Numeric.AD.Internal.Sparse

Unsafe and often partial combinators intended for internal usage.

Handle with care.

Documentation

newtype Monomial

Constructors

Monomial (IntMap Int)

emptyMonomial :: Monomial

addToMonomial :: Int -> Monomial -> Monomial

indices :: Monomial -> [Int]

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.

Constructors

Sparse !a (IntMap (Sparse a))

Zero

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 :: (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) :: 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)

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

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

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

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

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

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

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

skeleton :: Traversable f => f a -> f Int

spartial :: Num a => [Int] -> Sparse a -> Maybe a

partial :: Num a => [Int] -> Sparse a -> a

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

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

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

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

Methods

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

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

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

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'

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

Methods

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

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

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

terms :: Monomial -> [(Integer, Monomial, Monomial)]

The value of the derivative of (f*g) of order mi is

sum [a * primal (partialS (indices b) f) * primal (partialS (indices c) g) | (a,b,c) <- terms mi ]

It is a bit more complicated in mul below, since we build the whole tree of derivatives and want to prune the tree with Zeros as much as possible. The number of terms in the sum for order mi as of differentiation has sum (map (+1) as) terms, so this is *much* more efficient than the naive recursive differentiation with 2^sum as terms. The coefficients a, which collect equivalent derivatives, are suitable products of binomial coefficients.

primal :: Num a => Sparse a -> a