Safe Haskell	Safe
Language	Haskell98

Numeric.Coalgebra.Dual

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 t => Infinitesimal t where #

Minimal complete definition

d

Methods

d :: t #

Instances

Infinitesimal DualBasis' #
Methods d :: DualBasis' #
Infinitesimal DualBasis #
Methods d :: DualBasis #
Rig r => Infinitesimal (Dual' r) #
Methods d :: Dual' r #
Rig r => Infinitesimal (Dual r) #
Methods d :: Dual r #
Rig r => Infinitesimal (DualBasis' -> r) #
Methods d :: DualBasis' -> r #
Rig r => Infinitesimal (DualBasis -> r) #
Methods d :: DualBasis -> r #
Infinitesimal a => Infinitesimal (Covector r a) #
Methods d :: Covector r a #

data DualBasis' #

dual number basis, D^2 = 0. D /= 0.

Constructors

E
D

Instances

Bounded DualBasis' #
Methods minBound :: DualBasis' # maxBound :: DualBasis' #
Enum DualBasis' #
Methods succ :: DualBasis' -> DualBasis' # pred :: DualBasis' -> DualBasis' # toEnum :: Int -> DualBasis' # fromEnum :: DualBasis' -> Int # enumFrom :: DualBasis' -> [DualBasis'] # enumFromThen :: DualBasis' -> DualBasis' -> [DualBasis'] # enumFromTo :: DualBasis' -> DualBasis' -> [DualBasis'] # enumFromThenTo :: DualBasis' -> DualBasis' -> DualBasis' -> [DualBasis'] #
Eq DualBasis' #
Methods (==) :: DualBasis' -> DualBasis' -> Bool # (/=) :: DualBasis' -> DualBasis' -> Bool #
Data DualBasis' #
Methods gfoldl :: (forall d b. Data d => c (d -> b) -> d -> c b) -> (forall g. g -> c g) -> DualBasis' -> c DualBasis' # gunfold :: (forall b r. Data b => c (b -> r) -> c r) -> (forall r. r -> c r) -> Constr -> c DualBasis' # toConstr :: DualBasis' -> Constr # dataTypeOf :: DualBasis' -> DataType # dataCast1 :: Typeable (* -> ) t => (forall d. Data d => c (t d)) -> Maybe (c DualBasis') # dataCast2 :: Typeable ( -> * -> *) t => (forall d e. (Data d, Data e) => c (t d e)) -> Maybe (c DualBasis') # gmapT :: (forall b. Data b => b -> b) -> DualBasis' -> DualBasis' # gmapQl :: (r -> r' -> r) -> r -> (forall d. Data d => d -> r') -> DualBasis' -> r # gmapQr :: (r' -> r -> r) -> r -> (forall d. Data d => d -> r') -> DualBasis' -> r # gmapQ :: (forall d. Data d => d -> u) -> DualBasis' -> [u] # gmapQi :: Int -> (forall d. Data d => d -> u) -> DualBasis' -> u # gmapM :: Monad m => (forall d. Data d => d -> m d) -> DualBasis' -> m DualBasis' # gmapMp :: MonadPlus m => (forall d. Data d => d -> m d) -> DualBasis' -> m DualBasis' # gmapMo :: MonadPlus m => (forall d. Data d => d -> m d) -> DualBasis' -> m DualBasis' #
Ord DualBasis' #
Methods compare :: DualBasis' -> DualBasis' -> Ordering # (<) :: DualBasis' -> DualBasis' -> Bool # (<=) :: DualBasis' -> DualBasis' -> Bool # (>) :: DualBasis' -> DualBasis' -> Bool # (>=) :: DualBasis' -> DualBasis' -> Bool # max :: DualBasis' -> DualBasis' -> DualBasis' # min :: DualBasis' -> DualBasis' -> DualBasis' #
Read DualBasis' #
Methods readsPrec :: Int -> ReadS DualBasis' # readList :: ReadS [DualBasis'] # readPrec :: ReadPrec DualBasis' # readListPrec :: ReadPrec [DualBasis'] #
Show DualBasis' #
Methods showsPrec :: Int -> DualBasis' -> ShowS # show :: DualBasis' -> String # showList :: [DualBasis'] -> ShowS #
Ix DualBasis' #
Methods range :: (DualBasis', DualBasis') -> [DualBasis'] # index :: (DualBasis', DualBasis') -> DualBasis' -> Int # unsafeIndex :: (DualBasis', DualBasis') -> DualBasis' -> Int inRange :: (DualBasis', DualBasis') -> DualBasis' -> Bool # rangeSize :: (DualBasis', DualBasis') -> Int # unsafeRangeSize :: (DualBasis', DualBasis') -> Int
Distinguished DualBasis' #
Methods e :: DualBasis' #
Infinitesimal DualBasis' #
Methods d :: DualBasis' #
MonadReader DualBasis' Dual' #
Methods ask :: Dual' DualBasis' # local :: (DualBasis' -> DualBasis') -> Dual' a -> Dual' a # reader :: (DualBasis' -> a) -> Dual' a #
Rng k => Coalgebra k DualBasis' #
Methods comult :: (DualBasis' -> k) -> DualBasis' -> DualBasis' -> k #
Semiring k => Algebra k DualBasis' #
Methods mult :: (DualBasis' -> DualBasis' -> k) -> DualBasis' -> k #
Rng k => Bialgebra k DualBasis' #

