Copyright	(c) Edward Kmett 2010-2015
License	BSD3
Maintainer	ekmett@gmail.com
Stability	experimental
Portability	GHC only
Safe Haskell	None
Language	Haskell2010

Numeric.AD.Rank1.Tower

Contents

Taylor Series
Maclaurin Series
Derivatives
Directional Derivatives

Description

Higher order derivatives via a "dual number tower".

Synopsis

data Tower a
auto :: Mode t => Scalar t -> t
taylor :: Fractional a => (Tower a -> Tower a) -> a -> a -> [a]
taylor0 :: Fractional a => (Tower a -> Tower a) -> a -> a -> [a]
maclaurin :: Fractional a => (Tower a -> Tower a) -> a -> [a]
maclaurin0 :: Fractional a => (Tower a -> Tower a) -> a -> [a]
diff :: Num a => (Tower a -> Tower a) -> a -> a
diff' :: Num a => (Tower a -> Tower a) -> a -> (a, a)
diffs :: Num a => (Tower a -> Tower a) -> a -> [a]
diffs0 :: Num a => (Tower a -> Tower a) -> a -> [a]
diffsF :: (Functor f, Num a) => (Tower a -> f (Tower a)) -> a -> f [a]
diffs0F :: (Functor f, Num a) => (Tower a -> f (Tower a)) -> a -> f [a]
du :: (Functor f, Num a) => (f (Tower a) -> Tower a) -> f (a, a) -> a
du' :: (Functor f, Num a) => (f (Tower a) -> Tower a) -> f (a, a) -> (a, a)
dus :: (Functor f, Num a) => (f (Tower a) -> Tower a) -> f [a] -> [a]
dus0 :: (Functor f, Num a) => (f (Tower a) -> Tower a) -> f [a] -> [a]
duF :: (Functor f, Functor g, Num a) => (f (Tower a) -> g (Tower a)) -> f (a, a) -> g a
duF' :: (Functor f, Functor g, Num a) => (f (Tower a) -> g (Tower a)) -> f (a, a) -> g (a, a)
dusF :: (Functor f, Functor g, Num a) => (f (Tower a) -> g (Tower a)) -> f [a] -> g [a]
dus0F :: (Functor f, Functor g, Num a) => (f (Tower a) -> g (Tower a)) -> f [a] -> g [a]

Documentation

data Tower a Source #

Tower is an AD Mode that calculates a tangent tower by forward AD, and provides fast diffsUU, diffsUF

