Numeric.Coalgebra.Dual

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

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'