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.Sparse

Contents

Sparse Gradients
Sparse Jacobians (synonyms)
Sparse Hessians

Description

Higher order derivatives via a "dual number tower".

Synopsis

data AD s a
data Sparse a
auto :: Mode t => Scalar t -> t
grad :: (Traversable f, Num a) => (forall s. f (AD s (Sparse a)) -> AD s (Sparse a)) -> f a -> f a
grad' :: (Traversable f, Num a) => (forall s. f (AD s (Sparse a)) -> AD s (Sparse a)) -> f a -> (a, f a)
grads :: (Traversable f, Num a) => (forall s. f (AD s (Sparse a)) -> AD s (Sparse a)) -> f a -> Cofree f a
gradWith :: (Traversable f, Num a) => (a -> a -> b) -> (forall s. f (AD s (Sparse a)) -> AD s (Sparse a)) -> f a -> f b
gradWith' :: (Traversable f, Num a) => (a -> a -> b) -> (forall s. f (AD s (Sparse a)) -> AD s (Sparse a)) -> f a -> (a, f b)
jacobian :: (Traversable f, Functor g, Num a) => (forall s. f (AD s (Sparse a)) -> g (AD s (Sparse a))) -> f a -> g (f a)
jacobian' :: (Traversable f, Functor g, Num a) => (forall s. f (AD s (Sparse a)) -> g (AD s (Sparse a))) -> f a -> g (a, f a)
jacobianWith :: (Traversable f, Functor g, Num a) => (a -> a -> b) -> (forall s. f (AD s (Sparse a)) -> g (AD s (Sparse a))) -> f a -> g (f b)
jacobianWith' :: (Traversable f, Functor g, Num a) => (a -> a -> b) -> (forall s. f (AD s (Sparse a)) -> g (AD s (Sparse a))) -> f a -> g (a, f b)
jacobians :: (Traversable f, Functor g, Num a) => (forall s. f (AD s (Sparse a)) -> g (AD s (Sparse a))) -> f a -> g (Cofree f a)
hessian :: (Traversable f, Num a) => (forall s. f (AD s (Sparse a)) -> AD s (Sparse a)) -> f a -> f (f a)
hessian' :: (Traversable f, Num a) => (forall s. f (AD s (Sparse a)) -> AD s (Sparse a)) -> f a -> (a, f (a, f a))
hessianF :: (Traversable f, Functor g, Num a) => (forall s. f (AD s (Sparse a)) -> g (AD s (Sparse a))) -> f a -> g (f (f a))
hessianF' :: (Traversable f, Functor g, Num a) => (forall s. f (AD s (Sparse a)) -> g (AD s (Sparse a))) -> f a -> g (a, f (a, f a))

Documentation

data AD s a #

Instances

Bounded a => Bounded (AD s a) #
Instance details Defined in Numeric.AD.Internal.Type Methods minBound :: AD s a # maxBound :: AD s a #
Enum a => Enum (AD s a) #
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) #
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) #
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) #
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) #
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) #
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) #
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) #
Instance details Defined in Numeric.AD.Internal.Type Methods toRational :: AD s a -> Rational #
RealFloat a => RealFloat (AD s a) #
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) #
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) #
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) #
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) #
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) #
Instance details Defined in Numeric.AD.Internal.Type Associated Types type Scalar (AD s a) :: Type # Methods isKnownConstant :: AD s a -> Bool # isKnownZero :: AD s a -> Bool # auto :: Scalar (AD s a) -> AD s a # (^) :: Scalar (AD s a) -> AD s a -> AD s a # (^) :: AD s a -> Scalar (AD s a) -> AD s a # (^/) :: AD s a -> Scalar (AD s a) -> AD s a # zero :: AD s a #
type Scalar (AD s a) #
Instance details Defined in Numeric.AD.Internal.Type type Scalar (AD s a) = Scalar a

data Sparse a #

We only store partials in sorted order, so the map contained in a partial will only contain partials with equal or greater keys to that of the map in which it was found. This should be key for efficiently computing sparse hessians. there are only n + k - 1 choose k distinct nth partial derivatives of a function with k inputs.

Instances

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

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

Embed a constant

Sparse Gradients

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

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

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

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

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

grads :: (Traversable f, Num a) => (forall s. f (AD s (Sparse a)) -> AD s (Sparse a)) -> f a -> Cofree f a #

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

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

Sparse Jacobians (synonyms)

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

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

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

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

jacobians :: (Traversable f, Functor g, Num a) => (forall s. f (AD s (Sparse a)) -> g (AD s (Sparse a))) -> f a -> g (Cofree f a) #

Sparse Hessians

hessian :: (Traversable f, Num a) => (forall s. f (AD s (Sparse a)) -> AD s (Sparse a)) -> f a -> f (f a) #

hessian' :: (Traversable f, Num a) => (forall s. f (AD s (Sparse a)) -> AD s (Sparse a)) -> f a -> (a, f (a, f a)) #

hessianF :: (Traversable f, Functor g, Num a) => (forall s. f (AD s (Sparse a)) -> g (AD s (Sparse a))) -> f a -> g (f (f a)) #

hessianF' :: (Traversable f, Functor g, Num a) => (forall s. f (AD s (Sparse a)) -> g (AD s (Sparse a))) -> f a -> g (a, f (a, f a)) #