Haskell Unit 5: map and filter - ? Haskell Unit 5: map and filter Antoni Diller 26 July 2011 The functions

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Haskell Unit 5: map and filterAntoni Diller26 July 2011The functions map and filterThe higher-order function map can be defined like this:map :: (a -> b) -> [a] -> [b]map f [] = []map f (x:xs) = f x : map f xsIntuitively, what map does is to apply the function f to each element of the list that itis applied to:map f [ x1, x2, ..., xn ] = [ f x1, f x2, ..., f xn ]The higher-order function filter can be defined like this:filter :: (a -> Bool) -> [a] -> [a]filter _ [] = []filter pred (x:xs)| pred x = x:xs| otherwise = xs where xs = filter pred xsFor example,filter even [ 1 .. 20 ] = [ 2, 4, 6, 8, 10, 12, 14, 16, 18, 20 ]Every list that can be defined using ZF-expressions can also be defined using map andfilter and visa versa. First, I define map and filter using ZF-expressions:map f xs = [ f x | x Newtons method for finding positive square rootsLet x be the positive number whose square root you are trying to find. Then if y > 0is a guess, then (y + x/y)/2 is a better guess. For example, say we want to find thepositive square root of 27.3. Let us guess 1. Applying Newtons method, a better guessis 14.15. Applying Newtons method again, a still better guess is 8.03966. ApplyingNewtons method again, a still better guess is 5.71766. Applying Newtons methodagain, a still better guess is 5.24617. And so on. Newtons method can be programmedstraightforwardly in Haskell as follows:root :: Float -> Floatroot x = rootiter x 1rootiter :: Float -> Float -> Floatrootiter x y| satisfactory x y = y| otherwise = rootiter x (improve x y)satisfactory :: Float -> Float -> Boolsatisfactory x y = abs (y*y - x) < 0.01improve :: Float -> Float -> Floatimprove x y = (y + x/y)/2This, however, is quite an imperative solution. A more functional solution usesthe predefined Haskell function iterate:iterate :: (a -> a) -> a -> [a]iterate f x = x : iterate f (f x)The function iterate generates an infinte list. For example, iterate sqInt 2 wouldproduce: [ 2, 4, 16, 256, 65536, 4294967296, .. ]. A more functional so-lution is, therefore:root :: Float -> Floatroot x = head (filter (satisfactory x) (iterate (improve x) 1))satisfactory :: Float -> Float -> Boolsatisfactory x y = abs (y*y - x) < 0.01improve :: Float -> Float -> Floatimprove x y = (y + x/y)/22The sieve of Eratosthesnes for generating primes(1) Make a list of all the positive integers starting at 2.(2) The first number on the list is prime; call it p.(3) Construct a new list in which all multiples of p have been removed.(4) Repeat the above from step (2).For example,[ [ 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, ... ],[ 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, ... ],[ 5, 7, 11, 13, 17, 19, 23, 25, 29, 31, 35, ... ],[ 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, ... ],[ 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, ... ], ... ]In Haskell the sieve can be programmed like this:primes = map head (iterate sieve [2..])sieve (p:xs) = [ x | x b) -> (c -> a) -> c -> b.Function composition is useful for many reasons. One of them is that f (g ( h (i (j(k x))))), say, can be written as (f . g . h . i . j . k ) x; noting thatfunction composition is associative. This usefulness can be illustrated by means of thefollowing problem: Find the sum of the cubes of all the numbers divisible by 7 in alist xs of integers. The solution is as follows:answer :: [Int] -> Intanswer xs = sum (map cube (filter by7 xs))cube :: Int -> Intcube x = x * x * xby7 :: Int -> Boolby7 x = x mod 7 == 0But using function composition this can be written more clearly as follows:answer :: [Int] -> Intanswer = sum . map cube . filter by73MemoisationIn mathematics the Fibonacci numbers are usually defined like this:fib 0 = 0fib 1 = 1fib i = fib (i - 1) + fib (i - 2)Although this works in Haskell it is extremely inefficient. A more efficient definitionprevents the re-evaluation of the same Fibonacci number. The values are stored in alist. The definition is as follows:fib j = fiblist !! jfiblist = map f [ 0 .. ]wheref 0 = 0f 1 = 1f i = fiblist !! (i - 1) + fiblist !! (i - 2)Intuitively, fiblist contains the infinite list of Fibonacci numbers. Each element, saythe ith can be expressed in at least two ways, namely as fib i and as fiblist !!i. This version of the Fibonacci numbers is very much more efficient.4

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