{-# LANGUAGE Rank2Types #-}
{-# LANGUAGE BangPatterns #-}
-----------------------------------------------------------------------------
-- |
-- Copyright : (c) Edward Kmett 2010-2015
-- License : BSD3
-- Maintainer : ekmett@gmail.com
-- Stability : experimental
-- Portability : GHC only
--
-- Higher order derivatives via a \"dual number tower\".
--
-----------------------------------------------------------------------------
module Numeric.AD.Rank1.Tower
( Tower
, auto
-- * Taylor Series
, taylor
, taylor0
-- * Maclaurin Series
, maclaurin
, maclaurin0
-- * Derivatives
, diff -- first derivative of (a -> a)
, diff' -- answer and first derivative of (a -> a)
, diffs -- answer and all derivatives of (a -> a)
, diffs0 -- zero padded derivatives of (a -> a)
, diffsF -- answer and all derivatives of (a -> f a)
, diffs0F -- zero padded derivatives of (a -> f a)
-- * Directional Derivatives
, du -- directional derivative of (f a -> a)
, du' -- answer and directional derivative of (f a -> a)
, dus -- answer and all directional derivatives of (f a -> a)
, dus0 -- answer and all zero padded directional derivatives of (f a -> a)
, duF -- directional derivative of (f a -> g a)
, duF' -- answer and directional derivative of (f a -> g a)
, dusF -- answer and all directional derivatives of (f a -> g a)
, dus0F -- answer and all zero padded directional derivatives of (f a -> g a)
) where
import Control.Applicative ((<$>))
import Numeric.AD.Internal.Tower
import Numeric.AD.Mode
-- | Compute the answer and all derivatives of a function @(a -> a)@
diffs :: Num a => (Tower a -> Tower a) -> a -> [a]
diffs f a = getADTower $ apply f a
{-# INLINE diffs #-}
-- | Compute the zero-padded derivatives of a function @(a -> a)@
diffs0 :: Num a => (Tower a -> Tower a) -> a -> [a]
diffs0 f a = zeroPad (diffs f a)
{-# INLINE diffs0 #-}
-- | Compute the answer and all derivatives of a function @(a -> f a)@
diffsF :: (Functor f, Num a) => (Tower a -> f (Tower a)) -> a -> f [a]
diffsF f a = getADTower <$> apply f a
{-# INLINE diffsF #-}
-- | Compute the zero-padded derivatives of a function @(a -> f a)@
diffs0F :: (Functor f, Num a) => (Tower a -> f (Tower a)) -> a -> f [a]
diffs0F f a = (zeroPad . getADTower) <$> apply f a
{-# INLINE diffs0F #-}
-- | @taylor f x@ compute the Taylor series of @f@ around @x@.
taylor :: Fractional a => (Tower a -> Tower a) -> a -> a -> [a]
taylor f x dx = go 1 1 (diffs f x) where
go !n !acc (a:as) = a * acc : go (n + 1) (acc * dx / n) as
go _ _ [] = []
-- | @taylor0 f x@ compute the Taylor series of @f@ around @x@, zero-padded.
taylor0 :: Fractional a => (Tower a -> Tower a) -> a -> a -> [a]
taylor0 f x dx = zeroPad (taylor f x dx)
{-# INLINE taylor0 #-}
-- | @maclaurin f@ compute the Maclaurin series of @f@
maclaurin :: Fractional a => (Tower a -> Tower a) -> a -> [a]
maclaurin f = taylor f 0
{-# INLINE maclaurin #-}
-- | @maclaurin f@ compute the Maclaurin series of @f@, zero-padded
maclaurin0 :: Fractional a => (Tower a -> Tower a) -> a -> [a]
maclaurin0 f = taylor0 f 0
{-# INLINE maclaurin0 #-}
-- | Compute the first derivative of a function @(a -> a)@
diff :: Num a => (Tower a -> Tower a) -> a -> a
diff f = d . diffs f
{-# INLINE diff #-}
-- | Compute the answer and first derivative of a function @(a -> a)@
diff' :: Num a => (Tower a -> Tower a) -> a -> (a, a)
diff' f = d' . diffs f
{-# INLINE diff' #-}
-- | Compute a directional derivative of a function @(f a -> a)@
du :: (Functor f, Num a) => (f (Tower a) -> Tower a) -> f (a, a) -> a
du f = d . getADTower . f . fmap withD
{-# INLINE du #-}
-- | Compute the answer and a directional derivative of a function @(f a -> a)@
du' :: (Functor f, Num a) => (f (Tower a) -> Tower a) -> f (a, a) -> (a, a)
du' f = d' . getADTower . f . fmap withD
{-# INLINE du' #-}
-- | 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
duF f = fmap (d . getADTower) . f . fmap withD
{-# INLINE duF #-}
-- | Compute the answer and 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)
duF' f = fmap (d' . getADTower) . f . fmap withD
{-# INLINE duF' #-}
-- | Given a function @(f a -> a)@, and a tower of derivatives, compute the corresponding directional derivatives.
dus :: (Functor f, Num a) => (f (Tower a) -> Tower a) -> f [a] -> [a]
dus f = getADTower . f . fmap tower
{-# INLINE dus #-}
-- | Given a function @(f a -> a)@, and a tower of derivatives, compute the corresponding directional derivatives, zero-padded
dus0 :: (Functor f, Num a) => (f (Tower a) -> Tower a) -> f [a] -> [a]
dus0 f = zeroPad . getADTower . f . fmap tower
{-# INLINE dus0 #-}
-- | Given a function @(f a -> g a)@, and a tower of derivatives, compute the corresponding directional derivatives
dusF :: (Functor f, Functor g, Num a) => (f (Tower a) -> g (Tower a)) -> f [a] -> g [a]
dusF f = fmap getADTower . f . fmap tower
{-# INLINE dusF #-}
-- | Given a function @(f a -> g a)@, and a tower of derivatives, compute the corresponding directional derivatives, zero-padded
dus0F :: (Functor f, Functor g, Num a) => (f (Tower a) -> g (Tower a)) -> f [a] -> g [a]
dus0F f = fmap getADTower . f . fmap tower
{-# INLINE dus0F #-}