```{-# LANGUAGE MultiParamTypeClasses, ScopedTypeVariables, FlexibleContexts #-}
module Math.Root.Finder
( RootFinder(..)
, getDefaultNSteps
, runRootFinder
, traceRoot
, findRoot, findRootN
, eps
, realFloatDefaultNSteps
) where

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

-- |Convenience function to access 'defaultNSteps' for a root finder,
-- which requires a little bit of type-gymnastics.
--
-- This function does not evaluate its argument.
getDefaultNSteps :: RootFinder r a b => r a b -> Int
getDefaultNSteps rf = nSteps
where
Tagged nSteps =
(const :: Tagged a b -> a -> Tagged a b)
defaultNSteps rf

-- |General-purpose driver for stepping a root finder.  Given a \"control\"
-- function, the function being searched, and an initial 'RootFinder' state,
-- @runRootFinder step f state@ repeatedly steps the root-finder and passes
-- each intermediate state, along with a count of steps taken, to @step@.
--
-- The @step@ funtion will be called with the following arguments:
--
-- [@ n :: 'Int' @]
--  The number of steps taken thus far
--
-- [@ currentState :: r a b @]
--  The current state of the root finder
--
-- [@ continue :: c @]
--  The result of the \"rest\" of the iteration
--
-- For example, the following function simply iterates a root finder
-- and returns every intermediate state (similar to 'traceRoot'):
--
-- > iterateRoot :: RootFinder r a b => (a -> b) -> a -> a -> [r a b]
-- > iterateRoot f a b = runRootFinder (const (:)) f (initRootFinder f a b)
--
-- And the following function simply iterates the root finder to
-- convergence or throws an error after a given number of steps:
--
-- > solve :: (RootFinder r a b, RealFloat a)
-- >       => Int -> (a -> b) -> a -> a -> r a b
-- > solve maxN f a b = runRootFinder step f (initRootFinder f a b)
-- >    where
-- >        step n x continue
-- >            | converged eps x   = x
-- >            | n > maxN          = error "solve: step limit exceeded"
-- >            | otherwise         = continue
--
runRootFinder :: (RootFinder r a b) =>
(Int -> r a b -> c -> c) -> (a -> b) -> r a b -> c
runRootFinder cons f = go 0
where
go n x = n `seq` cons n x (go (n+1) (stepRootFinder f x))

-- |@traceRoot f x0 x1 mbEps@ initializes a root finder and repeatedly
-- steps it, returning each step of the process in a list.  No step limit
-- is imposed.
--
-- 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 mbEps = runRootFinder cons f start
where
start = initRootFinder f a b

cons _n x rest = x : if done x rest then [] else rest

-- 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.
done = case mbEps of
Nothing     -> \x (next:_)  -> x == next
Just xacc   -> \x _rest     -> converged xacc x

-- |@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 = result
where
result = findRootN nSteps f a b xacc
nSteps = getDefaultNSteps (either id id result)

-- |Like 'findRoot' but with a specified limit on the number of steps (rather
-- than using 'defaultNSteps').
findRootN :: (RootFinder r a b, Num a, Ord a) =>
Int -> (a -> b) -> a -> a -> a -> Either (r a b) (r a b)
findRootN nSteps f a b xacc = runRootFinder step f start
where
start = initRootFinder f a b

step n x continue
| converged xacc x  = Right x
| n > nSteps        = Left  x
| otherwise         = continue

-- |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')

-- |For 'RealFloat' types, computes a suitable default step limit based
-- on the precision of the type and a margin of error.
realFloatDefaultNSteps :: RealFloat a => Float -> Tagged (r a b) Int
realFloatDefaultNSteps margin = nSteps
where
f :: (Int -> Tagged (r a b) Int) -> (a -> Int) -> a -> Tagged (r a b) Int
f = (.)

nSteps :: RealFloat a => Tagged (r a b) Int
nSteps = f Tagged n 0

n :: RealFloat a => a -> Int
n x = round \$ product
[ margin
, realToFrac (floatDigits x)
, logBase 2 (realToFrac (floatRadix x))
]

```