

Methods

mult :: (QuaternionBasis' -> QuaternionBasis' -> r) -> QuaternionBasis' -> r

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

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

Methods

counit :: (QuaternionBasis' -> r) -> r

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

Methods

unit :: r -> QuaternionBasis' -> r

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

Methods

antipode :: (QuaternionBasis' -> r) -> QuaternionBasis' -> r

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

Methods

coinv :: (QuaternionBasis' -> r) -> QuaternionBasis' -> r

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

Methods

inv :: (QuaternionBasis' -> r) -> QuaternionBasis' -> r

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

Methods

e :: QuaternionBasis' -> r

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

Methods

i :: QuaternionBasis' -> r

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

Methods

j :: QuaternionBasis' -> r

k :: QuaternionBasis' -> r

data Quaternion' a

Constructors

Quaternion' a a a a

Instances

Monad Quaternion'

Methods

(>>=) :: Quaternion' a -> (a -> Quaternion' b) -> Quaternion' b

(>>) :: Quaternion' a -> Quaternion' b -> Quaternion' b

return :: a -> Quaternion' a

fail :: String -> Quaternion' a

Functor Quaternion'

Methods

fmap :: (a -> b) -> Quaternion' a -> Quaternion' b

(<$) :: a -> Quaternion' b -> Quaternion' a

Applicative Quaternion'

Methods

pure :: a -> Quaternion' a

(<*>) :: Quaternion' (a -> b) -> Quaternion' a -> Quaternion' b

liftA2 :: (a -> b -> c) -> Quaternion' a -> Quaternion' b -> Quaternion' c

(*>) :: Quaternion' a -> Quaternion' b -> Quaternion' b

(<*) :: Quaternion' a -> Quaternion' b -> Quaternion' a

Foldable Quaternion'

Methods

fold :: Monoid m => Quaternion' m -> m

foldMap :: Monoid m => (a -> m) -> Quaternion' a -> m

foldr :: (a -> b -> b) -> b -> Quaternion' a -> b

foldr' :: (a -> b -> b) -> b -> Quaternion' a -> b

foldl :: (b -> a -> b) -> b -> Quaternion' a -> b

foldl' :: (b -> a -> b) -> b -> Quaternion' a -> b

foldr1 :: (a -> a -> a) -> Quaternion' a -> a

foldl1 :: (a -> a -> a) -> Quaternion' a -> a

toList :: Quaternion' a -> [a]

null :: Quaternion' a -> Bool

length :: Quaternion' a -> Int

elem :: Eq a => a -> Quaternion' a -> Bool

maximum :: Ord a => Quaternion' a -> a

minimum :: Ord a => Quaternion' a -> a

sum :: Num a => Quaternion' a -> a

product :: Num a => Quaternion' a -> a

Traversable Quaternion'

Methods

traverse :: Applicative f => (a -> f b) -> Quaternion' a -> f (Quaternion' b)

sequenceA :: Applicative f => Quaternion' (f a) -> f (Quaternion' a)

mapM :: Monad m => (a -> m b) -> Quaternion' a -> m (Quaternion' b)

sequence :: Monad m => Quaternion' (m a) -> m (Quaternion' a)

Distributive Quaternion'

Methods

distribute :: Functor f => f (Quaternion' a) -> Quaternion' (f a)

collect :: Functor f => (a -> Quaternion' b) -> f a -> Quaternion' (f b)

distributeM :: Monad m => m (Quaternion' a) -> Quaternion' (m a)

collectM :: Monad m => (a -> Quaternion' b) -> m a -> Quaternion' (m b)

Representable Quaternion'

Associated Types

type Rep (Quaternion' :: * -> *) :: *

Methods

tabulate :: (Rep Quaternion' -> a) -> Quaternion' a

index :: Quaternion' a -> Rep Quaternion' -> a

Traversable1 Quaternion'

Methods

traverse1 :: Apply f => (a -> f b) -> Quaternion' a -> f (Quaternion' b)

sequence1 :: Apply f => Quaternion' (f b) -> f (Quaternion' b)

Foldable1 Quaternion'

Methods

fold1 :: Semigroup m => Quaternion' m -> m

foldMap1 :: Semigroup m => (a -> m) -> Quaternion' a -> m

toNonEmpty :: Quaternion' a -> NonEmpty a

Apply Quaternion'

Methods

(<.>) :: Quaternion' (a -> b) -> Quaternion' a -> Quaternion' b

(.>) :: Quaternion' a -> Quaternion' b -> Quaternion' b

(<.) :: Quaternion' a -> Quaternion' b -> Quaternion' a

Bind Quaternion'

Methods

(>>-) :: Quaternion' a -> (a -> Quaternion' b) -> Quaternion' b

join :: Quaternion' (Quaternion' a) -> Quaternion' a

MonadReader QuaternionBasis' Quaternion'

Methods

ask :: Quaternion' QuaternionBasis'

local :: (QuaternionBasis' -> QuaternionBasis') -> Quaternion' a -> Quaternion' a

reader :: (QuaternionBasis' -> a) -> Quaternion' a

RightModule r s => RightModule r (Quaternion' s)

Methods

(*.) :: Quaternion' s -> r -> Quaternion' s

LeftModule r s => LeftModule r (Quaternion' s)

