Copyright	(c) Edward Kmett 2010-2015
License	BSD3
Maintainer	ekmett@gmail.com
Stability	experimental
Portability	GHC only
Safe Haskell	None
Language	Haskell2010

Numeric.AD.Mode.Forward

Contents

Gradient
Jacobian
Transposed Jacobian
Hessian Product
Derivatives
Directional Derivatives

Description

Forward mode automatic differentiation

Synopsis

data AD s a
data Forward a
auto :: Mode t => Scalar t -> t
grad :: (Traversable f, Num a) => (forall s. f (AD s (Forward a)) -> AD s (Forward a)) -> f a -> f a
grad' :: (Traversable f, Num a) => (forall s. f (AD s (Forward a)) -> AD s (Forward a)) -> f a -> (a, f a)
gradWith :: (Traversable f, Num a) => (a -> a -> b) -> (forall s. f (AD s (Forward a)) -> AD s (Forward a)) -> f a -> f b
gradWith' :: (Traversable f, Num a) => (a -> a -> b) -> (forall s. f (AD s (Forward a)) -> AD s (Forward a)) -> f a -> (a, f b)
jacobian :: (Traversable f, Traversable g, Num a) => (forall s. f (AD s (Forward a)) -> g (AD s (Forward a))) -> f a -> g (f a)
jacobian' :: (Traversable f, Traversable g, Num a) => (forall s. f (AD s (Forward a)) -> g (AD s (Forward a))) -> f a -> g (a, f a)
jacobianWith :: (Traversable f, Traversable g, Num a) => (a -> a -> b) -> (forall s. f (AD s (Forward a)) -> g (AD s (Forward a))) -> f a -> g (f b)
jacobianWith' :: (Traversable f, Traversable g, Num a) => (a -> a -> b) -> (forall s. f (AD s (Forward a)) -> g (AD s (Forward a))) -> f a -> g (a, f b)
jacobianT :: (Traversable f, Functor g, Num a) => (forall s. f (AD s (Forward a)) -> g (AD s (Forward a))) -> f a -> f (g a)
jacobianWithT :: (Traversable f, Functor g, Num a) => (a -> a -> b) -> (forall s. f (AD s (Forward a)) -> g (AD s (Forward a))) -> f a -> f (g b)
hessianProduct :: (Traversable f, Num a) => (forall s. f (AD s (On (Forward (Forward a)))) -> AD s (On (Forward (Forward a)))) -> f (a, a) -> f a
hessianProduct' :: (Traversable f, Num a) => (forall s. f (AD s (On (Forward (Forward a)))) -> AD s (On (Forward (Forward a)))) -> f (a, a) -> f (a, a)
diff :: Num a => (forall s. AD s (Forward a) -> AD s (Forward a)) -> a -> a
diff' :: Num a => (forall s. AD s (Forward a) -> AD s (Forward a)) -> a -> (a, a)
diffF :: (Functor f, Num a) => (forall s. AD s (Forward a) -> f (AD s (Forward a))) -> a -> f a
diffF' :: (Functor f, Num a) => (forall s. AD s (Forward a) -> f (AD s (Forward a))) -> a -> f (a, a)
du :: (Functor f, Num a) => (forall s. f (AD s (Forward a)) -> AD s (Forward a)) -> f (a, a) -> a
du' :: (Functor f, Num a) => (forall s. f (AD s (Forward a)) -> AD s (Forward a)) -> f (a, a) -> (a, a)
duF :: (Functor f, Functor g, Num a) => (forall s. f (AD s (Forward a)) -> g (AD s (Forward a))) -> f (a, a) -> g a
duF' :: (Functor f, Functor g, Num a) => (forall s. f (AD s (Forward a)) -> g (AD s (Forward a))) -> f (a, a) -> g (a, a)

