{-# LANGUAGE BangPatterns, DeriveDataTypeable #-} -- | -- Module : Statistics.Math.RootFinding -- Copyright : (c) 2011 Bryan O'Sullivan -- License : BSD3 -- -- Maintainer : bos@serpentine.com -- Stability : experimental -- Portability : portable -- -- Haskell functions for finding the roots of mathematical functions. module Statistics.Math.RootFinding ( Root(..) , fromRoot , ridders -- * References -- $references ) where import Statistics.Function.Comparison import Control.Applicative import Control.Monad (MonadPlus(..), ap) import Data.Typeable (Typeable) -- | The result of searching for a root of a mathematical function. data Root a = NotBracketed -- ^ The function does not have opposite signs when -- evaluated at the lower and upper bounds of the search. | SearchFailed -- ^ The search failed to converge to within the given -- error tolerance after the given number of iterations. | Root a -- ^ A root was successfully found. deriving (Eq, Read, Show, Typeable) instance Functor Root where fmap _ NotBracketed = NotBracketed fmap _ SearchFailed = SearchFailed fmap f (Root a) = Root (f a) instance Monad Root where NotBracketed >>= _ = NotBracketed SearchFailed >>= _ = SearchFailed Root a >>= m = m a return = Root instance MonadPlus Root where mzero = SearchFailed r@(Root _) `mplus` _ = r _ `mplus` p = p instance Applicative Root where pure = Root (<*>) = ap instance Alternative Root where empty = SearchFailed r@(Root _) <|> _ = r _ <|> p = p -- | Returns either the result of a search for a root, or the default -- value if the search failed. fromRoot :: a -- ^ Default value. -> Root a -- ^ Result of search for a root. -> a fromRoot _ (Root a) = a fromRoot a _ = a -- | Use the method of Ridders to compute a root of a function. -- -- The function must have opposite signs when evaluated at the lower -- and upper bounds of the search (i.e. the root must be bracketed). ridders :: Double -- ^ Absolute error tolerance. -> (Double,Double) -- ^ Lower and upper bounds for the search. -> (Double -> Double) -- ^ Function to find the roots of. -> Root Double ridders tol (lo,hi) f | flo == 0 = Root lo | fhi == 0 = Root hi | flo*fhi > 0 = NotBracketed -- root is not bracketed | otherwise = go lo flo hi fhi 0 where go !a !fa !b !fb !i -- Root is bracketed within 1 ulp. No improvement could be made | within 1 a b = Root a -- Root is found. Check that f(m) == 0 is nessesary to ensure -- that root is never passed to 'go' | fm == 0 = Root m | fn == 0 = Root n | d < tol = Root n -- Too many iterations performed. Fail | i >= (100 :: Int) = SearchFailed -- Ridder's approximation coincide with one of old -- bounds. Revert to bisection | n == a || n == b = case () of _| fm*fa < 0 -> go a fa m fm (i+1) | otherwise -> go m fm b fb (i+1) -- Proceed as usual | fn*fm < 0 = go n fn m fm (i+1) | fn*fa < 0 = go a fa n fn (i+1) | otherwise = go n fn b fb (i+1) where d = abs (b - a) dm = (b - a) * 0.5 !m = a + dm !fm = f m !dn = signum (fb - fa) * dm * fm / sqrt(fm*fm - fa*fb) !n = m - signum dn * min (abs dn) (abs dm - 0.5 * tol) !fn = f n !flo = f lo !fhi = f hi -- $references -- -- * Ridders, C.F.J. (1979) A new algorithm for computing a single -- root of a real continuous function. -- /IEEE Transactions on Circuits and Systems/ 26:979–980.