Numeric.AD.Mode.Reverse.Double

Reverse-mode automatic differentiation using Wengert lists and Data.Reflection

This version is specialized to Double enabling the entire structure to be unboxed.

Documentation

data ReverseDouble s

Instances

Reifies s Tape => Enum (ReverseDouble s)

Defined in Numeric.AD.Internal.Reverse.Double

Methods

succ :: ReverseDouble s -> ReverseDouble s

pred :: ReverseDouble s -> ReverseDouble s

toEnum :: Int -> ReverseDouble s

fromEnum :: ReverseDouble s -> Int

enumFrom :: ReverseDouble s -> [ReverseDouble s]

enumFromThen :: ReverseDouble s -> ReverseDouble s -> [ReverseDouble s]

enumFromTo :: ReverseDouble s -> ReverseDouble s -> [ReverseDouble s]

enumFromThenTo :: ReverseDouble s -> ReverseDouble s -> ReverseDouble s -> [ReverseDouble s]

Reifies s Tape => Eq (ReverseDouble s)

Defined in Numeric.AD.Internal.Reverse.Double

Methods

(==) :: ReverseDouble s -> ReverseDouble s -> Bool

(/=) :: ReverseDouble s -> ReverseDouble s -> Bool

Reifies s Tape => Floating (ReverseDouble s)

Defined in Numeric.AD.Internal.Reverse.Double

Methods

pi :: ReverseDouble s

exp :: ReverseDouble s -> ReverseDouble s

log :: ReverseDouble s -> ReverseDouble s

sqrt :: ReverseDouble s -> ReverseDouble s

(**) :: ReverseDouble s -> ReverseDouble s -> ReverseDouble s

logBase :: ReverseDouble s -> ReverseDouble s -> ReverseDouble s

sin :: ReverseDouble s -> ReverseDouble s

cos :: ReverseDouble s -> ReverseDouble s

tan :: ReverseDouble s -> ReverseDouble s

asin :: ReverseDouble s -> ReverseDouble s

acos :: ReverseDouble s -> ReverseDouble s

atan :: ReverseDouble s -> ReverseDouble s

sinh :: ReverseDouble s -> ReverseDouble s

cosh :: ReverseDouble s -> ReverseDouble s

tanh :: ReverseDouble s -> ReverseDouble s

asinh :: ReverseDouble s -> ReverseDouble s

acosh :: ReverseDouble s -> ReverseDouble s

atanh :: ReverseDouble s -> ReverseDouble s

log1p :: ReverseDouble s -> ReverseDouble s

expm1 :: ReverseDouble s -> ReverseDouble s

log1pexp :: ReverseDouble s -> ReverseDouble s

log1mexp :: ReverseDouble s -> ReverseDouble s

Reifies s Tape => Fractional (ReverseDouble s)

Defined in Numeric.AD.Internal.Reverse.Double

Methods

(/) :: ReverseDouble s -> ReverseDouble s -> ReverseDouble s

recip :: ReverseDouble s -> ReverseDouble s

fromRational :: Rational -> ReverseDouble s

Reifies s Tape => Num (ReverseDouble s)

Defined in Numeric.AD.Internal.Reverse.Double

Methods

(+) :: ReverseDouble s -> ReverseDouble s -> ReverseDouble s

(-) :: ReverseDouble s -> ReverseDouble s -> ReverseDouble s

(*) :: ReverseDouble s -> ReverseDouble s -> ReverseDouble s

negate :: ReverseDouble s -> ReverseDouble s

abs :: ReverseDouble s -> ReverseDouble s

signum :: ReverseDouble s -> ReverseDouble s

fromInteger :: Integer -> ReverseDouble s

Reifies s Tape => Ord (ReverseDouble s)

Defined in Numeric.AD.Internal.Reverse.Double

Methods

compare :: ReverseDouble s -> ReverseDouble s -> Ordering

(<) :: ReverseDouble s -> ReverseDouble s -> Bool

(<=) :: ReverseDouble s -> ReverseDouble s -> Bool

(>) :: ReverseDouble s -> ReverseDouble s -> Bool

(>=) :: ReverseDouble s -> ReverseDouble s -> Bool

max :: ReverseDouble s -> ReverseDouble s -> ReverseDouble s

min :: ReverseDouble s -> ReverseDouble s -> ReverseDouble s

Reifies s Tape => Real (ReverseDouble s)

