ad-4.4: Automatic Differentiation

Copyright(c) Edward Kmett 2010-2015
LicenseBSD3
Maintainerekmett@gmail.com
Stabilityexperimental
PortabilityGHC only
Safe HaskellNone
LanguageHaskell2010

Numeric.AD.Rank1.Sparse

Contents

Description

Higher order derivatives via a "dual number tower".

Synopsis

Documentation

data Sparse a Source #

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) Source # 
Instance details

Defined in Numeric.AD.Internal.Sparse

Methods

minBound :: Sparse a #

maxBound :: Sparse a #

(Num a, Enum a) => Enum (Sparse a) Source # 
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) Source # 
Instance details

Defined in Numeric.AD.Internal.Sparse

Methods

(==) :: Sparse a -> Sparse a -> Bool #

(/=) :: Sparse a -> Sparse a -> Bool #

Floating a => Floating (Sparse a) Source # 
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) Source # 
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) Source # 
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) Source # 
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) Source # 
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) Source # 
Instance details

Defined in Numeric.AD.Internal.Sparse

Methods

toRational :: Sparse a -> Rational #

RealFloat a => RealFloat (Sparse a) Source # 
Instance details

Defined in Numeric.AD.Internal.Sparse

RealFrac a => RealFrac (Sparse a) Source # 
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) Source # 
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) Source # 
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) Source # 
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) Source # 
Instance details

Defined in Numeric.AD.Internal.Sparse

Associated Types

type Scalar (Sparse a) :: Type Source #

Num a => Jacobian (Sparse a) Source # 
Instance details

Defined in Numeric.AD.Internal.Sparse

Associated Types

type D (Sparse a) :: Type Source #

Methods

unary :: (Scalar (Sparse a) -> Scalar (Sparse a)) -> D (Sparse a) -> Sparse a -> Sparse a Source #

lift1 :: (Scalar (Sparse a) -> Scalar (Sparse a)) -> (D (Sparse a) -> D (Sparse a)) -> Sparse a -> Sparse a Source #

lift1_ :: (Scalar (Sparse a) -> Scalar (Sparse a)) -> (D (Sparse a) -> D (Sparse a) -> D (Sparse a)) -> Sparse a -> Sparse a Source #

binary :: (Scalar (Sparse a) -> Scalar (Sparse a) -> Scalar (Sparse a)) -> D (Sparse a) -> D (Sparse a) -> Sparse a -> Sparse a -> Sparse a Source #

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 Source #

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 Source #

Num a => Grad (Sparse a) [a] (a, [a]) a Source # 
Instance details

Defined in Numeric.AD.Internal.Sparse

Methods

pack :: Sparse a -> [Sparse a] -> Sparse a Source #

unpack :: ([a] -> [a]) -> [a] Source #

unpack' :: ([a] -> (a, [a])) -> (a, [a]) Source #

Num a => Grads (Sparse a) (Cofree [] a) a Source # 
Instance details

Defined in Numeric.AD.Internal.Sparse

Methods

packs :: Sparse a -> [Sparse a] -> Sparse a Source #

unpacks :: ([a] -> Cofree [] a) -> Cofree [] a Source #

Grads i o a => Grads (Sparse a -> i) (a -> o) a Source # 
Instance details

Defined in Numeric.AD.Internal.Sparse

Methods

packs :: (Sparse a -> i) -> [Sparse a] -> Sparse a Source #

unpacks :: ([a] -> Cofree [] a) -> a -> o Source #

Grad i o o' a => Grad (Sparse a -> i) (a -> o) (a -> o') a Source # 
Instance details

Defined in Numeric.AD.Internal.Sparse

Methods

pack :: (Sparse a -> i) -> [Sparse a] -> Sparse a Source #

unpack :: ([a] -> [a]) -> a -> o Source #

unpack' :: ([a] -> (a, [a])) -> a -> o' Source #

type Scalar (Sparse a) Source # 
Instance details

Defined in Numeric.AD.Internal.Sparse

type Scalar (Sparse a) = a
type D (Sparse a) Source # 
Instance details

Defined in Numeric.AD.Internal.Sparse

type D (Sparse a) = Sparse a

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

Embed a constant

Sparse Gradients

grad :: (Traversable f, Num a) => (f (Sparse a) -> Sparse a) -> f a -> f a Source #

grad' :: (Traversable f, Num a) => (f (Sparse a) -> Sparse a) -> f a -> (a, f a) Source #

gradWith :: (Traversable f, Num a) => (a -> a -> b) -> (f (Sparse a) -> Sparse a) -> f a -> f b Source #

gradWith' :: (Traversable f, Num a) => (a -> a -> b) -> (f (Sparse a) -> Sparse a) -> f a -> (a, f b) Source #

Variadic Gradients

Variadic combinators for variadic mixed-mode automatic differentiation.

