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

Numeric.AD.Mode.Tower

Contents

Taylor Series
Maclaurin Series
Derivatives
Directional Derivatives

Description

Higher order derivatives via a "dual number tower".

Synopsis

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

Documentation

data AD s a Source #

Instances

Bounded a => Bounded (AD s a) Source #
Instance details Defined in Numeric.AD.Internal.Type Methods minBound :: AD s a # maxBound :: AD s a #
Enum a => Enum (AD s a) Source #
Instance details Defined in Numeric.AD.Internal.Type Methods succ :: AD s a -> AD s a # pred :: AD s a -> AD s a # toEnum :: Int -> AD s a # fromEnum :: AD s a -> Int # enumFrom :: AD s a -> [AD s a] # enumFromThen :: AD s a -> AD s a -> [AD s a] # enumFromTo :: AD s a -> AD s a -> [AD s a] # enumFromThenTo :: AD s a -> AD s a -> AD s a -> [AD s a] #
Eq a => Eq (AD s a) Source #
Instance details Defined in Numeric.AD.Internal.Type Methods (==) :: AD s a -> AD s a -> Bool # (/=) :: AD s a -> AD s a -> Bool #
Floating a => Floating (AD s a) Source #
Instance details Defined in Numeric.AD.Internal.Type Methods pi :: AD s a # exp :: AD s a -> AD s a # log :: AD s a -> AD s a # sqrt :: AD s a -> AD s a # (**) :: AD s a -> AD s a -> AD s a # logBase :: AD s a -> AD s a -> AD s a # sin :: AD s a -> AD s a # cos :: AD s a -> AD s a # tan :: AD s a -> AD s a # asin :: AD s a -> AD s a # acos :: AD s a -> AD s a # atan :: AD s a -> AD s a # sinh :: AD s a -> AD s a # cosh :: AD s a -> AD s a # tanh :: AD s a -> AD s a # asinh :: AD s a -> AD s a # acosh :: AD s a -> AD s a # atanh :: AD s a -> AD s a # log1p :: AD s a -> AD s a # expm1 :: AD s a -> AD s a # log1pexp :: AD s a -> AD s a # log1mexp :: AD s a -> AD s a #
Fractional a => Fractional (AD s a) Source #
Instance details Defined in Numeric.AD.Internal.Type Methods (/) :: AD s a -> AD s a -> AD s a # recip :: AD s a -> AD s a # fromRational :: Rational -> AD s a #
Num a => Num (AD s a) Source #
Instance details Defined in Numeric.AD.Internal.Type Methods (+) :: AD s a -> AD s a -> AD s a # (-) :: AD s a -> AD s a -> AD s a # (*) :: AD s a -> AD s a -> AD s a # negate :: AD s a -> AD s a # abs :: AD s a -> AD s a # signum :: AD s a -> AD s a # fromInteger :: Integer -> AD s a #
Ord a => Ord (AD s a) Source #
Instance details Defined in Numeric.AD.Internal.Type Methods compare :: AD s a -> AD s a -> Ordering # (<) :: AD s a -> AD s a -> Bool # (<=) :: AD s a -> AD s a -> Bool # (>) :: AD s a -> AD s a -> Bool # (>=) :: AD s a -> AD s a -> Bool # max :: AD s a -> AD s a -> AD s a # min :: AD s a -> AD s a -> AD s a #
Read a => Read (AD s a) Source #
Instance details Defined in Numeric.AD.Internal.Type Methods readsPrec :: Int -> ReadS (AD s a) # readList :: ReadS [AD s a] # readPrec :: ReadPrec (AD s a) # readListPrec :: ReadPrec [AD s a] #
Real a => Real (AD s a) Source #
Instance details Defined in Numeric.AD.Internal.Type Methods toRational :: AD s a -> Rational #
RealFloat a => RealFloat (AD s a) Source #
Instance details Defined in Numeric.AD.Internal.Type Methods floatRadix :: AD s a -> Integer # floatDigits :: AD s a -> Int # floatRange :: AD s a -> (Int, Int) # decodeFloat :: AD s a -> (Integer, Int) # encodeFloat :: Integer -> Int -> AD s a # exponent :: AD s a -> Int # significand :: AD s a -> AD s a # scaleFloat :: Int -> AD s a -> AD s a # isNaN :: AD s a -> Bool # isInfinite :: AD s a -> Bool # isDenormalized :: AD s a -> Bool # isNegativeZero :: AD s a -> Bool # isIEEE :: AD s a -> Bool # atan2 :: AD s a -> AD s a -> AD s a #
RealFrac a => RealFrac (AD s a) Source #
Instance details Defined in Numeric.AD.Internal.Type Methods properFraction :: Integral b => AD s a -> (b, AD s a) # truncate :: Integral b => AD s a -> b # round :: Integral b => AD s a -> b # ceiling :: Integral b => AD s a -> b # floor :: Integral b => AD s a -> b #
Show a => Show (AD s a) Source #
Instance details Defined in Numeric.AD.Internal.Type Methods showsPrec :: Int -> AD s a -> ShowS # show :: AD s a -> String # showList :: [AD s a] -> ShowS #
Erf a => Erf (AD s a) Source #
Instance details Defined in Numeric.AD.Internal.Type Methods erf :: AD s a -> AD s a # erfc :: AD s a -> AD s a # erfcx :: AD s a -> AD s a # normcdf :: AD s a -> AD s a #
InvErf a => InvErf (AD s a) Source #
Instance details Defined in Numeric.AD.Internal.Type Methods inverf :: AD s a -> AD s a # inverfc :: AD s a -> AD s a # invnormcdf :: AD s a -> AD s a #
Mode a => Mode (AD s a) Source #
Instance details Defined in Numeric.AD.Internal.Type Associated Types type Scalar (AD s a) :: Type Source # Methods isKnownConstant :: AD s a -> Bool Source # isKnownZero :: AD s a -> Bool Source # auto :: Scalar (AD s a) -> AD s a Source # (^) :: Scalar (AD s a) -> AD s a -> AD s a Source # (^) :: AD s a -> Scalar (AD s a) -> AD s a Source # (^/) :: AD s a -> Scalar (AD s a) -> AD s a Source # zero :: AD s a Source #
type Scalar (AD s a) Source #
Instance details Defined in Numeric.AD.Internal.Type type Scalar (AD s a) = Scalar a

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 => (forall s. AD s (Tower a) -> AD s (Tower a)) -> a -> a -> [a] Source #

taylor0 :: Fractional a => (forall s. AD s (Tower a) -> AD s (Tower a)) -> a -> a -> [a] Source #

Maclaurin Series

maclaurin :: Fractional a => (forall s. AD s (Tower a) -> AD s (Tower a)) -> a -> [a] Source #

maclaurin0 :: Fractional a => (forall s. AD s (Tower a) -> AD s (Tower a)) -> a -> [a] Source #

Derivatives

diff :: Num a => (forall s. AD s (Tower a) -> AD s (Tower a)) -> a -> a Source #

diff' :: Num a => (forall s. AD s (Tower a) -> AD s (Tower a)) -> a -> (a, a) Source #

diffs :: Num a => (forall s. AD s (Tower a) -> AD s (Tower a)) -> a -> [a] Source #

diffs0 :: Num a => (forall s. AD s (Tower a) -> AD s (Tower a)) -> a -> [a] Source #

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

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

Directional Derivatives

du :: (Functor f, Num a) => (forall s. f (AD s (Tower a)) -> AD s (Tower a)) -> f (a, a) -> a Source #

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

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

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

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

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

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

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