ad-4.3.2.1: Automatic Differentiation

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

Numeric.AD.Internal.Sparse

Description

Unsafe and often partial combinators intended for internal usage.

Handle with care.

Synopsis

Documentation

newtype Monomial Source #

Constructors

Monomial (IntMap Int) 

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 - 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) Source # 

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) # 
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) Source # 

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) Source # 

Associated Types

type Scalar (Sparse a) :: * Source #

Num a => Jacobian (Sparse a) Source # 

Associated Types

type D (Sparse a) :: * 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 # 

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 # 

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 # 

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 # 

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

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

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

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

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

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

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

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

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

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

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

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

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

Minimal complete definition

pack, unpack, unpack'

Methods

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

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

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

Instances

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

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 # 

Methods

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

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

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

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

Minimal complete definition

packs, unpacks

Methods

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

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

Instances

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

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 # 

Methods

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

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

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

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 Source #