Documentation

data AD s a Source #

Instances

Bounded a => Bounded (AD s a) Source #
Instance details Defined in Numeric.AD.Internal.Type Methods minBound :: AD s a # maxBound :: AD s a #
Enum a => Enum (AD s a) Source #
Instance details Defined in Numeric.AD.Internal.Type Methods succ :: AD s a -> AD s a # pred :: AD s a -> AD s a # toEnum :: Int -> AD s a # fromEnum :: AD s a -> Int # enumFrom :: AD s a -> [AD s a] # enumFromThen :: AD s a -> AD s a -> [AD s a] # enumFromTo :: AD s a -> AD s a -> [AD s a] # enumFromThenTo :: AD s a -> AD s a -> AD s a -> [AD s a] #
Eq a => Eq (AD s a) Source #
Instance details Defined in Numeric.AD.Internal.Type Methods (==) :: AD s a -> AD s a -> Bool # (/=) :: AD s a -> AD s a -> Bool #
Floating a => Floating (AD s a) Source #
Instance details Defined in Numeric.AD.Internal.Type Methods pi :: AD s a # exp :: AD s a -> AD s a # log :: AD s a -> AD s a # sqrt :: AD s a -> AD s a # (**) :: AD s a -> AD s a -> AD s a # logBase :: AD s a -> AD s a -> AD s a # sin :: AD s a -> AD s a # cos :: AD s a -> AD s a # tan :: AD s a -> AD s a # asin :: AD s a -> AD s a # acos :: AD s a -> AD s a # atan :: AD s a -> AD s a # sinh :: AD s a -> AD s a # cosh :: AD s a -> AD s a # tanh :: AD s a -> AD s a # asinh :: AD s a -> AD s a # acosh :: AD s a -> AD s a # atanh :: AD s a -> AD s a # log1p :: AD s a -> AD s a # expm1 :: AD s a -> AD s a # log1pexp :: AD s a -> AD s a # log1mexp :: AD s a -> AD s a #
Fractional a => Fractional (AD s a) Source #
Instance details Defined in Numeric.AD.Internal.Type Methods (/) :: AD s a -> AD s a -> AD s a # recip :: AD s a -> AD s a # fromRational :: Rational -> AD s a #
Num a => Num (AD s a) Source #
Instance details Defined in Numeric.AD.Internal.Type Methods (+) :: AD s a -> AD s a -> AD s a # (-) :: AD s a -> AD s a -> AD s a # (*) :: AD s a -> AD s a -> AD s a # negate :: AD s a -> AD s a # abs :: AD s a -> AD s a # signum :: AD s a -> AD s a # fromInteger :: Integer -> AD s a #
Ord a => Ord (AD s a) Source #
Instance details Defined in Numeric.AD.Internal.Type Methods compare :: AD s a -> AD s a -> Ordering # (<) :: AD s a -> AD s a -> Bool # (<=) :: AD s a -> AD s a -> Bool # (>) :: AD s a -> AD s a -> Bool # (>=) :: AD s a -> AD s a -> Bool # max :: AD s a -> AD s a -> AD s a # min :: AD s a -> AD s a -> AD s a #
Read a => Read (AD s a) Source #
Instance details Defined in Numeric.AD.Internal.Type Methods readsPrec :: Int -> ReadS (AD s a) # readList :: ReadS [AD s a] # readPrec :: ReadPrec (AD s a) # readListPrec :: ReadPrec [AD s a] #
Real a => Real (AD s a) Source #
Instance details Defined in Numeric.AD.Internal.Type Methods toRational :: AD s a -> Rational #
RealFloat a => RealFloat (AD s a) Source #
Instance details Defined in Numeric.AD.Internal.Type Methods floatRadix :: AD s a -> Integer # floatDigits :: AD s a -> Int # floatRange :: AD s a -> (Int, Int) # decodeFloat :: AD s a -> (Integer, Int) # encodeFloat :: Integer -> Int -> AD s a # exponent :: AD s a -> Int # significand :: AD s a -> AD s a # scaleFloat :: Int -> AD s a -> AD s a # isNaN :: AD s a -> Bool # isInfinite :: AD s a -> Bool # isDenormalized :: AD s a -> Bool # isNegativeZero :: AD s a -> Bool # isIEEE :: AD s a -> Bool # atan2 :: AD s a -> AD s a -> AD s a #
RealFrac a => RealFrac (AD s a) Source #
Instance details Defined in Numeric.AD.Internal.Type Methods properFraction :: Integral b => AD s a -> (b, AD s a) # truncate :: Integral b => AD s a -> b # round :: Integral b => AD s a -> b # ceiling :: Integral b => AD s a -> b # floor :: Integral b => AD s a -> b #
Show a => Show (AD s a) Source #
Instance details Defined in Numeric.AD.Internal.Type Methods showsPrec :: Int -> AD s a -> ShowS # show :: AD s a -> String # showList :: [AD s a] -> ShowS #
Erf a => Erf (AD s a) Source #
Instance details Defined in Numeric.AD.Internal.Type Methods erf :: AD s a -> AD s a # erfc :: AD s a -> AD s a # erfcx :: AD s a -> AD s a # normcdf :: AD s a -> AD s a #
InvErf a => InvErf (AD s a) Source #
Instance details Defined in Numeric.AD.Internal.Type Methods inverf :: AD s a -> AD s a # inverfc :: AD s a -> AD s a # invnormcdf :: AD s a -> AD s a #
Mode a => Mode (AD s a) Source #
Instance details Defined in Numeric.AD.Internal.Type Associated Types type Scalar (AD s a) :: Type Source # Methods isKnownConstant :: AD s a -> Bool Source # isKnownZero :: AD s a -> Bool Source # auto :: Scalar (AD s a) -> AD s a Source # (^) :: Scalar (AD s a) -> AD s a -> AD s a Source # (^) :: AD s a -> Scalar (AD s a) -> AD s a Source # (^/) :: AD s a -> Scalar (AD s a) -> AD s a Source # zero :: AD s a Source #
type Scalar (AD s a) Source #
Instance details Defined in Numeric.AD.Internal.Type type Scalar (AD s a) = Scalar a

