(make Matrix definition portable)
(Added subsection "Treating pairs and lists the same way")
Revision as of 16:03, 12 May 2007
1 Sorted lists
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)
See also point-free programming.
2.1 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)
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
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]
4 Data.Either extensions
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