```{-# LANGUAGE Rank2Types, BangPatterns, ScopedTypeVariables #-}
-----------------------------------------------------------------------------
-- |
-- Module      :  Numeric.AD.Newton
-- Copyright   :  (c) Edward Kmett 2010
-- License     :  BSD3
-- Maintainer  :  ekmett@gmail.com
-- Stability   :  experimental
-- Portability :  GHC only
--
-----------------------------------------------------------------------------

module Numeric.AD.Newton
(
-- * Newton's Method (Forward AD)
findZero
, inverse
, fixedPoint
, extremum
-- * Gradient Ascent/Descent (Reverse AD)
, gradientDescent
, gradientAscent
-- * Exposed Types
, UU, UF, FU, FF
, AD(..)
, Mode(..)
) where

import Prelude hiding (all)
import Data.Foldable (all)
import Data.Traversable (Traversable)
import Numeric.AD.Types
import Numeric.AD.Classes
import Numeric.AD.Mode.Forward (diff, diff')
import Numeric.AD.Mode.Reverse (gradWith')
import Numeric.AD.Internal.Composition

-- | The 'findZero' function finds a zero of a scalar function using
-- Newton's method; its output is a stream of increasingly accurate
-- results.  (Modulo the usual caveats.)
--
-- Examples:
--
--  > take 10 \$ findZero (\\x->x^2-4) 1  -- converge to 2.0
--
--  > module Data.Complex
--  > take 10 \$ findZero ((+1).(^2)) (1 :+ 1)  -- converge to (0 :+ 1)@
--
findZero :: Fractional a => UU a -> a -> [a]
findZero f = go
where
go x = x : if y == 0 then [] else go (x - y/y')
where
(y,y') = diff' f x
{-# INLINE findZero #-}

-- | The 'inverseNewton' function inverts a scalar function using
-- Newton's method; its output is a stream of increasingly accurate
-- results.  (Modulo the usual caveats.)
--
-- Example:
--
-- > take 10 \$ inverseNewton sqrt 1 (sqrt 10)  -- converges to 10
--
inverse :: Fractional a => UU a -> a -> a -> [a]
inverse f x0 y = findZero (\x -> f x - lift y) x0
{-# INLINE inverse  #-}

-- | The 'fixedPoint' function find a fixedpoint of a scalar
-- function using Newton's method; its output is a stream of
-- increasingly accurate results.  (Modulo the usual caveats.)
--
-- > take 10 \$ fixedPoint cos 1 -- converges to 0.7390851332151607
fixedPoint :: Fractional a => UU a -> a -> [a]
fixedPoint f = findZero (\x -> f x - x)
{-# INLINE fixedPoint #-}

-- | The 'extremum' function finds an extremum of a scalar
-- function using Newton's method; produces a stream of increasingly
-- accurate results.  (Modulo the usual caveats.)
--
-- > take 10 \$ extremum cos 1 -- convert to 0
extremum :: Fractional a => UU a -> a -> [a]
extremum f = findZero (diff (decomposeMode . f . composeMode))
{-# INLINE extremum #-}

-- | The 'gradientDescent' function performs a multivariate
-- optimization, based on the naive-gradient-descent in the file
-- @stalingrad\/examples\/flow-tests\/pre-saddle-1a.vlad@ from the
-- VLAD compiler Stalingrad sources.  Its output is a stream of
-- increasingly accurate results.  (Modulo the usual caveats.)
--
-- It uses reverse mode automatic differentiation to compute the gradient.
gradientDescent :: (Traversable f, Fractional a, Ord a) => FU f a -> f a -> [f a]
gradientDescent f x0 = go x0 fx0 xgx0 0.1 (0 :: Int)
where
(fx0, xgx0) = gradWith' (,) f x0
go x fx xgx !eta !i
| eta == 0     = [] -- step size is 0
| fx1 > fx     = go x fx xgx (eta/2) 0 -- we stepped too far
| zeroGrad xgx = [] -- gradient is 0
| otherwise    = x1 : if i == 10
then go x1 fx1 xgx1 (eta*2) 0
else go x1 fx1 xgx1 eta (i+1)
where
zeroGrad = all (\(_,g) -> g == 0)
x1 = fmap (\(xi,gxi) -> xi - eta * gxi) xgx
(fx1, xgx1) = gradWith' (,) f x1
{-# INLINE gradientDescent #-}

gradientAscent :: (Traversable f, Fractional a, Ord a) => FU f a -> f a -> [f a]
gradientAscent f = gradientDescent (negate . f)
{-# INLINE gradientAscent #-}
```