Rng k => CounitalCoalgebra k DualBasis' #
Methods counit :: (DualBasis' -> k) -> k #
Semiring k => UnitalAlgebra k DualBasis' #
Methods unit :: k -> DualBasis' -> k #
(InvolutiveSemiring k, Rng k) => HopfAlgebra k DualBasis' #
Methods antipode :: (DualBasis' -> k) -> DualBasis' -> k #
(InvolutiveSemiring k, Rng k) => InvolutiveCoalgebra k DualBasis' #
Methods coinv :: (DualBasis' -> k) -> DualBasis' -> k #
(InvolutiveSemiring k, Rng k) => InvolutiveAlgebra k DualBasis' #
Methods inv :: (DualBasis' -> k) -> DualBasis' -> k #
Rig r => Distinguished (DualBasis' -> r) #
Methods e :: DualBasis' -> r #
Rig r => Infinitesimal (DualBasis' -> r) #
Methods d :: DualBasis' -> r #

data Dual' a #

Constructors

Dual' a a

Instances

Monad Dual' #
Methods (>>=) :: Dual' a -> (a -> Dual' b) -> Dual' b # (>>) :: Dual' a -> Dual' b -> Dual' b # return :: a -> Dual' a # fail :: String -> Dual' a #
Functor Dual' #
Methods fmap :: (a -> b) -> Dual' a -> Dual' b # (<$) :: a -> Dual' b -> Dual' a #
Applicative Dual' #
Methods pure :: a -> Dual' a # (<>) :: Dual' (a -> b) -> Dual' a -> Dual' b # liftA2 :: (a -> b -> c) -> Dual' a -> Dual' b -> Dual' c # (>) :: Dual' a -> Dual' b -> Dual' b # (<*) :: Dual' a -> Dual' b -> Dual' a #
Foldable Dual' #
Methods fold :: Monoid m => Dual' m -> m # foldMap :: Monoid m => (a -> m) -> Dual' a -> m # foldr :: (a -> b -> b) -> b -> Dual' a -> b # foldr' :: (a -> b -> b) -> b -> Dual' a -> b # foldl :: (b -> a -> b) -> b -> Dual' a -> b # foldl' :: (b -> a -> b) -> b -> Dual' a -> b # foldr1 :: (a -> a -> a) -> Dual' a -> a # foldl1 :: (a -> a -> a) -> Dual' a -> a # toList :: Dual' a -> [a] # null :: Dual' a -> Bool # length :: Dual' a -> Int # elem :: Eq a => a -> Dual' a -> Bool # maximum :: Ord a => Dual' a -> a # minimum :: Ord a => Dual' a -> a # sum :: Num a => Dual' a -> a # product :: Num a => Dual' a -> a #
Traversable Dual' #
Methods traverse :: Applicative f => (a -> f b) -> Dual' a -> f (Dual' b) # sequenceA :: Applicative f => Dual' (f a) -> f (Dual' a) # mapM :: Monad m => (a -> m b) -> Dual' a -> m (Dual' b) # sequence :: Monad m => Dual' (m a) -> m (Dual' a) #
Distributive Dual' #
Methods distribute :: Functor f => f (Dual' a) -> Dual' (f a) # collect :: Functor f => (a -> Dual' b) -> f a -> Dual' (f b) # distributeM :: Monad m => m (Dual' a) -> Dual' (m a) # collectM :: Monad m => (a -> Dual' b) -> m a -> Dual' (m b) #
Representable Dual' #
Associated Types type Rep (Dual' :: * -> ) :: # Methods tabulate :: (Rep Dual' -> a) -> Dual' a # index :: Dual' a -> Rep Dual' -> a #
Traversable1 Dual' #
