Numeric.AD.Rank1.Sparse

Higher order derivatives via a "dual number tower".

Documentation

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)

Defined in Numeric.AD.Internal.Sparse

Methods

minBound :: Sparse a

maxBound :: Sparse a

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

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)

Defined in Numeric.AD.Internal.Sparse

Methods

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

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

Floating a => Floating (Sparse a)

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)

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)

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 :: forall r r'. (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)

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)

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)

Defined in Numeric.AD.Internal.Sparse

Methods

toRational :: Sparse a -> Rational

RealFloat a => RealFloat (Sparse a)

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)

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)

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)

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)

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)

Defined in Numeric.AD.Internal.Sparse

Associated Types

type Scalar (Sparse a)

Methods

isKnownConstant :: Sparse a -> Bool

asKnownConstant :: Sparse a -> Maybe (Scalar (Sparse a))

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)

Defined in Numeric.AD.Internal.Sparse

Associated Types

type D (Sparse a)

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

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

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

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

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)

Defined in Numeric.AD.Internal.Sparse

type Scalar (Sparse a) = a

type D (Sparse a)

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) => (f (Sparse a) -> Sparse a) -> f a -> f a

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

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

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

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

Minimal complete definition

pack, unpack, unpack'

Instances

Num a => Grad (Sparse a) [a] (a, [a]) a

Defined in Numeric.AD.Internal.Sparse

Methods

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

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

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

Grad i o o' a => Grad (Sparse a -> i) (a -> o) (a -> o') a

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'

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

Higher-Order Gradients

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

Variadic Higher-Order Gradients

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

Minimal complete definition

packs, unpacks

Instances

Num a => Grads (Sparse a) (Cofree [] a) a

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

Defined in Numeric.AD.Internal.Sparse

Methods

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

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

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

Sparse Jacobians (synonyms)

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

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

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

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

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

Sparse Hessians

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

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

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

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