module Numeric.Algebra.Quaternion
( Distinguished(..)
, Complicated(..)
, Hamiltonian(..)
, QuaternionBasis(..)
, Quaternion(..)
, complicate
, vectorPart
, scalarPart
) where
import Control.Applicative
import Control.Monad.Reader.Class
import Data.Ix hiding (index)
import Data.Key
import Data.Data
import Data.Distributive
import Data.Functor.Bind
import Data.Functor.Representable
import Data.Functor.Representable.Trie
import Data.Foldable
import Data.Traversable
import Data.Semigroup
import Data.Semigroup.Traversable
import Data.Semigroup.Foldable
import Numeric.Algebra
import Numeric.Algebra.Distinguished.Class
import Numeric.Algebra.Complex.Class
import Numeric.Algebra.Quaternion.Class
import Prelude hiding ((),(+),(*),negate,subtract, fromInteger)
instance Distinguished QuaternionBasis where
e = E
instance Complicated QuaternionBasis where
i = I
instance Hamiltonian QuaternionBasis where
j = J
k = K
instance Rig r => Distinguished (Quaternion r) where
e = Quaternion one zero zero zero
instance Rig r => Complicated (Quaternion r) where
i = Quaternion zero one zero zero
instance Rig r => Hamiltonian (Quaternion r) where
j = Quaternion zero zero one zero
k = Quaternion one zero zero one
instance Rig r => Distinguished (QuaternionBasis :->: r) where
e = Trie e
instance Rig r => Complicated (QuaternionBasis :->: r) where
i = Trie i
instance Rig r => Hamiltonian (QuaternionBasis :->: r) where
j = Trie j
k = Trie k
instance Rig r => Distinguished (QuaternionBasis -> r) where
e E = one
e _ = zero
instance Rig r => Complicated (QuaternionBasis -> r) where
i I = one
i _ = zero
instance Rig r => Hamiltonian (QuaternionBasis -> r) where
j J = one
j _ = zero
k K = one
k _ = zero
data QuaternionBasis = E | I | J | K deriving (Eq,Ord,Enum,Read,Show,Bounded,Ix,Data,Typeable)
data Quaternion a = Quaternion a a a a deriving (Eq,Show,Read,Data,Typeable)
type instance Key Quaternion = QuaternionBasis
instance Representable Quaternion where
tabulate f = Quaternion (f E) (f I) (f J) (f K)
instance Indexable Quaternion where
index (Quaternion a _ _ _) E = a
index (Quaternion _ b _ _) I = b
index (Quaternion _ _ c _) J = c
index (Quaternion _ _ _ d) K = d
instance Lookup Quaternion where
lookup = lookupDefault
instance Adjustable Quaternion where
adjust f E (Quaternion a b c d) = Quaternion (f a) b c d
adjust f I (Quaternion a b c d) = Quaternion a (f b) c d
adjust f J (Quaternion a b c d) = Quaternion a b (f c) d
adjust f K (Quaternion a b c d) = Quaternion a b c (f d)
instance Distributive Quaternion where
distribute = distributeRep
instance Functor Quaternion where
fmap = fmapRep
instance Zip Quaternion where
zipWith f (Quaternion a1 b1 c1 d1) (Quaternion a2 b2 c2 d2) =
Quaternion (f a1 a2) (f b1 b2) (f c1 c2) (f d1 d2)
instance ZipWithKey Quaternion where
zipWithKey f (Quaternion a1 b1 c1 d1) (Quaternion a2 b2 c2 d2) =
Quaternion (f E a1 a2) (f I b1 b2) (f J c1 c2) (f K d1 d2)
instance Keyed Quaternion where
mapWithKey = mapWithKeyRep
instance Apply Quaternion where
(<.>) = apRep
instance Applicative Quaternion where
pure = pureRep
(<*>) = apRep
instance Bind Quaternion where
(>>-) = bindRep
instance Monad Quaternion where
return = pureRep
(>>=) = bindRep
instance MonadReader QuaternionBasis Quaternion where
ask = askRep
local = localRep
instance Foldable Quaternion where
foldMap f (Quaternion a b c d) =
f a `mappend` f b `mappend` f c `mappend` f d
instance FoldableWithKey Quaternion where
foldMapWithKey f (Quaternion a b c d) =
f E a `mappend` f I b `mappend` f J c `mappend` f K d
instance Traversable Quaternion where
traverse f (Quaternion a b c d) =
Quaternion <$> f a <*> f b <*> f c <*> f d
instance TraversableWithKey Quaternion where
traverseWithKey f (Quaternion a b c d) =
Quaternion <$> f E a <*> f I b <*> f J c <*> f K d
instance Foldable1 Quaternion where
foldMap1 f (Quaternion a b c d) =
f a <> f b <> f c <> f d
instance FoldableWithKey1 Quaternion where
foldMapWithKey1 f (Quaternion a b c d) =
f E a <> f I b <> f J c <> f K d
