Numeric.Algebra.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

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

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