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 - 1) 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)

Methods

minBound :: Sparse a

maxBound :: Sparse a

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

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)

Methods

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

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

Floating a => Floating (Sparse a)

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)

Methods

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

recip :: Sparse a -> Sparse a

fromRational :: Rational -> Sparse a

Data a => Data (Sparse a)

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)

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)

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)

Methods

toRational :: Sparse a -> Rational

RealFloat a => RealFloat (Sparse a)

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)

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)

Methods

showsPrec :: Int -> Sparse a -> ShowS

show :: Sparse a -> String

showList :: [Sparse a] -> ShowS

Erf a => Erf (Sparse a)

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)

Methods

inverf :: Sparse a -> Sparse a

inverfc :: Sparse a -> Sparse a

invnormcdf :: Sparse a -> Sparse a

Num a => Mode (Sparse a)

Associated Types

type Scalar (Sparse a) :: *

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)

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

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

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

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

Methods

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

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

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

type Scalar (Sparse a)

type Scalar (Sparse a) = a

type D (Sparse a)

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

Minimal complete definition

pack, unpack, unpack'

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

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

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

Minimal complete definition

packs, unpacks

Methods

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

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

Instances

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

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

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