instance Traversable1 Quaternion where
traverse1 f (Quaternion a b c d) =
Quaternion <$> f a <.> f b <.> f c <.> f d
instance TraversableWithKey1 Quaternion where
traverseWithKey1 f (Quaternion a b c d) =
Quaternion <$> f E a <.> f I b <.> f J c <.> f K d
instance HasTrie QuaternionBasis where
type BaseTrie QuaternionBasis = Quaternion
embedKey = id
projectKey = id
instance Additive r => Additive (Quaternion r) where
(+) = addRep
sinnum1p = sinnum1pRep
instance LeftModule r s => LeftModule r (Quaternion s) where
r .* Quaternion a b c d =
Quaternion (r .* a) (r .* b) (r .* c) (r .* d)
instance RightModule r s => RightModule r (Quaternion s) where
Quaternion a b c d *. r =
Quaternion (a *. r) (b *. r) (c *. r) (d *. r)
instance Monoidal r => Monoidal (Quaternion r) where
zero = zeroRep
sinnum = sinnumRep
instance Group r => Group (Quaternion r) where
() = minusRep
negate = negateRep
subtract = subtractRep
times = timesRep
instance Abelian r => Abelian (Quaternion r)
instance Idempotent r => Idempotent (Quaternion r)
instance Partitionable r => Partitionable (Quaternion r) where
partitionWith f (Quaternion a b c d) = id =<<
partitionWith (\a1 a2 -> id =<<
partitionWith (\b1 b2 -> id =<<
partitionWith (\c1 c2 ->
partitionWith (\d1 d2 -> f (Quaternion a1 b1 c1 d1)
(Quaternion a2 b2 c2 d2)
) d) c) b) a
instance (TriviallyInvolutive r, Rng r) => Algebra r QuaternionBasis where
mult f = f' where
fe = f E E (f I I + f J J + f K K)
fi = f E I + f I E + f J K f K J
fj = f E J + f J E + f K I f I K
fk = f E K + f K E + f I J f J I
f' E = fe
f' I = fi
f' J = fj
f' K = fk
instance (TriviallyInvolutive r, Rng r) => UnitalAlgebra r QuaternionBasis where
unit x E = x
unit _ _ = zero
instance (TriviallyInvolutive r, Rng r) => Coalgebra r QuaternionBasis where
comult f = f' where
fe = f E
fi = f I
fj = f J
fk = f K
f' E E = fe
f' I I = fi
f' J J = fj
f' K K = fk
f' _ _ = zero
instance (TriviallyInvolutive r, Rng r) => CounitalCoalgebra r QuaternionBasis where
counit f = f E + f I + f J + f K
instance (TriviallyInvolutive r, Rng r) => Bialgebra r QuaternionBasis
instance (TriviallyInvolutive r, InvolutiveSemiring r, Rng r) => InvolutiveAlgebra r QuaternionBasis where
inv f E = f E
inv f b = negate (f b)
instance (TriviallyInvolutive r, InvolutiveSemiring r, Rng r) => InvolutiveCoalgebra r QuaternionBasis where
coinv = inv
instance (TriviallyInvolutive r, InvolutiveSemiring r, Rng r) => HopfAlgebra r QuaternionBasis where
antipode = inv
instance (TriviallyInvolutive r, Rng r) => Multiplicative (Quaternion r) where
(*) = mulRep
instance (TriviallyInvolutive r, Rng r) => Semiring (Quaternion r)
instance (TriviallyInvolutive r, Ring r) => Unital (Quaternion r) where
one = oneRep
instance (TriviallyInvolutive r, Ring r) => Rig (Quaternion r) where
fromNatural n = Quaternion (fromNatural n) zero zero zero
instance (TriviallyInvolutive r, Ring r) => Ring (Quaternion r) where
fromInteger n = Quaternion (fromInteger n) zero zero zero
instance ( TriviallyInvolutive r, Rng r) => LeftModule (Quaternion r) (Quaternion r) where
(.*) = (*)
instance (TriviallyInvolutive r, Rng r) => RightModule (Quaternion r) (Quaternion r) where
(*.) = (*)
instance (TriviallyInvolutive r, Rng r) => InvolutiveMultiplication (Quaternion r) where
adjoint (Quaternion a b c d) = Quaternion a (negate b) (negate c) (negate d)
complicate :: Complicated c => QuaternionBasis -> (c,c)
complicate E = (e, e)
complicate I = (i, e)
complicate J = (e, i)
complicate K = (i, i)
scalarPart :: (Representable f, Key f ~ QuaternionBasis) => f r -> r
scalarPart f = index f E
vectorPart :: (Representable f, Key f ~ QuaternionBasis) => f r -> (r,r,r)
vectorPart f = (index f I, index f J, index f K)
instance (TriviallyInvolutive r, Rng r) => Quadrance r (Quaternion r) where
quadrance n = scalarPart (adjoint n * n)
instance (TriviallyInvolutive r, Ring r, Division r) => Division (Quaternion r) where
recip q@(Quaternion a b c d) = Quaternion (qq \\ a) (qq \\ b) (qq \\ c) (qq \\ d)
where qq = quadrance q