Methods traverse1 :: Apply f => (a -> f b) -> Dual' a -> f (Dual' b) # sequence1 :: Apply f => Dual' (f b) -> f (Dual' b) #
Foldable1 Dual' #
Methods fold1 :: Semigroup m => Dual' m -> m # foldMap1 :: Semigroup m => (a -> m) -> Dual' a -> m # toNonEmpty :: Dual' a -> NonEmpty a #
Apply Dual' #
Methods (<.>) :: Dual' (a -> b) -> Dual' a -> Dual' b # (.>) :: Dual' a -> Dual' b -> Dual' b # (<.) :: Dual' a -> Dual' b -> Dual' a # liftF2 :: (a -> b -> c) -> Dual' a -> Dual' b -> Dual' c #
Bind Dual' #
Methods (>>-) :: Dual' a -> (a -> Dual' b) -> Dual' b # join :: Dual' (Dual' a) -> Dual' a #
MonadReader DualBasis' Dual' #
Methods ask :: Dual' DualBasis' # local :: (DualBasis' -> DualBasis') -> Dual' a -> Dual' a # reader :: (DualBasis' -> a) -> Dual' a #
RightModule r s => RightModule r (Dual' s) #
Methods (*.) :: Dual' s -> r -> Dual' s #
LeftModule r s => LeftModule r (Dual' s) #
Methods (.*) :: r -> Dual' s -> Dual' s #
(Commutative r, Rng r, InvolutiveSemiring r) => Quadrance r (Dual' r) #
Methods quadrance :: Dual' r -> r #
Eq a => Eq (Dual' a) #
Methods (==) :: Dual' a -> Dual' a -> Bool # (/=) :: Dual' a -> Dual' a -> Bool #
Data a => Data (Dual' a) #
Methods gfoldl :: (forall d b. Data d => c (d -> b) -> d -> c b) -> (forall g. g -> c g) -> Dual' a -> c (Dual' a) # gunfold :: (forall b r. Data b => c (b -> r) -> c r) -> (forall r. r -> c r) -> Constr -> c (Dual' a) # toConstr :: Dual' a -> Constr # dataTypeOf :: Dual' a -> DataType # dataCast1 :: Typeable (* -> ) t => (forall d. Data d => c (t d)) -> Maybe (c (Dual' a)) # dataCast2 :: Typeable ( -> * -> *) t => (forall d e. (Data d, Data e) => c (t d e)) -> Maybe (c (Dual' a)) # gmapT :: (forall b. Data b => b -> b) -> Dual' a -> Dual' a # gmapQl :: (r -> r' -> r) -> r -> (forall d. Data d => d -> r') -> Dual' a -> r # gmapQr :: (r' -> r -> r) -> r -> (forall d. Data d => d -> r') -> Dual' a -> r # gmapQ :: (forall d. Data d => d -> u) -> Dual' a -> [u] # gmapQi :: Int -> (forall d. Data d => d -> u) -> Dual' a -> u # gmapM :: Monad m => (forall d. Data d => d -> m d) -> Dual' a -> m (Dual' a) # gmapMp :: MonadPlus m => (forall d. Data d => d -> m d) -> Dual' a -> m (Dual' a) # gmapMo :: MonadPlus m => (forall d. Data d => d -> m d) -> Dual' a -> m (Dual' a) #
Read a => Read (Dual' a) #
Methods readsPrec :: Int -> ReadS (Dual' a) # readList :: ReadS [Dual' a] # readPrec :: ReadPrec (Dual' a) # readListPrec :: ReadPrec [Dual' a] #
Show a => Show (Dual' a) #
Methods showsPrec :: Int -> Dual' a -> ShowS # show :: Dual' a -> String # showList :: [Dual' a] -> ShowS #
Idempotent r => Idempotent (Dual' r) #

