The following are versions of standard prelude functions, but intended for sorted lists. The advantage is that they frequently reduce execution time by an O(n). The disadvantage is that the elements have to be members of Ord, and the lists have to be already sorted.
-- Eliminates duplicate entries from the list, where duplication is defined -- by the 'eq' function. The last value is kept. sortedNubBy :: (a -> a -> Bool) -> [a] -> [a] sortedNubBy eq (x1 : xs@(x2 : _)) = if eq x1 x2 then sortedNubBy eq xs else x1 : sortedNubBy eq xs sortedNubBy _ xs = xs sortedNub :: (Eq a) => [a] -> [a] sortedNub = sortedNubBy (==) -- Merge two sorted lists into a new sorted list. Where elements are equal -- the element from the first list is taken first. mergeBy :: (a -> a -> Ordering) -> [a] -> [a] -> [a] mergeBy cmp xs@(x1:xs1) ys@(y1:ys1) = if cmp x1 y1 == GT then y1 : mergeBy cmp xs ys1 else x1 : mergeBy cmp xs1 ys mergeBy _  ys = ys mergeBy _ xs  = xs merge :: (Ord a) => [a] -> [a] -> [a] merge = mergeBy compare
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) -> (c, b) 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