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Prelude extensions

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See also [[pointfree|point-free]] programming.
See also [[pointfree|point-free]] programming.
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== Matrix ==
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== Matrices ==
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Sometimes you just want to multiply 2 matrices, like
+
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+
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[[1,2],[3,4]]*[[1,2],[3,4]]
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-
The following makes it possible, but requires -fglasgow-exts :
+
 +
A simple representation of matrices is as lists of lists of numbers:
<haskell>
<haskell>
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instance Num a => Num [[a]] where
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newtype Matrix a = Matrix [[a]] deriving (Eq, Show)
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(+) = zipWith (zipWith (+))
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(-) = zipWith (zipWith (-))
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negate = map (map negate)
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(*) x y = map (matrixXvector x) y
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where
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matrixXvector m v = foldl vectorsum (repeat 0) $ zipWith vectorXnumber m v
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vectorXnumber v n = map (n*) v
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vectorsum x y = zipWith (+) x y
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</haskell>
</haskell>
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These matrices may be made an instance of <hask>Num</hask>
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Or, in a more elegant way:
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(though the definitions of <hask>abs</hask> and <hask>signum</hask> are just fillers):
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<haskell>
<haskell>
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import Data.List
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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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instance Num a => Num [[a]] where
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Matrix as - Matrix bs = Matrix (zipWith (zipWith (-)) as bs)
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(+) = zipWith (zipWith (+))
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Matrix as * Matrix bs =
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(-) = zipWith (zipWith (-))
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Matrix [[sum $ zipWith (*) a b | b <- transpose bs] | a <- as]
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negate = map (map negate)
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negate (Matrix as) = Matrix (map (map negate) as)
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n * m = [ [ sum $ zipWith (*) v w | w <- transpose n ] | v <- m ]
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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
 +
<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]
</haskell>
</haskell>

Revision as of 22:34, 9 May 2007

Contents


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


2 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) -> (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
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.

3 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]

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

5 See also

Prelude function suggestions

Retrieved from "http://www.haskell.org/haskellwiki/Prelude_extensions"

Category: Code