Defined in Numeric.AD.Internal.Reverse.Double

Methods

toRational :: ReverseDouble s -> Rational

Reifies s Tape => RealFloat (ReverseDouble s)

Defined in Numeric.AD.Internal.Reverse.Double

Methods

floatRadix :: ReverseDouble s -> Integer

floatDigits :: ReverseDouble s -> Int

floatRange :: ReverseDouble s -> (Int, Int)

decodeFloat :: ReverseDouble s -> (Integer, Int)

encodeFloat :: Integer -> Int -> ReverseDouble s

exponent :: ReverseDouble s -> Int

significand :: ReverseDouble s -> ReverseDouble s

scaleFloat :: Int -> ReverseDouble s -> ReverseDouble s

isNaN :: ReverseDouble s -> Bool

isInfinite :: ReverseDouble s -> Bool

isDenormalized :: ReverseDouble s -> Bool

isNegativeZero :: ReverseDouble s -> Bool

isIEEE :: ReverseDouble s -> Bool

atan2 :: ReverseDouble s -> ReverseDouble s -> ReverseDouble s

Reifies s Tape => RealFrac (ReverseDouble s)

Defined in Numeric.AD.Internal.Reverse.Double

Methods

properFraction :: Integral b => ReverseDouble s -> (b, ReverseDouble s)

truncate :: Integral b => ReverseDouble s -> b

round :: Integral b => ReverseDouble s -> b

ceiling :: Integral b => ReverseDouble s -> b

floor :: Integral b => ReverseDouble s -> b

Show (ReverseDouble s)

Defined in Numeric.AD.Internal.Reverse.Double

Methods

showsPrec :: Int -> ReverseDouble s -> ShowS

show :: ReverseDouble s -> String

showList :: [ReverseDouble s] -> ShowS

Reifies s Tape => Erf (ReverseDouble s)

Defined in Numeric.AD.Internal.Reverse.Double

Methods

erf :: ReverseDouble s -> ReverseDouble s

erfc :: ReverseDouble s -> ReverseDouble s

erfcx :: ReverseDouble s -> ReverseDouble s

normcdf :: ReverseDouble s -> ReverseDouble s

Reifies s Tape => InvErf (ReverseDouble s)

Defined in Numeric.AD.Internal.Reverse.Double

Methods

inverf :: ReverseDouble s -> ReverseDouble s

inverfc :: ReverseDouble s -> ReverseDouble s

invnormcdf :: ReverseDouble s -> ReverseDouble s

Reifies s Tape => Mode (ReverseDouble s)

Defined in Numeric.AD.Internal.Reverse.Double

Associated Types

type Scalar (ReverseDouble s)

Methods

isKnownConstant :: ReverseDouble s -> Bool

asKnownConstant :: ReverseDouble s -> Maybe (Scalar (ReverseDouble s))

isKnownZero :: ReverseDouble s -> Bool

auto :: Scalar (ReverseDouble s) -> ReverseDouble s

(*^) :: Scalar (ReverseDouble s) -> ReverseDouble s -> ReverseDouble s

(^*) :: ReverseDouble s -> Scalar (ReverseDouble s) -> ReverseDouble s

(^/) :: ReverseDouble s -> Scalar (ReverseDouble s) -> ReverseDouble s

zero :: ReverseDouble s

Reifies s Tape => Jacobian (ReverseDouble s)

Defined in Numeric.AD.Internal.Reverse.Double

Associated Types

type D (ReverseDouble s)