data Forward a Source #

Forward mode AD

Instances

(Num a, Bounded a) => Bounded (Forward a) Source #
Instance details Defined in Numeric.AD.Internal.Forward Methods minBound :: Forward a # maxBound :: Forward a #
(Num a, Enum a) => Enum (Forward a) Source #
Instance details Defined in Numeric.AD.Internal.Forward Methods succ :: Forward a -> Forward a # pred :: Forward a -> Forward a # toEnum :: Int -> Forward a # fromEnum :: Forward a -> Int # enumFrom :: Forward a -> [Forward a] # enumFromThen :: Forward a -> Forward a -> [Forward a] # enumFromTo :: Forward a -> Forward a -> [Forward a] # enumFromThenTo :: Forward a -> Forward a -> Forward a -> [Forward a] #
(Num a, Eq a) => Eq (Forward a) Source #
Instance details Defined in Numeric.AD.Internal.Forward Methods (==) :: Forward a -> Forward a -> Bool # (/=) :: Forward a -> Forward a -> Bool #
Floating a => Floating (Forward a) Source #
Instance details Defined in Numeric.AD.Internal.Forward Methods pi :: Forward a # exp :: Forward a -> Forward a # log :: Forward a -> Forward a # sqrt :: Forward a -> Forward a # (**) :: Forward a -> Forward a -> Forward a # logBase :: Forward a -> Forward a -> Forward a # sin :: Forward a -> Forward a # cos :: Forward a -> Forward a # tan :: Forward a -> Forward a # asin :: Forward a -> Forward a # acos :: Forward a -> Forward a # atan :: Forward a -> Forward a # sinh :: Forward a -> Forward a # cosh :: Forward a -> Forward a # tanh :: Forward a -> Forward a # asinh :: Forward a -> Forward a # acosh :: Forward a -> Forward a # atanh :: Forward a -> Forward a # log1p :: Forward a -> Forward a # expm1 :: Forward a -> Forward a # log1pexp :: Forward a -> Forward a # log1mexp :: Forward a -> Forward a #
Fractional a => Fractional (Forward a) Source #
Instance details Defined in Numeric.AD.Internal.Forward Methods (/) :: Forward a -> Forward a -> Forward a # recip :: Forward a -> Forward a # fromRational :: Rational -> Forward a #
Data a => Data (Forward a) Source #
Instance details Defined in Numeric.AD.Internal.Forward Methods gfoldl :: (forall d b. Data d => c (d -> b) -> d -> c b) -> (forall g. g -> c g) -> Forward a -> c (Forward a) # gunfold :: (forall b r. Data b => c (b -> r) -> c r) -> (forall r. r -> c r) -> Constr -> c (Forward a) # toConstr :: Forward a -> Constr # dataTypeOf :: Forward a -> DataType # dataCast1 :: Typeable t => (forall d. Data d => c (t d)) -> Maybe (c (Forward a)) # dataCast2 :: Typeable t => (forall d e. (Data d, Data e) => c (t d e)) -> Maybe (c (Forward a)) # gmapT :: (forall b. Data b => b -> b) -> Forward a -> Forward a # gmapQl :: (r -> r' -> r) -> r -> (forall d. Data d => d -> r') -> Forward a -> r # gmapQr :: (r' -> r -> r) -> r -> (forall d. Data d => d -> r') -> Forward a -> r # gmapQ :: (forall d. Data d => d -> u) -> Forward a -> [u] # gmapQi :: Int -> (forall d. Data d => d -> u) -> Forward a -> u # gmapM :: Monad m => (forall d. Data d => d -> m d) -> Forward a -> m (Forward a) # gmapMp :: MonadPlus m => (forall d. Data d => d -> m d) -> Forward a -> m (Forward a) # gmapMo :: MonadPlus m => (forall d. Data d => d -> m d) -> Forward a -> m (Forward a) #
Num a => Num (Forward a) Source #
Instance details Defined in Numeric.AD.Internal.Forward Methods (+) :: Forward a -> Forward a -> Forward a # (-) :: Forward a -> Forward a -> Forward a # (*) :: Forward a -> Forward a -> Forward a # negate :: Forward a -> Forward a # abs :: Forward a -> Forward a # signum :: Forward a -> Forward a # fromInteger :: Integer -> Forward a #
(Num a, Ord a) => Ord (Forward a) Source #
Instance details Defined in Numeric.AD.Internal.Forward Methods compare :: Forward a -> Forward a -> Ordering # (<) :: Forward a -> Forward a -> Bool # (<=) :: Forward a -> Forward a -> Bool # (>) :: Forward a -> Forward a -> Bool # (>=) :: Forward a -> Forward a -> Bool # max :: Forward a -> Forward a -> Forward a # min :: Forward a -> Forward a -> Forward a #
