Data.Number.Dual

newtype Dual a b

Constructors

• D (a -> b /\ (a -> b))

Instances

• Monoidal Dual
• Semigroupoid Dual
• Category Dual
• Cartesian Dual
• (Ring s) => RingCat Dual s
• (DivisionRing s, EuclideanRing s, RingCat Dual s) => DivisionRingCat Dual s
• NumCat Dual

class Monoidal :: (Type -> Type -> Type) -> Constraintclass Monoidal k  where

Members

• cross :: forall a b c d. k a c -> k b d -> k (a /\ b) (c /\ d)

Instances

• Monoidal Function
• Monoidal Dual

linearPropagation :: forall a b. (a -> b) -> (a -> b) -> Dual a b

class Cartesian :: (Type -> Type -> Type) -> Constraintclass (Category k) <= Cartesian k  where

Members

• exl :: forall a b. k (a /\ b) a
• exr :: forall a b. k (a /\ b) b
• dup :: forall a. k a (a /\ a)

Instances

• Cartesian Function
• Cartesian Dual

class Space a  where

Members

• scale :: a -> (a -> a)
• accum :: (a /\ a) -> a

Instances

• (Semiring s) => Space s

class RingCat :: (Type -> Type -> Type) -> Type -> Constraintclass RingCat k s  where

Members

• negate :: k s s
• add :: k (s /\ s) s
• mul :: k (s /\ s) s

Instances

• (Ring s) => RingCat Function s
• (Ring s) => RingCat Dual s

class DivisionRingCat :: (Type -> Type -> Type) -> Type -> Constraintclass DivisionRingCat k s  where

Members

• recip :: k s s
• div :: k (s /\ s) s

Instances

• (DivisionRing s, EuclideanRing s) => DivisionRingCat Function s
• (DivisionRing s, EuclideanRing s, RingCat Dual s) => DivisionRingCat Dual s

recipImpl :: forall b. EuclideanRing b => DivisionRing b => Dual b b

expImpl :: Dual Number Number

lnImpl :: Dual Number Number

sinImpl :: Dual Number Number

cosImpl :: Dual Number Number

sqrtImpl :: Dual Number Number

absImpl :: Dual Number Number

atanImpl :: Dual Number Number

coshImpl :: Dual Number Number

sinhImpl :: Dual Number Number

class NumCat :: (Type -> Type -> Type) -> Constraintclass NumCat k  where

Members

• exp :: k Number Number
• sqrt :: k Number Number
• sin :: k Number Number
• cos :: k Number Number
• tan :: k Number Number
• asin :: k Number Number
• acos :: k Number Number
• atan :: k Number Number
• sinh :: k Number Number
• cosh :: k Number Number
• tanh :: k Number Number
• asinh :: k Number Number
• acosh :: k Number Number
• atanh :: k Number Number
• ln :: k Number Number
• abs :: k Number Number
• sign :: k Number Number
• atan2 :: k (Number /\ Number) Number
• pow :: k (Number /\ Number) Number
• min :: k (Number /\ Number) Number
• max :: k (Number /\ Number) Number

Instances

• NumCat Dual

pair :: forall a c d k. Cartesian k => Monoidal k => k a c -> k a d -> k a (c /\ d)

Generalized pairing operator such that exl >>> f .:. exr >>> g is equivalent to cross f g

Operator alias for Data.Number.Dual.pair (right-associative / precedence 6)

cst :: forall a s. Semiring s => s -> Dual a s

axes :: forall @n a. Axes n a => a

class Axes :: Int -> Type -> Constraintclass Axes n a | n -> a where

Members

• axesImpl :: Proxy n -> a

Instances

• Axes 1 Number
• Axes 2 (Tuple (Tuple Number Number) (Tuple Number Number))
• (Axes n (Tuple h t), Add n 1 n1, Fmapable h (Tuple h t) (Tuple Number h) (Tuple h' t'), Fmapable Number h Number h) => Axes n1 (Tuple (Tuple Number h) (Tuple h' t'))

class Transposable a b | a -> b where

Members

• transpose :: Dual a b

Instances

• (Transposable (Tuple (Tuple b x) (Tuple d y)) e) => Transposable (Tuple (Tuple a (Tuple b x)) (Tuple c (Tuple d y))) (Tuple (Tuple a c) e)
• Transposable (Tuple (Tuple a b) (Tuple c d)) (Tuple (Tuple a c) (Tuple b d))
• Transposable a a

class Cumulative c a  where

Members

• cumulate :: Dual c a

Instances

• (Ring a, Cumulative c a) => Cumulative (Tuple a c) a
• (Ring a) => Cumulative (Tuple a a) a
• Cumulative a a

class Fmapable a c b k | b c -> k where

Members

• fmap :: Dual a b -> Dual c k

Instances

• Fmapable a a b b
• (Fmapable a c b k) => Fmapable a (Tuple a c) b (Tuple b k)

minimize :: forall a c u t v axs. Fmapable a axs Number v => Transposable (a /\ v) c => Fmapable (Number /\ Number) c Number a => Transposable (v /\ v) u => Fmapable (Number /\ Number) u Number t => Cumulative t Number => axs -> Dual a Number -> Number -> Number -> a -> a

distance2 :: forall a b c t. Transposable a b => Fmapable (Number /\ Number) b (Number /\ Number) c => Fmapable (Number /\ Number) c Number t => Cumulative t Number => Dual a Number

norm2 :: forall a s t u v. Fmapable Number a Number s => Transposable (a /\ s) t => Fmapable (Number /\ Number) t (Number /\ Number) u => Fmapable (Number /\ Number) u Number v => Cumulative v Number => Dual a Number

distance :: forall a b c t. Transposable a b => Fmapable (Number /\ Number) b (Number /\ Number) c => Fmapable (Number /\ Number) c Number (Number /\ t) => Cumulative t Number => Dual a Number