Instances

(Num a, Bounded a) => Bounded (Tower a) Source #
Instance details Defined in Numeric.AD.Internal.Tower Methods minBound :: Tower a # maxBound :: Tower a #
(Num a, Enum a) => Enum (Tower a) Source #
Instance details Defined in Numeric.AD.Internal.Tower Methods succ :: Tower a -> Tower a # pred :: Tower a -> Tower a # toEnum :: Int -> Tower a # fromEnum :: Tower a -> Int # enumFrom :: Tower a -> [Tower a] # enumFromThen :: Tower a -> Tower a -> [Tower a] # enumFromTo :: Tower a -> Tower a -> [Tower a] # enumFromThenTo :: Tower a -> Tower a -> Tower a -> [Tower a] #
(Num a, Eq a) => Eq (Tower a) Source #
Instance details Defined in Numeric.AD.Internal.Tower Methods (==) :: Tower a -> Tower a -> Bool # (/=) :: Tower a -> Tower a -> Bool #
Floating a => Floating (Tower a) Source #
Instance details Defined in Numeric.AD.Internal.Tower Methods pi :: Tower a # exp :: Tower a -> Tower a # log :: Tower a -> Tower a # sqrt :: Tower a -> Tower a # (**) :: Tower a -> Tower a -> Tower a # logBase :: Tower a -> Tower a -> Tower a # sin :: Tower a -> Tower a # cos :: Tower a -> Tower a # tan :: Tower a -> Tower a # asin :: Tower a -> Tower a # acos :: Tower a -> Tower a # atan :: Tower a -> Tower a # sinh :: Tower a -> Tower a # cosh :: Tower a -> Tower a # tanh :: Tower a -> Tower a # asinh :: Tower a -> Tower a # acosh :: Tower a -> Tower a # atanh :: Tower a -> Tower a # log1p :: Tower a -> Tower a # expm1 :: Tower a -> Tower a # log1pexp :: Tower a -> Tower a # log1mexp :: Tower a -> Tower a #
Fractional a => Fractional (Tower a) Source #
Instance details Defined in Numeric.AD.Internal.Tower Methods (/) :: Tower a -> Tower a -> Tower a # recip :: Tower a -> Tower a # fromRational :: Rational -> Tower a #
Data a => Data (Tower a) Source #
Instance details Defined in Numeric.AD.Internal.Tower Methods gfoldl :: (forall d b. Data d => c (d -> b) -> d -> c b) -> (forall g. g -> c g) -> Tower a -> c (Tower a) # gunfold :: (forall b r. Data b => c (b -> r) -> c r) -> (forall r. r -> c r) -> Constr -> c (Tower a) # toConstr :: Tower a -> Constr # dataTypeOf :: Tower a -> DataType # dataCast1 :: Typeable t => (forall d. Data d => c (t d)) -> Maybe (c (Tower a)) # dataCast2 :: Typeable t => (forall d e. (Data d, Data e) => c (t d e)) -> Maybe (c (Tower a)) # gmapT :: (forall b. Data b => b -> b) -> Tower a -> Tower a # gmapQl :: (r -> r' -> r) -> r -> (forall d. Data d => d -> r') -> Tower a -> r # gmapQr :: (r' -> r -> r) -> r -> (forall d. Data d => d -> r') -> Tower a -> r # gmapQ :: (forall d. Data d => d -> u) -> Tower a -> [u] # gmapQi :: Int -> (forall d. Data d => d -> u) -> Tower a -> u # gmapM :: Monad m => (forall d. Data d => d -> m d) -> Tower a -> m (Tower a) # gmapMp :: MonadPlus m => (forall d. Data d => d -> m d) -> Tower a -> m (Tower a) # gmapMo :: MonadPlus m => (forall d. Data d => d -> m d) -> Tower a -> m (Tower a) #
Num a => Num (Tower a) Source #
Instance details Defined in Numeric.AD.Internal.Tower Methods (+) :: Tower a -> Tower a -> Tower a # (-) :: Tower a -> Tower a -> Tower a # (*) :: Tower a -> Tower a -> Tower a # negate :: Tower a -> Tower a # abs :: Tower a -> Tower a # signum :: Tower a -> Tower a # fromInteger :: Integer -> Tower a #
(Num a, Ord a) => Ord (Tower a) Source #
Instance details Defined in Numeric.AD.Internal.Tower Methods compare :: Tower a -> Tower a -> Ordering # (<) :: Tower a -> Tower a -> Bool # (<=) :: Tower a -> Tower a -> Bool # (>) :: Tower a -> Tower a -> Bool # (>=) :: Tower a -> Tower a -> Bool # max :: Tower a -> Tower a -> Tower a # min :: Tower a -> Tower a -> Tower a #
Real a => Real (Tower a) Source #
Instance details Defined in Numeric.AD.Internal.Tower Methods toRational :: Tower a -> Rational #
RealFloat a => RealFloat (Tower a) Source #
Instance details Defined in Numeric.AD.Internal.Tower Methods floatRadix :: Tower a -> Integer # floatDigits :: Tower a -> Int # floatRange :: Tower a -> (Int, Int) # decodeFloat :: Tower a -> (Integer, Int) # encodeFloat :: Integer -> Int -> Tower a # exponent :: Tower a -> Int # significand :: Tower a -> Tower a # scaleFloat :: Int -> Tower a -> Tower a # isNaN :: Tower a -> Bool # isInfinite :: Tower a -> Bool # isDenormalized :: Tower a -> Bool # isNegativeZero :: Tower a -> Bool # isIEEE :: Tower a -> Bool # atan2 :: Tower a -> Tower a -> Tower a #
RealFrac a => RealFrac (Tower a) Source #
Instance details Defined in Numeric.AD.Internal.Tower Methods properFraction :: Integral b => Tower a -> (b, Tower a) # truncate :: Integral b => Tower a -> b # round :: Integral b => Tower a -> b # ceiling :: Integral b => Tower a -> b # floor :: Integral b => Tower a -> b #
Show a => Show (Tower a) Source #
Instance details Defined in Numeric.AD.Internal.Tower Methods showsPrec :: Int -> Tower a -> ShowS # show :: Tower a -> String # showList :: [Tower a] -> ShowS #
Erf a => Erf (Tower a) Source #
Instance details Defined in Numeric.AD.Internal.Tower Methods erf :: Tower a -> Tower a # erfc :: Tower a -> Tower a # erfcx :: Tower a -> Tower a # normcdf :: Tower a -> Tower a #
InvErf a => InvErf (Tower a) Source #
Instance details Defined in Numeric.AD.Internal.Tower Methods inverf :: Tower a -> Tower a # inverfc :: Tower a -> Tower a # invnormcdf :: Tower a -> Tower a #
Num a => Mode (Tower a) Source #
Instance details Defined in Numeric.AD.Internal.Tower Associated Types type Scalar (Tower a) :: Type Source # Methods isKnownConstant :: Tower a -> Bool Source # isKnownZero :: Tower a -> Bool Source # auto :: Scalar (Tower a) -> Tower a Source # (^) :: Scalar (Tower a) -> Tower a -> Tower a Source # (^) :: Tower a -> Scalar (Tower a) -> Tower a Source # (^/) :: Tower a -> Scalar (Tower a) -> Tower a Source # zero :: Tower a Source #
Num a => Jacobian (Tower a) Source #
Instance details Defined in Numeric.AD.Internal.Tower Associated Types type D (Tower a) :: Type Source # Methods unary :: (Scalar (Tower a) -> Scalar (Tower a)) -> D (Tower a) -> Tower a -> Tower a Source # lift1 :: (Scalar (Tower a) -> Scalar (Tower a)) -> (D (Tower a) -> D (Tower a)) -> Tower a -> Tower a Source # lift1_ :: (Scalar (Tower a) -> Scalar (Tower a)) -> (D (Tower a) -> D (Tower a) -> D (Tower a)) -> Tower a -> Tower a Source # binary :: (Scalar (Tower a) -> Scalar (Tower a) -> Scalar (Tower a)) -> D (Tower a) -> D (Tower a) -> Tower a -> Tower a -> Tower a Source # lift2 :: (Scalar (Tower a) -> Scalar (Tower a) -> Scalar (Tower a)) -> (D (Tower a) -> D (Tower a) -> (D (Tower a), D (Tower a))) -> Tower a -> Tower a -> Tower a Source # lift2_ :: (Scalar (Tower a) -> Scalar (Tower a) -> Scalar (Tower a)) -> (D (Tower a) -> D (Tower a) -> D (Tower a) -> (D (Tower a), D (Tower a))) -> Tower a -> Tower a -> Tower a Source #
type Scalar (Tower a) Source #
Instance details Defined in Numeric.AD.Internal.Tower type Scalar (Tower a) = a
type D (Tower a) Source #
Instance details Defined in Numeric.AD.Internal.Tower type D (Tower a) = Tower a