Methods

unary :: (Scalar (ReverseDouble s) -> Scalar (ReverseDouble s)) -> D (ReverseDouble s) -> ReverseDouble s -> ReverseDouble s

lift1 :: (Scalar (ReverseDouble s) -> Scalar (ReverseDouble s)) -> (D (ReverseDouble s) -> D (ReverseDouble s)) -> ReverseDouble s -> ReverseDouble s

lift1_ :: (Scalar (ReverseDouble s) -> Scalar (ReverseDouble s)) -> (D (ReverseDouble s) -> D (ReverseDouble s) -> D (ReverseDouble s)) -> ReverseDouble s -> ReverseDouble s

binary :: (Scalar (ReverseDouble s) -> Scalar (ReverseDouble s) -> Scalar (ReverseDouble s)) -> D (ReverseDouble s) -> D (ReverseDouble s) -> ReverseDouble s -> ReverseDouble s -> ReverseDouble s

lift2 :: (Scalar (ReverseDouble s) -> Scalar (ReverseDouble s) -> Scalar (ReverseDouble s)) -> (D (ReverseDouble s) -> D (ReverseDouble s) -> (D (ReverseDouble s), D (ReverseDouble s))) -> ReverseDouble s -> ReverseDouble s -> ReverseDouble s

lift2_ :: (Scalar (ReverseDouble s) -> Scalar (ReverseDouble s) -> Scalar (ReverseDouble s)) -> (D (ReverseDouble s) -> D (ReverseDouble s) -> D (ReverseDouble s) -> (D (ReverseDouble s), D (ReverseDouble s))) -> ReverseDouble s -> ReverseDouble s -> ReverseDouble s

type Scalar (ReverseDouble s)

Defined in Numeric.AD.Internal.Reverse.Double

type Scalar (ReverseDouble s) = Double

type D (ReverseDouble s)

Defined in Numeric.AD.Internal.Reverse.Double

type D (ReverseDouble s) = Id Double

auto :: Mode t => Scalar t -> t

Embed a constant

Gradient

grad :: Traversable f => (forall s. (Reifies s Tape, Typeable s) => f (ReverseDouble s) -> ReverseDouble s) -> f Double -> f Double

The grad function calculates the gradient of a non-scalar-to-scalar function with reverse-mode AD in a single pass.

>>> grad (\[x,y,z] -> x*y+z) [1,2,3]
[2.0,1.0,1.0]

>>> grad (\[x,y] -> x**y) [0,2]
[0.0,NaN]

grad' :: Traversable f => (forall s. (Reifies s Tape, Typeable s) => f (ReverseDouble s) -> ReverseDouble s) -> f Double -> (Double, f Double)

The grad' function calculates the result and gradient of a non-scalar-to-scalar function with reverse-mode AD ƒin a single pass.

>>> grad' (\[x,y,z] -> x*y+z) [1,2,3]
(5.0,[2.0,1.0,1.0])

gradWith :: Traversable f => (Double -> Double -> b) -> (forall s. (Reifies s Tape, Typeable s) => f (ReverseDouble s) -> ReverseDouble s) -> f Double -> f b

grad g f function calculates the gradient of a non-scalar-to-scalar function f with reverse-mode AD in a single pass. The gradient is combined element-wise with the argument using the function g.

grad == gradWith (_ dx -> dx)
id == gradWith const

gradWith' :: Traversable f => (Double -> Double -> b) -> (forall s. (Reifies s Tape, Typeable s) => f (ReverseDouble s) -> ReverseDouble s) -> f Double -> (Double, f b)

grad' g f calculates the result and gradient of a non-scalar-to-scalar function f with reverse-mode AD in a single pass the gradient is combined element-wise with the argument using the function g.

grad' == gradWith' (_ dx -> dx)

Jacobian

jacobian :: (Traversable f, Functor g) => (forall s. (Reifies s Tape, Typeable s) => f (ReverseDouble s) -> g (ReverseDouble s)) -> f Double -> g (f Double)

The jacobian function calculates the jacobian of a non-scalar-to-non-scalar function with reverse AD lazily in m passes for m outputs.

