Portability  MPTCs, fundeps 

Stability  provisional 
Maintainer  Edward Kmett <ekmett@gmail.com> 
Safe Haskell  None 
The coiterative comonad generated by a comonad
 newtype CoiterT w a = CoiterT {
 runCoiterT :: w (a, CoiterT w a)
 type Coiter = CoiterT Identity
 coiter :: a > Coiter a > Coiter a
 runCoiter :: Coiter a > (a, Coiter a)
 unfold :: Comonad w => (w a > a) > w a > CoiterT w a
 class (Functor f, Comonad w) => ComonadCofree f w  w > f where
 unwrap :: w a > f (w a)
Documentation
Coiterative comonads represent nonterminating, productive computations.
They are the dual notion of iterative monads. While iterative computations produce no values or eventually terminate with one, coiterative computations constantly produce values and they never terminate.
It's simpler form, Coiter
, is an infinite stream of data. CoiterT
extends this so that each step of the computation can be performed in
a comonadic context.
The coiterative comonad transformer
This is the coiterative comonad generated by a comonad
CoiterT  

ComonadHoist CoiterT  
ComonadTrans CoiterT  
ComonadTraced m w => ComonadTraced m (CoiterT w)  
ComonadStore s w => ComonadStore s (CoiterT w)  
ComonadEnv e w => ComonadEnv e (CoiterT w)  
Comonad w => ComonadCofree Identity (CoiterT w)  
Functor w => Functor (CoiterT w)  
Typeable1 w => Typeable1 (CoiterT w)  
Foldable w => Foldable (CoiterT w)  
Traversable w => Traversable (CoiterT w)  
Comonad w => Comonad (CoiterT w)  
Eq (w (a, CoiterT w a)) => Eq (CoiterT w a)  
(Typeable1 w, Typeable a, Data (w (a, CoiterT w a)), Data a) => Data (CoiterT w a)  
Ord (w (a, CoiterT w a)) => Ord (CoiterT w a)  
Read (w (a, CoiterT w a)) => Read (CoiterT w a)  
Show (w (a, CoiterT w a)) => Show (CoiterT w a) 
The coiterative comonad
coiter :: a > Coiter a > Coiter aSource
Prepends a result to a coiterative computation.
runCoiter . uncurry coiter == id
runCoiter :: Coiter a > (a, Coiter a)Source
Extracts the first result from a coiterative computation.
uncurry coiter . runCoiter == id
Generating coiterative comonads
unfold :: Comonad w => (w a > a) > w a > CoiterT w aSource
Unfold a CoiterT
comonad transformer from a cokleisli arrow and an initial comonadic seed.
Cofree comonads
class (Functor f, Comonad w) => ComonadCofree f w  w > f whereSource
Allows you to peel a layer off a cofree comonad.
ComonadCofree Maybe NonEmpty  
ComonadCofree f w => ComonadCofree f (IdentityT w)  
Functor f => ComonadCofree f (Cofree f)  
Comonad w => ComonadCofree Identity (CoiterT w)  
(ComonadCofree f w, Semigroup m, Monoid m) => ComonadCofree f (TracedT m w)  
ComonadCofree f w => ComonadCofree f (StoreT s w)  
ComonadCofree f w => ComonadCofree f (EnvT e w)  
(Functor f, Comonad w) => ComonadCofree f (CofreeT f w)  
ComonadCofree (Const b) ((,) b) 
Example
This is literate Haskell! To run the example, open the source and copy this comment block into a new file with '.lhs' extension.
Many numerical approximation methods compute infinite sequences of results; each, hopefully, more accurate than the previous one.
Newton's method to find zeroes of a function is one such algorithm.
{# LANGUAGE FlexibleInstances, MultiParamTypeClasses, UndecidableInstances #}
import Control.Comonad.Trans.Coiter import Control.Comonad.Env import Control.Applicative import Data.Foldable (toList, find)
data Function = Function {  Function to find zeroes of function :: Double > Double,  Derivative of the function derivative :: Double > Double } data Result = Result {  Estimated zero of the function value :: Double,  Estimated distance to the actual zero xerror :: Double,  How far is value from being an actual zero; that is,  the difference between @0@ and @f value@ ferror :: Double } deriving (Show) data Outlook = Outlook { result :: Result,  Whether the result improves in future steps progress :: Bool } deriving (Show)
To make our lives easier, we will store the problem at hand using the Env environment comonad.
type Solution a = CoiterT (Env Function) a
Problems consist of a function and its derivative as the environment, and an initial value.
type Problem = Env Function Double
We can express an iterative algorithm using unfold over an initial environment.
newton :: Problem > Solution Double newton = unfold (\wd > let f = asks function wd in let df = asks derivative wd in let x = extract wd in x  f x / df x)
To estimate the error, we look forward one position in the stream. The next value will be much more precise than the current one, so we can consider it as the actual result.
We know that the exact value of a function at one of it's zeroes is 0. So,
ferror
can be computed exactly as abs (f a  f 0) == abs (f a)
estimateError :: Solution Double > Result estimateError s = let a:a':_ = toList s in let f = asks function s in Result { value = a, xerror = abs $ a  a', ferror = abs $ f a }
To get a sense of when the algorithm is making any progress, we can sample the future and check if the result improves at all.
estimateOutlook :: Int > Solution Result > Outlook estimateOutlook sampleSize solution = let sample = map ferror $ take sampleSize $ tail $ toList solution in let result = extract solution in Outlook { result = result, progress = ferror result > minimum sample }
To compute the square root of c
, we solve the equation x*x  c = 0
. We will
stop whenever the accuracy of the result doesn't improve in the next 5 steps.
The starting value for our algorithm is c
itself. One could compute a better
estimate, but the algorithm converges fast enough that it's not really worth it.
squareRoot :: Double > Maybe Result squareRoot c = let problem = flip env c (Function { function = (\x > x*x  c), derivative = (\x > 2*x) }) in fmap result $ find (not . progress) $ newton problem =>> estimateError =>> estimateOutlook 5
This program will output the result together with the error.
main :: IO () main = putStrLn $ show $ squareRoot 4