Safe Haskell	Safe
Language	Haskell98

Numeric.Coalgebra.Quaternion

Synopsis

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 # liftF2 :: (a -> b -> c) -> Quaternion' a -> Quaternion' b -> Quaternion' c #
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 #