Methods

(.*) :: r -> Quaternion' s -> Quaternion' s

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

Methods

quadrance :: Quaternion' r -> r

Eq a => Eq (Quaternion' a)

Methods

(==) :: Quaternion' a -> Quaternion' a -> Bool

(/=) :: Quaternion' a -> Quaternion' a -> Bool

Data a => Data (Quaternion' a)

Methods

gfoldl :: (forall d b. Data d => c (d -> b) -> d -> c b) -> (forall g. g -> c g) -> Quaternion' a -> c (Quaternion' a)

gunfold :: (forall b r. Data b => c (b -> r) -> c r) -> (forall r. r -> c r) -> Constr -> c (Quaternion' a)

toConstr :: Quaternion' a -> Constr

dataTypeOf :: Quaternion' a -> DataType

dataCast1 :: Typeable (* -> *) t => (forall d. Data d => c (t d)) -> Maybe (c (Quaternion' a))

dataCast2 :: Typeable (* -> * -> *) t => (forall d e. (Data d, Data e) => c (t d e)) -> Maybe (c (Quaternion' a))

gmapT :: (forall b. Data b => b -> b) -> Quaternion' a -> Quaternion' a

gmapQl :: (r -> r' -> r) -> r -> (forall d. Data d => d -> r') -> Quaternion' a -> r

gmapQr :: (r' -> r -> r) -> r -> (forall d. Data d => d -> r') -> Quaternion' a -> r

gmapQ :: (forall d. Data d => d -> u) -> Quaternion' a -> [u]

gmapQi :: Int -> (forall d. Data d => d -> u) -> Quaternion' a -> u

gmapM :: Monad m => (forall d. Data d => d -> m d) -> Quaternion' a -> m (Quaternion' a)

gmapMp :: MonadPlus m => (forall d. Data d => d -> m d) -> Quaternion' a -> m (Quaternion' a)

gmapMo :: MonadPlus m => (forall d. Data d => d -> m d) -> Quaternion' a -> m (Quaternion' a)

Read a => Read (Quaternion' a)

Methods

readsPrec :: Int -> ReadS (Quaternion' a)

readList :: ReadS [Quaternion' a]

readPrec :: ReadPrec (Quaternion' a)

readListPrec :: ReadPrec [Quaternion' a]

Show a => Show (Quaternion' a)

Methods

showsPrec :: Int -> Quaternion' a -> ShowS

show :: Quaternion' a -> String

showList :: [Quaternion' a] -> ShowS

Idempotent r => Idempotent (Quaternion' r)

Abelian r => Abelian (Quaternion' r)

Partitionable r => Partitionable (Quaternion' r)

Methods

partitionWith :: (Quaternion' r -> Quaternion' r -> r) -> Quaternion' r -> NonEmpty r

Additive r => Additive (Quaternion' r)

Methods

(+) :: Quaternion' r -> Quaternion' r -> Quaternion' r

sinnum1p :: Natural -> Quaternion' r -> Quaternion' r

sumWith1 :: Foldable1 f => (a -> Quaternion' r) -> f a -> Quaternion' r

Monoidal r => Monoidal (Quaternion' r)

Methods

zero :: Quaternion' r

sinnum :: Natural -> Quaternion' r -> Quaternion' r

sumWith :: Foldable f => (a -> Quaternion' r) -> f a -> Quaternion' r

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

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

Methods

(*) :: Quaternion' r -> Quaternion' r -> Quaternion' r

pow1p :: Quaternion' r -> Natural -> Quaternion' r

productWith1 :: Foldable1 f => (a -> Quaternion' r) -> f a -> Quaternion' r

Group r => Group (Quaternion' r)

Methods

(-) :: Quaternion' r -> Quaternion' r -> Quaternion' r

negate :: Quaternion' r -> Quaternion' r

subtract :: Quaternion' r -> Quaternion' r -> Quaternion' r

times :: Integral n => n -> Quaternion' r -> Quaternion' r

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

Methods

one :: Quaternion' r

pow :: Quaternion' r -> Natural -> Quaternion' r

productWith :: Foldable f => (a -> Quaternion' r) -> f a -> Quaternion' r

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

Methods

recip :: Quaternion' r -> Quaternion' r

(/) :: Quaternion' r -> Quaternion' r -> Quaternion' r

(\\) :: Quaternion' r -> Quaternion' r -> Quaternion' r

(^) :: Integral n => Quaternion' r -> n -> Quaternion' r

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

Methods

adjoint :: Quaternion' r -> Quaternion' r

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

Methods

fromNatural :: Natural -> Quaternion' r

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

Methods

fromInteger :: Integer -> Quaternion' r

Rig r => Distinguished (Quaternion' r)

Methods

e :: Quaternion' r

Rig r => Complicated (Quaternion' r)

Methods

i :: Quaternion' r

Rig r => Hamiltonian (Quaternion' r)

Methods

j :: Quaternion' r

k :: Quaternion' r

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

Methods

(*.) :: Quaternion' r -> Quaternion' r -> Quaternion' r

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

Methods

(.*) :: Quaternion' r -> Quaternion' r -> Quaternion' r

type Rep Quaternion'

type Rep Quaternion' = QuaternionBasis'

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

Cayley-Dickson quaternion isomorphism (one way)

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

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