>>> jacobian (\[x,y] -> [y,x,x*y]) [2,1]
[[0.0,1.0],[1.0,0.0],[1.0,2.0]]

jacobian' :: (Traversable f, Functor g) => (forall s. (Reifies s Tape, Typeable s) => f (ReverseDouble s) -> g (ReverseDouble s)) -> f Double -> g (Double, f Double)

The jacobian' function calculates both the result and the Jacobian of a nonscalar-to-nonscalar function, using m invocations of reverse AD, where m is the output dimensionality. Applying fmap snd to the result will recover the result of jacobian | An alias for gradF'

>>> jacobian' (\[x,y] -> [y,x,x*y]) [2,1]
[(1.0,[0.0,1.0]),(2.0,[1.0,0.0]),(2.0,[1.0,2.0])]

jacobianWith :: (Traversable f, Functor g) => (Double -> Double -> b) -> (forall s. (Reifies s Tape, Typeable s) => f (ReverseDouble s) -> g (ReverseDouble s)) -> f Double -> g (f b)

'jacobianWith g f' calculates the Jacobian of a non-scalar-to-non-scalar function f with reverse AD lazily in m passes for m outputs.

Instead of returning the Jacobian matrix, the elements of the matrix are combined with the input using the g.

jacobian == jacobianWith (_ dx -> dx)
jacobianWith const == (f x -> const x <$> f x)

jacobianWith' :: (Traversable f, Functor g) => (Double -> Double -> b) -> (forall s. (Reifies s Tape, Typeable s) => f (ReverseDouble s) -> g (ReverseDouble s)) -> f Double -> g (Double, f b)

jacobianWith g f' calculates both the result and the Jacobian of a nonscalar-to-nonscalar function f, using m invocations of reverse AD, where m is the output dimensionality. Applying fmap snd to the result will recover the result of jacobianWith

Instead of returning the Jacobian matrix, the elements of the matrix are combined with the input using the g.

jacobian' == jacobianWith' (_ dx -> dx)

Hessian

hessian :: Traversable f => (forall s s'. (Reifies s Tape, Typeable s, Reifies s' Tape, Typeable s') => f (On (Reverse s (ReverseDouble s'))) -> On (Reverse s (ReverseDouble s'))) -> f Double -> f (f Double)

Compute the hessian via the jacobian of the gradient. gradient is computed in reverse mode and then the jacobian is computed in reverse mode.

However, since the grad f :: f a -> f a is square this is not as fast as using the forward-mode Jacobian of a reverse mode gradient provided by hessian.

>>> hessian (\[x,y] -> x*y) [1,2]
[[0.0,1.0],[1.0,0.0]]

hessianF :: (Traversable f, Functor g) => (forall s s'. (Reifies s Tape, Typeable s, Reifies s' Tape, Typeable s') => f (On (Reverse s (ReverseDouble s'))) -> g (On (Reverse s (ReverseDouble s')))) -> f Double -> g (f (f Double))

Compute the order 3 Hessian tensor on a non-scalar-to-non-scalar function via the reverse-mode Jacobian of the reverse-mode Jacobian of the function.

Less efficient than hessianF.

>>> hessianF (\[x,y] -> [x*y,x+y,exp x*cos y]) [1,2 :: Double]
[[[0.0,1.0],[1.0,0.0]],[[0.0,0.0],[0.0,0.0]],[[-1.1312043837568135,-2.4717266720048188],[-2.4717266720048188,1.1312043837568135]]]

Derivatives

diff :: (forall s. (Reifies s Tape, Typeable s) => ReverseDouble s -> ReverseDouble s) -> Double -> Double

Compute the derivative of a function.

>>> diff sin 0
1.0

diff' :: (forall s. (Reifies s Tape, Typeable s) => ReverseDouble s -> ReverseDouble s) -> Double -> (Double, Double)

The diff' function calculates the result and derivative, as a pair, of a scalar-to-scalar function.

>>> diff' sin 0
(0.0,1.0)

>>> diff' exp 0
(1.0,1.0)

diffF :: Functor f => (forall s. (Reifies s Tape, Typeable s) => ReverseDouble s -> f (ReverseDouble s)) -> Double -> f Double

Compute the derivatives of each result of a scalar-to-vector function with regards to its input.

>>> diffF (\a -> [sin a, cos a]) 0
[1.0,0.0]

diffF' :: Functor f => (forall s. (Reifies s Tape, Typeable s) => ReverseDouble s -> f (ReverseDouble s)) -> Double -> f (Double, Double)

Compute the derivatives of each result of a scalar-to-vector function with regards to its input along with the answer.

>>> diffF' (\a -> [sin a, cos a]) 0
[(0.0,1.0),(1.0,0.0)]