Difference between revisions of "Prelude extensions"
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+ | __TOC__ |
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− | == Sorted lists == |
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− | |||
− | 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. |
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− | |||
− | <haskell> |
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− | -- Eliminates duplicate entries from the list, where duplication is defined |
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− | -- by the 'eq' function. The last value is kept. |
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− | sortedNubBy :: (a -> a -> Bool) -> [a] -> [a] |
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− | sortedNubBy eq (x1 : xs@(x2 : _)) = |
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− | if eq x1 x2 then sortedNubBy eq xs else x1 : sortedNubBy eq xs |
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− | sortedNubBy _ xs = xs |
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− | |||
− | sortedNub :: (Eq a) => [a] -> [a] |
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− | sortedNub = sortedNubBy (==) |
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− | |||
− | -- Merge two sorted lists into a new sorted list. Where elements are equal |
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− | -- the element from the first list is taken first. |
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− | mergeBy :: (a -> a -> Ordering) -> [a] -> [a] -> [a] |
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− | mergeBy cmp xs@(x1:xs1) ys@(y1:ys1) = |
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− | if cmp x1 y1 == GT |
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− | then y1 : mergeBy cmp xs ys1 |
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− | else x1 : mergeBy cmp xs1 ys |
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− | mergeBy _ [] ys = ys |
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− | mergeBy _ xs [] = xs |
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− | |||
− | merge :: (Ord a) => [a] -> [a] -> [a] |
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− | merge = mergeBy compare |
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− | </haskell> |
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− | |||
== Tuples == |
== Tuples == |
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-- | Apply a function to the second element of a pair |
-- | Apply a function to the second element of a pair |
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− | mapSnd :: (b -> c) -> (a, b) -> ( |
+ | mapSnd :: (b -> c) -> (a, b) -> (a, c) |
mapSnd f (a, b) = (a, f b) |
mapSnd f (a, b) = (a, f b) |
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</haskell> |
</haskell> |
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+ | [http://haskell.org/ghc/docs/latest/html/libraries/fgl/Data-Graph-Inductive-Query-Monad.html#1 Data.Graph.Inductive.Query.Monad module (section ''Additional Graph Utilities'')] contains <hask>mapFst</hask>, <hask>mapSnd</hask>, and also a function <hask>><</hask> corresponding to <hask>mapPair</hask>. Another implementation of these functions in the standard libraries: using <hask>first</hask>, <hask>second</hask>, <hask>***</hask> arrow operations overloaded for functions (as special arrows), see [http://haskell.org/ghc/docs/latest/html/libraries/base/Control-Arrow.html Control.Arrow] module, or [[Arrow]] HaskellWiki page. |
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− | == Matrix == |
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+ | See also [[pointfree|point-free]] programming. |
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− | Sometimes you just want to multiply 2 matrices, like |
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+ | === Treating pairs and lists in the same way === |
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− | [[1,2],[3,4]]*[[1,2],[3,4]] |
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+ | We can define a Pair class which allows us to process both pairs and non-empty lists using the same operator: |
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− | The following makes it possible, but requires -fglasgow-exts : |
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<haskell> |
<haskell> |
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+ | import Control.Arrow ((***)) |
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− | instance Num a => Num [[a]] where |
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+ | |||
− | abs x = map (map abs) x |
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+ | infixl 4 <**> |
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− | (+) x y = zipWith (zipWith (+)) x y |
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+ | |||
− | (*) x y = map (matrixXvector x) y |
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− | where |
+ | class Pair p x y | p -> x, p -> y where |
− | + | toPair :: p -> (x, y) |
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+ | (<**>) :: (x -> a -> b) -> (y -> a) -> p -> b |
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− | matrixXvector m v = foldl vectorsum [] $ zipWith vectorXnumber m v |
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+ | (<**>) f g = uncurry id . (f *** g) . toPair |
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− | vectorXnumber :: [a] -> a -> [a] |
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+ | |||
− | vectorXnumber v n = map (n*) v |
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+ | instance Pair (a, b) a b where |
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− | vectorsum :: [a] -> [a] -> [a] |
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− | + | toPair = id |
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+ | |||
− | vectorsum x [] = x |
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+ | instance Pair [a] a [a] where |
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− | vectorsum x y = zipWith (+) x y |
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+ | toPair l = (head l, tail l) |
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</haskell> |