Real a => Real (Forward a) Source #
Instance details Defined in Numeric.AD.Internal.Forward Methods toRational :: Forward a -> Rational #
RealFloat a => RealFloat (Forward a) Source #
Instance details Defined in Numeric.AD.Internal.Forward Methods floatRadix :: Forward a -> Integer # floatDigits :: Forward a -> Int # floatRange :: Forward a -> (Int, Int) # decodeFloat :: Forward a -> (Integer, Int) # encodeFloat :: Integer -> Int -> Forward a # exponent :: Forward a -> Int # significand :: Forward a -> Forward a # scaleFloat :: Int -> Forward a -> Forward a # isNaN :: Forward a -> Bool # isInfinite :: Forward a -> Bool # isDenormalized :: Forward a -> Bool # isNegativeZero :: Forward a -> Bool # isIEEE :: Forward a -> Bool # atan2 :: Forward a -> Forward a -> Forward a #
RealFrac a => RealFrac (Forward a) Source #
Instance details Defined in Numeric.AD.Internal.Forward Methods properFraction :: Integral b => Forward a -> (b, Forward a) # truncate :: Integral b => Forward a -> b # round :: Integral b => Forward a -> b # ceiling :: Integral b => Forward a -> b # floor :: Integral b => Forward a -> b #
Show a => Show (Forward a) Source #
Instance details Defined in Numeric.AD.Internal.Forward Methods showsPrec :: Int -> Forward a -> ShowS # show :: Forward a -> String # showList :: [Forward a] -> ShowS #
Erf a => Erf (Forward a) Source #
Instance details Defined in Numeric.AD.Internal.Forward Methods erf :: Forward a -> Forward a # erfc :: Forward a -> Forward a # erfcx :: Forward a -> Forward a # normcdf :: Forward a -> Forward a #
InvErf a => InvErf (Forward a) Source #
Instance details Defined in Numeric.AD.Internal.Forward Methods inverf :: Forward a -> Forward a # inverfc :: Forward a -> Forward a # invnormcdf :: Forward a -> Forward a #
Num a => Mode (Forward a) Source #
Instance details Defined in Numeric.AD.Internal.Forward Associated Types type Scalar (Forward a) :: Type Source # Methods isKnownConstant :: Forward a -> Bool Source # isKnownZero :: Forward a -> Bool Source # auto :: Scalar (Forward a) -> Forward a Source # (^) :: Scalar (Forward a) -> Forward a -> Forward a Source # (^) :: Forward a -> Scalar (Forward a) -> Forward a Source # (^/) :: Forward a -> Scalar (Forward a) -> Forward a Source # zero :: Forward a Source #
Num a => Jacobian (Forward a) Source #
Instance details Defined in Numeric.AD.Internal.Forward Associated Types type D (Forward a) :: Type Source # Methods unary :: (Scalar (Forward a) -> Scalar (Forward a)) -> D (Forward a) -> Forward a -> Forward a Source # lift1 :: (Scalar (Forward a) -> Scalar (Forward a)) -> (D (Forward a) -> D (Forward a)) -> Forward a -> Forward a Source # lift1_ :: (Scalar (Forward a) -> Scalar (Forward a)) -> (D (Forward a) -> D (Forward a) -> D (Forward a)) -> Forward a -> Forward a Source # binary :: (Scalar (Forward a) -> Scalar (Forward a) -> Scalar (Forward a)) -> D (Forward a) -> D (Forward a) -> Forward a -> Forward a -> Forward a Source # lift2 :: (Scalar (Forward a) -> Scalar (Forward a) -> Scalar (Forward a)) -> (D (Forward a) -> D (Forward a) -> (D (Forward a), D (Forward a))) -> Forward a -> Forward a -> Forward a Source # lift2_ :: (Scalar (Forward a) -> Scalar (Forward a) -> Scalar (Forward a)) -> (D (Forward a) -> D (Forward a) -> D (Forward a) -> (D (Forward a), D (Forward a))) -> Forward a -> Forward a -> Forward a Source #
type Scalar (Forward a) Source #
Instance details Defined in Numeric.AD.Internal.Forward type Scalar (Forward a) = a
type D (Forward a) Source #
Instance details Defined in Numeric.AD.Internal.Forward type D (Forward a) = Id a

