Numeric.AD.Rank1.Tower

Higher order derivatives via a "dual number tower".

Documentation

data Tower a

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)

Methods

minBound :: Tower a

maxBound :: Tower a

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

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)

Methods

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

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

Floating a => Floating (Tower a)

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)

Methods

(/) :: Tower a -> Tower a -> Tower a

recip :: Tower a -> Tower a

fromRational :: Rational -> Tower a

Data a => Data (Tower a)

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)

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)

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)

Methods

toRational :: Tower a -> Rational

RealFloat a => RealFloat (Tower a)

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)

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)

Methods

showsPrec :: Int -> Tower a -> ShowS

show :: Tower a -> String

showList :: [Tower a] -> ShowS

Erf a => Erf (Tower a)

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)

Methods

inverf :: Tower a -> Tower a

inverfc :: Tower a -> Tower a

invnormcdf :: Tower a -> Tower a

Num a => Mode (Tower a)

Associated Types

type Scalar (Tower a) :: *

Methods

isKnownConstant :: Tower a -> Bool

isKnownZero :: Tower a -> Bool

auto :: Scalar (Tower a) -> Tower a

(*^) :: Scalar (Tower a) -> Tower a -> Tower a

(^*) :: Tower a -> Scalar (Tower a) -> Tower a

(^/) :: Tower a -> Scalar (Tower a) -> Tower a

zero :: Tower a

Num a => Jacobian (Tower a)

Associated Types

type D (Tower a) :: *

Methods

unary :: (Scalar (Tower a) -> Scalar (Tower a)) -> D (Tower a) -> Tower a -> Tower a

lift1 :: (Scalar (Tower a) -> Scalar (Tower a)) -> (D (Tower a) -> D (Tower a)) -> Tower a -> Tower a

lift1_ :: (Scalar (Tower a) -> Scalar (Tower a)) -> (D (Tower a) -> D (Tower a) -> D (Tower a)) -> Tower a -> Tower a

binary :: (Scalar (Tower a) -> Scalar (Tower a) -> Scalar (Tower a)) -> D (Tower a) -> D (Tower a) -> Tower a -> Tower a -> Tower a

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

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

type Scalar (Tower a)

type Scalar (Tower a) = a

type D (Tower a)

type D (Tower a) = Tower a

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

Embed a constant

Taylor Series

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

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

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

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

Maclaurin Series

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

maclaurin f compute the Maclaurin series of f

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

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

Derivatives

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

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

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

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

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

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

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

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

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

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]

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

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)

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]

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]

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

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)

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]

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]

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