It is often necessary to apply functions to either the first or the second part of a pair. This is often considered a form of mapping (like map from Data.List).
-- | Apply a function to the first element of a pair mapFst :: (a -> c) -> (a, b) -> (c, b) mapFst f (a, b) = (f a, b) -- | Apply a function to the second element of a pair mapSnd :: (b -> c) -> (a, b) -> (a, c) mapSnd f (a, b) = (a, f b) -- | Apply a function to both elements of a pair mapPair :: (a -> c, b -> d) -> (a, b) -> (c, d) mapPair (f, g) (a, b) = (f a, g b)
Data.Graph.Inductive.Query.Monad module (section Additional Graph Utilities) contains
mapSnd, and also a function
>< corresponding to
mapPair. Another implementation of these functions in the standard libraries: using
*** arrow operations overloaded for functions (as special arrows), see Control.Arrow module, or Arrow HaskellWiki page.
See also point-free programming.
Treating pairs and lists in the same way
We can define a Pair class which allows us to process both pairs and non-empty lists using the same operator:
import Control.Arrow ((***)) infixl 4 <**> class Pair p x y | p -> x, p -> y where toPair :: p -> (x, y) (<**>) :: (x -> a -> b) -> (y -> a) -> p -> b (<**>) f g = uncurry id . (f *** g) . toPair instance Pair (a, b) a b where toPair = id instance Pair [a] a [a] where toPair l = (head l, tail l)
A simple representation of matrices is as lists of lists of numbers:
newtype Matrix a = Matrix [[a]] deriving (Eq, Show)
These matrices may be made an instance of
(though the definitions of
signum are just fillers):
instance Num a => Num (Matrix a) where Matrix as + Matrix bs = Matrix (zipWith (zipWith (+)) as bs) Matrix as - Matrix bs = Matrix (zipWith (zipWith (-)) as bs) Matrix as * Matrix bs = Matrix [[sum $ zipWith (*) a b | b <- transpose bs] | a <- as] negate (Matrix as) = Matrix (map (map negate) as) fromInteger x = Matrix (iterate (0:) (fromInteger x : repeat 0)) abs m = m signum _ = 1
fromInteger method builds an infinite matrix, but addition and subtraction work even with infinite matrices, and multiplication works as long as either the first matrix is of finite width or the second is of finite height.
Applying the linear transformation defined by a matrix to a vector is
apply :: Num a => Matrix a -> [a] -> [a] apply (Matrix as) b = [sum (zipWith (*) a b) | a <- as]
import Data.Either either', trigger, trigger_, switch :: (a -> b) -> (a -> b) -> Either a a -> Either b b either' f g (Left x) = Left (f x) either' f g (Right x) = Right (g x) trigger f g (Left x) = Left (f x) trigger f g (Right x) = Left (g x) trigger_ f g (Left x) = Right (f x) trigger_ f g (Right x) = Right (g x) switch f g (Left x) = Right (f x) switch f g (Right x) = Left (g x) sure :: (a->b) -> Either a a -> b sure f = either f f sure' :: (a->b) -> Either a a -> Either b b sure' f = either' f f
Schönfinkel & Curry's amalgamation combinator, for Haskell
sperse :: (a -> b -> c) -> (a -> b) -> a -> c sperse f g x = f x (g x)
Curry and Feys's paradoxical combinator, for Haskell
yet :: (a -> a) -> a yet f = f (yet f)