Numeric.AD.Rank1.Forward

Forward mode automatic differentiation

Documentation

data Forward a

Forward mode AD

Instances

(Num a, Bounded a) => Bounded (Forward a)

Methods

minBound :: Forward a

maxBound :: Forward a

(Num a, Enum a) => Enum (Forward a)

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)

Methods

(==) :: Forward a -> Forward a -> Bool

(/=) :: Forward a -> Forward a -> Bool

Floating a => Floating (Forward a)

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)

Methods

(/) :: Forward a -> Forward a -> Forward a

recip :: Forward a -> Forward a

fromRational :: Rational -> Forward a

Data a => Data (Forward a)

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)

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)

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)

Methods

toRational :: Forward a -> Rational

RealFloat a => RealFloat (Forward a)

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)

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)

Methods

showsPrec :: Int -> Forward a -> ShowS

show :: Forward a -> String

showList :: [Forward a] -> ShowS

Erf a => Erf (Forward a)

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)

Methods

inverf :: Forward a -> Forward a

inverfc :: Forward a -> Forward a

invnormcdf :: Forward a -> Forward a

Num a => Mode (Forward a)

Associated Types

type Scalar (Forward a) :: *

Methods

isKnownConstant :: Forward a -> Bool

isKnownZero :: Forward a -> Bool

auto :: Scalar (Forward a) -> Forward a

(*^) :: Scalar (Forward a) -> Forward a -> Forward a

(^*) :: Forward a -> Scalar (Forward a) -> Forward a

(^/) :: Forward a -> Scalar (Forward a) -> Forward a

zero :: Forward a

Num a => Jacobian (Forward a)

Associated Types

type D (Forward a) :: *

Methods

unary :: (Scalar (Forward a) -> Scalar (Forward a)) -> D (Forward a) -> Forward a -> Forward a

lift1 :: (Scalar (Forward a) -> Scalar (Forward a)) -> (D (Forward a) -> D (Forward a)) -> Forward a -> Forward a

lift1_ :: (Scalar (Forward a) -> Scalar (Forward a)) -> (D (Forward a) -> D (Forward a) -> D (Forward a)) -> Forward a -> Forward a

binary :: (Scalar (Forward a) -> Scalar (Forward a) -> Scalar (Forward a)) -> D (Forward a) -> D (Forward a) -> Forward a -> Forward a -> Forward a

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

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

type Scalar (Forward a)

type Scalar (Forward a) = a

type D (Forward a)

type D (Forward a) = Id a

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

Embed a constant

Gradient

grad :: (Traversable f, Num a) => (f (Forward a) -> Forward a) -> f a -> f a

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) => (f (Forward a) -> Forward a) -> f a -> (a, f a)

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) -> (f (Forward a) -> Forward a) -> f a -> f b

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) -> (f (Forward a) -> Forward a) -> f a -> (a, f b)

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) => (f (Forward a) -> g (Forward a)) -> f a -> g (f a)

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) => (f (Forward a) -> g (Forward a)) -> f a -> g (a, f a)

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

jacobianWith :: (Traversable f, Traversable g, Num a) => (a -> a -> b) -> (f (Forward a) -> g (Forward a)) -> f a -> g (f b)

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) -> (f (Forward a) -> g (Forward a)) -> f a -> g (a, f b)

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) => (f (Forward a) -> g (Forward a)) -> f a -> f (g a)

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

jacobianWithT :: (Traversable f, Functor g, Num a) => (a -> a -> b) -> (f (Forward a) -> g (Forward a)) -> f a -> f (g b)

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

Hessian Product

hessianProduct :: (Traversable f, Num a) => (f (On (Forward (Forward a))) -> On (Forward (Forward a))) -> f (a, a) -> f a

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

hessianProduct' :: (Traversable f, Num a) => (f (On (Forward (Forward a))) -> On (Forward (Forward a))) -> f (a, a) -> f (a, a)

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

Derivatives

diff :: Num a => (Forward a -> Forward a) -> a -> a

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 => (Forward a -> Forward a) -> a -> (a, a)

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) => (Forward a -> f (Forward a)) -> a -> f a

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) => (Forward a -> f (Forward a)) -> a -> f (a, a)

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) => (f (Forward a) -> Forward a) -> f (a, a) -> a

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

du' :: (Functor f, Num a) => (f (Forward a) -> Forward a) -> f (a, a) -> (a, a)

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) => (f (Forward a) -> g (Forward a)) -> f (a, a) -> g a

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) => (f (Forward a) -> g (Forward a)) -> f (a, a) -> g (a, a)

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