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

Embed a constant

Gradient

grad :: (Traversable f, Num a) => (forall s. f (AD s (Forward a)) -> AD s (Forward a)) -> f a -> f a Source #

Compute the gradient of a function using forward mode AD.

Note, this performs O(n) worse than grad for n inputs, in exchange for better space utilization.

grad' :: (Traversable f, Num a) => (forall s. f (AD s (Forward a)) -> AD s (Forward a)) -> f a -> (a, f a) Source #

Compute the gradient and answer to a function using forward mode AD.

Note, this performs O(n) worse than grad' for n inputs, in exchange for better space utilization.

gradWith :: (Traversable f, Num a) => (a -> a -> b) -> (forall s. f (AD s (Forward a)) -> AD s (Forward a)) -> f a -> f b Source #

Compute the gradient of a function using forward mode AD and combine the result with the input using a user-specified function.

Note, this performs O(n) worse than gradWith for n inputs, in exchange for better space utilization.

gradWith' :: (Traversable f, Num a) => (a -> a -> b) -> (forall s. f (AD s (Forward a)) -> AD s (Forward a)) -> f a -> (a, f b) Source #

Compute the gradient of a function using forward mode AD and the answer, and combine the result with the input using a user-specified function.

Note, this performs O(n) worse than gradWith' for n inputs, in exchange for better space utilization.

>>> gradWith' (,) sum [0..4]
(10,[(0,1),(1,1),(2,1),(3,1),(4,1)])

Jacobian

jacobian :: (Traversable f, Traversable g, Num a) => (forall s. f (AD s (Forward a)) -> g (AD s (Forward a))) -> f a -> g (f a) Source #

Compute the Jacobian using Forward mode AD. This must transpose the result, so jacobianT is faster and allows more result types.

