Numeric.Algebra.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

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

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