{-# LANGUAGE CPP #-} {-# LANGUAGE BangPatterns #-} {-# LANGUAGE FlexibleContexts #-} {-# LANGUAGE ScopedTypeVariables #-} {-# LANGUAGE TypeFamilies #-} ----------------------------------------------------------------------------- -- | -- Copyright : (c) Edward Kmett 2015 -- License : BSD3 -- Maintainer : ekmett@gmail.com -- Stability : experimental -- Portability : GHC only -- ----------------------------------------------------------------------------- module Numeric.AD.Rank1.Newton.Double ( -- * Newton's Method (Forward) findZero , inverse , fixedPoint , extremum ) where import Prelude hiding (all, mapM) import Numeric.AD.Mode import Numeric.AD.Rank1.Forward (Forward) import qualified Numeric.AD.Rank1.Forward as Forward import Numeric.AD.Rank1.Forward.Double (ForwardDouble, diff') import Numeric.AD.Internal.On -- | 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.) If the stream becomes constant -- ("it converges"), no further elements are returned. -- -- Examples: -- -- >>> take 10 \$ findZero (\x->x^2-4) 1 -- [1.0,2.5,2.05,2.000609756097561,2.0000000929222947,2.000000000000002,2.0] findZero :: (ForwardDouble -> ForwardDouble) -> Double -> [Double] findZero f = go where go x = x : if x == xn then [] else go xn where (y,y') = diff' f x xn = x - y/y' {-# INLINE findZero #-} -- | The 'inverse' function inverts a scalar function using -- Newton's method; its output is a stream of increasingly accurate -- results. (Modulo the usual caveats.) If the stream becomes -- constant ("it converges"), no further elements are returned. -- -- Example: -- -- >>> last \$ take 10 \$ inverse sqrt 1 (sqrt 10) -- 10.0 inverse :: (ForwardDouble -> ForwardDouble) -> Double -> Double -> [Double] inverse f x0 y = findZero (\x -> f x - auto 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.) -- -- If the stream becomes constant ("it converges"), no further -- elements are returned. -- -- >>> last \$ take 10 \$ fixedPoint cos 1 -- 0.7390851332151607 fixedPoint :: (ForwardDouble -> ForwardDouble) -> Double -> [Double] 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.) If the stream -- becomes constant ("it converges"), no further elements are returned. -- -- >>> last \$ take 10 \$ extremum cos 1 -- 0.0 extremum :: (On (Forward ForwardDouble) -> On (Forward ForwardDouble)) -> Double -> [Double] extremum f = findZero (Forward.diff (off . f . On)) {-# INLINE extremum #-}