>>> jacobian (\[x,y] -> [y,x,x+y,x*y,exp x * sin y]) [pi,1]
[[0.0,1.0],[1.0,0.0],[1.0,1.0],[1.0,3.141592653589793],[19.472221418841606,12.502969588876512]]

jacobian' :: (Traversable f, Traversable g, Num a) => (forall s. f (AD s (Forward a)) -> g (AD s (Forward a))) -> f a -> g (a, f a) Source #

Compute the Jacobian using Forward mode AD along with the actual answer.

jacobianWith :: (Traversable f, Traversable g, Num a) => (a -> a -> b) -> (forall s. f (AD s (Forward a)) -> g (AD s (Forward a))) -> f a -> g (f b) Source #

Compute the Jacobian using Forward mode AD and combine the output with the input. This must transpose the result, so jacobianWithT is faster, and allows more result types.

jacobianWith' :: (Traversable f, Traversable g, Num a) => (a -> a -> b) -> (forall s. f (AD s (Forward a)) -> g (AD s (Forward a))) -> f a -> g (a, f b) Source #

Compute the Jacobian using Forward mode AD combined with the input using a user specified function, along with the actual answer.

Transposed Jacobian

jacobianT :: (Traversable f, Functor g, Num a) => (forall s. f (AD s (Forward a)) -> g (AD s (Forward a))) -> f a -> f (g a) Source #

A fast, simple, transposed Jacobian computed with forward-mode AD.

jacobianWithT :: (Traversable f, Functor g, Num a) => (a -> a -> b) -> (forall s. f (AD s (Forward a)) -> g (AD s (Forward a))) -> f a -> f (g b) Source #

A fast, simple, transposed Jacobian computed with Forward mode AD that combines the output with the input.

Hessian Product

hessianProduct :: (Traversable f, Num a) => (forall s. f (AD s (On (Forward (Forward a)))) -> AD s (On (Forward (Forward a)))) -> f (a, a) -> f a Source #

Compute the product of a vector with the Hessian using forward-on-forward-mode AD.

hessianProduct' :: (Traversable f, Num a) => (forall s. f (AD s (On (Forward (Forward a)))) -> AD s (On (Forward (Forward a)))) -> f (a, a) -> f (a, a) Source #

Compute the gradient and hessian product using forward-on-forward-mode AD.

Derivatives

diff :: Num a => (forall s. AD s (Forward a) -> AD s (Forward a)) -> a -> a Source #

The diff function calculates the first derivative of a scalar-to-scalar function by forward-mode AD

>>> diff sin 0
1.0

diff' :: Num a => (forall s. AD s (Forward a) -> AD s (Forward a)) -> a -> (a, a) Source #

The diff' function calculates the result and first derivative of scalar-to-scalar function by Forward mode AD

diff' sin == sin &&& cos
diff' f = f &&& d f

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

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

diffF :: (Functor f, Num a) => (forall s. AD s (Forward a) -> f (AD s (Forward a))) -> a -> f a Source #

The diffF function calculates the first derivatives of scalar-to-nonscalar function by Forward mode AD

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

diffF' :: (Functor f, Num a) => (forall s. AD s (Forward a) -> f (AD s (Forward a))) -> a -> f (a, a) Source #

The diffF' function calculates the result and first derivatives of a scalar-to-non-scalar function by Forward mode AD

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

Directional Derivatives

du :: (Functor f, Num a) => (forall s. f (AD s (Forward a)) -> AD s (Forward a)) -> f (a, a) -> a Source #

Compute the directional derivative of a function given a zipped up Functor of the input values and their derivatives

du' :: (Functor f, Num a) => (forall s. f (AD s (Forward a)) -> AD s (Forward a)) -> f (a, a) -> (a, a) Source #

Compute the answer and directional derivative of a function given a zipped up Functor of the input values and their derivatives

duF :: (Functor f, Functor g, Num a) => (forall s. f (AD s (Forward a)) -> g (AD s (Forward a))) -> f (a, a) -> g a Source #

Compute a vector of directional derivatives for a function given a zipped up Functor of the input values and their derivatives.

duF' :: (Functor f, Functor g, Num a) => (forall s. f (AD s (Forward a)) -> g (AD s (Forward a))) -> f (a, a) -> g (a, a) Source #

Compute a vector of answers and directional derivatives for a function given a zipped up Functor of the input values and their derivatives.