Safe Haskell	Safe
Language	Haskell98

Numeric.Algebra.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 #	the trivial diagonal coalgebra
Methods comult :: (QuaternionBasis -> r) -> QuaternionBasis -> QuaternionBasis -> r #
(TriviallyInvolutive r, Rng r) => Algebra r QuaternionBasis #	the quaternion 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, Rng 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, Rng r) => Semiring (Quaternion r) #

(TriviallyInvolutive r, Rng 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 #