{-# LANGUAGE MultiParamTypeClasses, ScopedTypeVariables, FlexibleContexts #-} module Math.Root.Finder where import Control.Monad.Instances () import Data.Tagged -- |General interface for numerical root finders. class RootFinder r a b where -- |@initRootFinder f x0 x1@: Initialize a root finder for the given -- function with the initial bracketing interval (x0,x1). initRootFinder :: (a -> b) -> a -> a -> r a b -- |Step a root finder for the given function (which should generally -- be the same one passed to @initRootFinder@), refining the finder's -- estimate of the location of a root. stepRootFinder :: (a -> b) -> r a b -> r a b -- |Extract the finder's current estimate of the position of a root. estimateRoot :: r a b -> a -- |Extract the finder's current estimate of the upper bound of the -- distance from @estimateRoot@ to an actual root in the function. -- -- Generally, @estimateRoot r@ +- @estimateError r@ should bracket -- a root of the function. estimateError :: r a b -> a -- |Test whether a root finding algorithm has converged to a given -- relative accuracy. converged :: (Num a, Ord a) => a -> r a b -> Bool converged xacc r = abs (estimateError r) <= abs xacc -- |Default number of steps after which root finding will be deemed -- to have failed. Purely a convenience used to control the behavior -- of built-in functions such as 'findRoot' and 'traceRoot'. The -- default value is 250. defaultNSteps :: Tagged (r a b) Int defaultNSteps = Tagged 250 -- |@traceRoot f x0 x1 mbEps@ initializes a root finder and repeatedly -- steps it, returning each step of the process in a list. When the algorithm -- terminates or the 'defaultNSteps' limit is exceeded, the list ends. -- Termination criteria depends on @mbEps@; if it is of the form @Just eps@ -- then convergence to @eps@ is used (using the @converged@ method of the -- root finder). Otherwise, the trace is not terminated until subsequent -- states are equal (according to '=='). This is a stricter condition than -- convergence to 0; subsequent states may have converged to zero but as long -- as any internal state changes the trace will continue. traceRoot :: (Eq (r a b), RootFinder r a b, Num a, Ord a) => (a -> b) -> a -> a -> Maybe a -> [r a b] traceRoot f a b xacc = go nSteps start (stepRootFinder f start) where Tagged nSteps = (const :: Tagged a b -> a -> Tagged a b) defaultNSteps start start = initRootFinder f a b -- lookahead 1; if tracing with no convergence test, apply a -- naive test to bail out if the root stops changing. This is -- provided because that's not always the same as convergence to 0, -- and the main purpose of this function is to watch what actually -- happens inside the root finder. go n x next | maybe (x==next) (flip converged x) xacc = [x] | n <= 0 = [] | otherwise = x : go (n-1) next (stepRootFinder f next) -- |@findRoot f x0 x1 eps@ initializes a root finder and repeatedly -- steps it. When the algorithm converges to @eps@ or the 'defaultNSteps' -- limit is exceeded, the current best guess is returned, with the @Right@ -- constructor indicating successful convergence or the @Left@ constructor -- indicating failure to converge. findRoot :: (RootFinder r a b, Num a, Ord a) => (a -> b) -> a -> a -> a -> Either (r a b) (r a b) findRoot f a b xacc = go nSteps start where Tagged nSteps = (const :: Tagged a b -> a -> Tagged a b) defaultNSteps start start = initRootFinder f a b go n x | converged xacc x = Right x | n <= 0 = Left x | otherwise = go (n-1) (stepRootFinder f x) -- |A useful constant: 'eps' is (for most 'RealFloat' types) the smallest -- positive number such that @1 + eps /= 1@. eps :: RealFloat a => a eps = eps' where eps' = encodeFloat 1 (1 - floatDigits eps')