Safe Haskell	Safe
Language	Haskell98

Numeric.Algebra.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 #
Rng 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 #
Rng 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