Unfortunately, variadicity comes at the expense of being able to use quantification to avoid sensitivity confusion, so be careful when counting the number of auto calls you use when taking the gradient of a function that takes gradients!

class Num a => Grad i o o' a | i -> a o o', o -> a i o', o' -> a i o Source #

Minimal complete definition

pack, unpack, unpack'

Instances
Num a => Grad (Sparse a) [a] (a, [a]) a Source # 
Instance details

Defined in Numeric.AD.Internal.Sparse

Methods

pack :: Sparse a -> [Sparse a] -> Sparse a Source #

unpack :: ([a] -> [a]) -> [a] Source #

unpack' :: ([a] -> (a, [a])) -> (a, [a]) Source #

Grad i o o' a => Grad (Sparse a -> i) (a -> o) (a -> o') a Source # 
Instance details

Defined in Numeric.AD.Internal.Sparse

Methods

pack :: (Sparse a -> i) -> [Sparse a] -> Sparse a Source #

unpack :: ([a] -> [a]) -> a -> o Source #

unpack' :: ([a] -> (a, [a])) -> a -> o' Source #

vgrad :: Grad i o o' a => i -> o Source #

Higher-Order Gradients

grads :: (Traversable f, Num a) => (f (Sparse a) -> Sparse a) -> f a -> Cofree f a Source #

Variadic Higher-Order Gradients

class Num a => Grads i o a | i -> a o, o -> a i Source #

Minimal complete definition

packs, unpacks

Instances
Num a => Grads (Sparse a) (Cofree [] a) a Source # 
Instance details

Defined in Numeric.AD.Internal.Sparse

Methods

packs :: Sparse a -> [Sparse a] -> Sparse a Source #

unpacks :: ([a] -> Cofree [] a) -> Cofree [] a Source #

Grads i o a => Grads (Sparse a -> i) (a -> o) a Source # 
Instance details

Defined in Numeric.AD.Internal.Sparse

Methods

packs :: (Sparse a -> i) -> [Sparse a] -> Sparse a Source #

unpacks :: ([a] -> Cofree [] a) -> a -> o Source #

vgrads :: Grads i o a => i -> o Source #

Sparse Jacobians (synonyms)

jacobian :: (Traversable f, Functor g, Num a) => (f (Sparse a) -> g (Sparse a)) -> f a -> g (f a) Source #

jacobian' :: (Traversable f, Functor g, Num a) => (f (Sparse a) -> g (Sparse a)) -> f a -> g (a, f a) Source #

jacobianWith :: (Traversable f, Functor g, Num a) => (a -> a -> b) -> (f (Sparse a) -> g (Sparse a)) -> f a -> g (f b) Source #

jacobianWith' :: (Traversable f, Functor g, Num a) => (a -> a -> b) -> (f (Sparse a) -> g (Sparse a)) -> f a -> g (a, f b) Source #

jacobians :: (Traversable f, Functor g, Num a) => (f (Sparse a) -> g (Sparse a)) -> f a -> g (Cofree f a) Source #

Sparse Hessians

hessian :: (Traversable f, Num a) => (f (Sparse a) -> Sparse a) -> f a -> f (f a) Source #

hessian' :: (Traversable f, Num a) => (f (Sparse a) -> Sparse a) -> f a -> (a, f (a, f a)) Source #

hessianF :: (Traversable f, Functor g, Num a) => (f (Sparse a) -> g (Sparse a)) -> f a -> g (f (f a)) Source #

hessianF' :: (Traversable f, Functor g, Num a) => (f (Sparse a) -> g (Sparse a)) -> f a -> g (a, f (a, f a)) Source #