</haskell> |
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+ | |||
+ | == Matrices == |
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+ | |||
+ | A simple representation of matrices is as lists of lists of numbers: |
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+ | <haskell> |
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+ | newtype Matrix a = Matrix [[a]] deriving (Eq, Show) |
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+ | </haskell> |
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+ | These matrices may be made an instance of <hask>Num</hask> |
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+ | (though the definitions of <hask>abs</hask> and <hask>signum</hask> are just fillers): |
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+ | <haskell> |
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+ | instance Num a => Num (Matrix a) where |
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+ | Matrix as + Matrix bs = Matrix (zipWith (zipWith (+)) as bs) |
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+ | Matrix as - Matrix bs = Matrix (zipWith (zipWith (-)) as bs) |
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+ | Matrix as * Matrix bs = |
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+ | Matrix [[sum $ zipWith (*) a b | b <- transpose bs] | a <- as] |
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+ | negate (Matrix as) = Matrix (map (map negate) as) |
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+ | fromInteger x = Matrix (iterate (0:) (fromInteger x : repeat 0)) |
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+ | abs m = m |
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+ | signum _ = 1 |
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+ | </haskell> |
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+ | The <hask>fromInteger</hask> 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. |
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+ | Applying the linear transformation defined by a matrix to a vector is |
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+ | <haskell> |
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+ | apply :: Num a => Matrix a -> [a] -> [a] |
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+ | apply (Matrix as) b = [sum (zipWith (*) a b) | a <- as] |
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+ | </haskell> |
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+ | |||
+ | == Data.Either extensions == |
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+ | |||
+ | |||
+ | <haskell> |
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+ | import Data.Either |
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+ | |||
+ | either', trigger, trigger_, switch :: (a -> b) -> (a -> b) -> Either a a -> Either b b |
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+ | |||
+ | either' f g (Left x) = Left (f x) |
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+ | either' f g (Right x) = Right (g x) |
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+ | |||
+ | trigger f g (Left x) = Left (f x) |
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+ | trigger f g (Right x) = Left (g x) |
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+ | |||
+ | trigger_ f g (Left x) = Right (f x) |
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+ | trigger_ f g (Right x) = Right (g x) |
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+ | |||
+ | switch f g (Left x) = Right (f x) |
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+ | switch f g (Right x) = Left (g x) |
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+ | |||
+ | sure :: (a->b) -> Either a a -> b |
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+ | sure f = either f f |
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+ | |||
+ | sure' :: (a->b) -> Either a a -> Either b b |
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+ | sure' f = either' f f |
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+ | </haskell> |
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+ | |||
+ | == Schönfinkel & Curry's amalgamation combinator, for Haskell == |
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+ | |||
+ | |||
+ | <haskell> |
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+ | sperse :: (a -> b -> c) -> (a -> b) -> a -> c |
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+ | sperse f g x = f x (g x) |
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+ | </haskell> |
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+ | |||
+ | == Curry and Feys's paradoxical combinator, for Haskell == |
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+ | |||
+ | |||
+ | <haskell> |
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+ | yet :: (a -> a) -> a |
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+ | yet f = f (yet f) |
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+ | </haskell> |
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+ | |||
+ | == Hyperstrict evaluation[http://foldoc.org/hyperstrict <span></span>] == |
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+ | |||
+ | |||
+ | <haskell> |
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+ | compel :: a -> a -- primitive |
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+ | </haskell> |
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+ | |||
+ | == See also == |
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+ | [[List function suggestions]] |
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+ | |||
+ | |||
+ | [[Category:Code]] |
Latest revision as of 02:22, 18 May 2020
Tuples
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 mapFst
, mapSnd
, and also a function ><
corresponding to mapPair
. Another implementation of these functions in the standard libraries: using first
, second
, ***
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)
Matrices
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 Num
(though the definitions of abs
and 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
The 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]
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
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)
Hyperstrict evaluation
compel :: a -> a -- primitive