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

Embed a constant

Taylor Series

taylor :: Fractional a => (Tower a -> Tower a) -> a -> a -> [a] Source #

taylor f x compute the Taylor series of f around x.

taylor0 :: Fractional a => (Tower a -> Tower a) -> a -> a -> [a] Source #

taylor0 f x compute the Taylor series of f around x, zero-padded.

Maclaurin Series

maclaurin :: Fractional a => (Tower a -> Tower a) -> a -> [a] Source #

maclaurin f compute the Maclaurin series of f

maclaurin0 :: Fractional a => (Tower a -> Tower a) -> a -> [a] Source #

maclaurin f compute the Maclaurin series of f, zero-padded

Derivatives

diff :: Num a => (Tower a -> Tower a) -> a -> a Source #

Compute the first derivative of a function (a -> a)

diff' :: Num a => (Tower a -> Tower a) -> a -> (a, a) Source #

Compute the answer and first derivative of a function (a -> a)

diffs :: Num a => (Tower a -> Tower a) -> a -> [a] Source #

Compute the answer and all derivatives of a function (a -> a)

diffs0 :: Num a => (Tower a -> Tower a) -> a -> [a] Source #

Compute the zero-padded derivatives of a function (a -> a)

diffsF :: (Functor f, Num a) => (Tower a -> f (Tower a)) -> a -> f [a] Source #

Compute the answer and all derivatives of a function (a -> f a)

diffs0F :: (Functor f, Num a) => (Tower a -> f (Tower a)) -> a -> f [a] Source #

Compute the zero-padded derivatives of a function (a -> f a)

Directional Derivatives

du :: (Functor f, Num a) => (f (Tower a) -> Tower a) -> f (a, a) -> a Source #

Compute a directional derivative of a function (f a -> a)

du' :: (Functor f, Num a) => (f (Tower a) -> Tower a) -> f (a, a) -> (a, a) Source #

Compute the answer and a directional derivative of a function (f a -> a)

dus :: (Functor f, Num a) => (f (Tower a) -> Tower a) -> f [a] -> [a] Source #

Given a function (f a -> a), and a tower of derivatives, compute the corresponding directional derivatives.

dus0 :: (Functor f, Num a) => (f (Tower a) -> Tower a) -> f [a] -> [a] Source #

Given a function (f a -> a), and a tower of derivatives, compute the corresponding directional derivatives, zero-padded

duF :: (Functor f, Functor g, Num a) => (f (Tower a) -> g (Tower a)) -> f (a, a) -> g a Source #

Compute a directional derivative of a function (f a -> g a)

duF' :: (Functor f, Functor g, Num a) => (f (Tower a) -> g (Tower a)) -> f (a, a) -> g (a, a) Source #

Compute the answer and a directional derivative of a function (f a -> g a)

dusF :: (Functor f, Functor g, Num a) => (f (Tower a) -> g (Tower a)) -> f [a] -> g [a] Source #

Given a function (f a -> g a), and a tower of derivatives, compute the corresponding directional derivatives

dus0F :: (Functor f, Functor g, Num a) => (f (Tower a) -> g (Tower a)) -> f [a] -> g [a] Source #

Given a function (f a -> g a), and a tower of derivatives, compute the corresponding directional derivatives, zero-padded