Abelian r => Abelian (Dual' r) #

Partitionable r => Partitionable (Dual' r) #
Methods partitionWith :: (Dual' r -> Dual' r -> r) -> Dual' r -> NonEmpty r #
Additive r => Additive (Dual' r) #
Methods (+) :: Dual' r -> Dual' r -> Dual' r # sinnum1p :: Natural -> Dual' r -> Dual' r # sumWith1 :: Foldable1 f => (a -> Dual' r) -> f a -> Dual' r #
Monoidal r => Monoidal (Dual' r) #
Methods zero :: Dual' r # sinnum :: Natural -> Dual' r -> Dual' r # sumWith :: Foldable f => (a -> Dual' r) -> f a -> Dual' r #
(Commutative r, Rng r) => Semiring (Dual' r) #

(Commutative r, Rng r) => Multiplicative (Dual' r) #
Methods (*) :: Dual' r -> Dual' r -> Dual' r # pow1p :: Dual' r -> Natural -> Dual' r # productWith1 :: Foldable1 f => (a -> Dual' r) -> f a -> Dual' r #
Group r => Group (Dual' r) #
Methods (-) :: Dual' r -> Dual' r -> Dual' r # negate :: Dual' r -> Dual' r # subtract :: Dual' r -> Dual' r -> Dual' r # times :: Integral n => n -> Dual' r -> Dual' r #
(Commutative r, Ring r) => Unital (Dual' r) #
Methods one :: Dual' r # pow :: Dual' r -> Natural -> Dual' r # productWith :: Foldable f => (a -> Dual' r) -> f a -> Dual' r #
(Commutative r, InvolutiveSemiring r, DivisionRing r) => Division (Dual' r) #
Methods recip :: Dual' r -> Dual' r # (/) :: Dual' r -> Dual' r -> Dual' r # (\\) :: Dual' r -> Dual' r -> Dual' r # (^) :: Integral n => Dual' r -> n -> Dual' r #
(TriviallyInvolutive r, Rng r) => Commutative (Dual' r) #

(Commutative r, Rng r, InvolutiveSemiring r) => InvolutiveSemiring (Dual' r) #

(Commutative r, Rng r, InvolutiveSemiring r) => InvolutiveMultiplication (Dual' r) #
Methods adjoint :: Dual' r -> Dual' r #
(Commutative r, Ring r) => Rig (Dual' r) #
Methods fromNatural :: Natural -> Dual' r #
(Commutative r, Ring r) => Ring (Dual' r) #
Methods fromInteger :: Integer -> Dual' r #
Rig r => Distinguished (Dual' r) #
Methods e :: Dual' r #
Rig r => Infinitesimal (Dual' r) #
Methods d :: Dual' r #
(Commutative r, Rng r) => RightModule (Dual' r) (Dual' r) #
Methods (*.) :: Dual' r -> Dual' r -> Dual' r #
(Commutative r, Rng r) => LeftModule (Dual' r) (Dual' r) #
Methods (.*) :: Dual' r -> Dual' r -> Dual' r #
type Rep Dual' #
type Rep Dual' = DualBasis'