# Blow your mind

### From HaskellWiki

(Simplify and generalize obscure polynomial algorithms) |
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=== Polynomials === | === Polynomials === | ||

− | In abstract algebra you learn that polynomials can be used the same way integers are used given the right assumptions about their coefficients and roots. Specifically, polynomials support addition, subtraction, multiplication and sometimes division. It also turns out that one way to think of polynomials is that they are just lists of numbers (their coefficients). | + | In abstract algebra you learn that polynomials can be used the same way integers are used given the right assumptions about their coefficients and roots. Specifically, polynomials support addition, subtraction, multiplication and sometimes division. It also turns out that one way to think of polynomials is that they are just lists of numbers (their coefficients). |

− | + | instance Num a => Num [a] where -- (1) | |

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− | + | (f:fs) + (g:gs) = f+g : fs+gs -- (2) | |

− | + | fs + [] = fs -- (3a) | |

− | + | [] + gs = gs -- (3b) | |

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− | + | (f:fs) * (g:gs) = f*g : [f]*gs + fs*(g:gs) -- (4) | |

+ | _ * _ = [] -- (5) | ||

+ | |||

+ | ====Explanation==== | ||

+ | (1) puts lists into type class Num, the class to which operators + and * belong, provided the list elements are in class Num. | ||

+ | |||

+ | Lists are ordered by increasing powers. Thus <tt>f:fs</tt> means <tt>f+x*fs</tt> in algebraic notation. (2) and (4) follow from these algebraic identities: | ||

+ | |||

+ | (f+x*fs) + (g+x*gs) = f+g + x*(fs+gs) | ||

+ | (f+x*fs) * (g+x*gs) = f*g + x*(f*gs + fs*(g+x*gs)) | ||

+ | |||

+ | (3) and (5) handle list ends. | ||

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+ | The bracketed <tt>[f]</tt> in (4) avoids mixed arithmetic, which Haskell doesn't support. | ||

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+ | ====Comments==== | ||

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+ | The methods are qualitatively different from ordinary array-based methods; there is no vestige of subscripting or counting of terms. | ||

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+ | The methods are suitable for on-line computation. Only | ||

+ | <i>n</i> terms of each input must be seen before the <i>n</i>-th term | ||

+ | of output is produced. | ||

+ | |||

+ | Thus the methods work on infinite series as well as polynomials. | ||

+ | |||

+ | Integer power comes for free. This example tests the cubing of (1+x): | ||

+ | |||

+ | [1, 1]^3 == [1, 3, 3, 1] | ||

See also | See also |

## Revision as of 14:04, 28 December 2006

Useful Idioms that will blow your mind (unless you already know them :)

This collection is supposed to be comprised of short, useful, cool, magical examples, which should incite the reader's curiosity and (hopefully) lead him to a deeper understanding of advanced Haskell concepts. At a later time I might add explanations to the more obscure solutions. I've also started providing several alternatives to give more insight into the interrelations of solutions.

More examples are always welcome, especially "obscure" monadic ones.

## Contents |

## 1 List/String Operations

-- split at whitespace -- "hello world" -> ["hello","world"] words takeWhile (not . null) . unfoldr (Just . (second $ drop 1) . break (==' ')) fix (\f l -> if null l then [] else let (s,e) = break (==' ') l in s:f (drop 1 e)) -- splitting in two (alternating) -- "1234567" -> ("1357", "246") -- the lazy match with ~ is necessary for efficiency, especially enabling processing of infinite lists foldr (\a ~(x,y) -> (a:y,x)) ([],[]) (map snd *** map snd) . partition (even . fst) . zip [0..] transpose . unfoldr (\a -> toMaybe (null a) (splitAt 2 a)) -- this one uses the solution to the next problem in a nice way :) toMaybe b x = if b then Just x else Nothing -- splitting into lists of length N -- "1234567" -> ["12", "34", "56", "7"] unfoldr (\a -> toMaybe (null a) (splitAt 2 a)) takeWhile (not . null) . unfoldr (Just . splitAt 2) -- sorting by a custom function -- length -> ["abc", "ab", "a"] -> ["a", "ab", "abc"] comparing f x y = compare (f x) (f y) sortBy (comparing length) map snd . sortBy (comparing fst) . map (length &&& id) -- the so called "Schwartzian Transform" for computationally more expensive -- functions. -- comparing adjacent elements rises xs = zipWith (<) xs (tail xs) -- lazy substring search -- "ell" -> "hello" -> True substr a b = any (a `isPrefixOf`) $ tails b -- multiple splitAt's: -- splitAts [2,5,0,3] [1..15] == [[1,2],[3,4,5,6,7],[],[8,9,10],[11,12,13,14,15]] splitAts = foldr (\n r -> splitAt n >>> second r >>> uncurry (:)) return

## 2 Mathematical Sequences, etc

-- factorial -- 6 -> 720 product [1..6] foldl1 (*) [1..6] (!!6) $ scanl (*) 1 [1..] fix (\f n -> if n <= 0 then 1 else n * f (n-1)) -- powers of two sequence iterate (*2) 1 unfoldr (\z -> Just (z,2*z)) 1 -- fibonacci sequence unfoldr (\(f1,f2) -> Just (f1,(f2,f1+f2))) (0,1) fibs = 0:1:zipWith (+) fibs (tail fibs) fib = 0:scanl (+) 1 fib -- pascal triangle pascal = iterate (\row -> zipWith (+) ([0] ++ row) (row ++ [0])) [1] -- prime numbers -- example of a memoising caf (??) primes = sieve [2..] where sieve (p:x) = p : sieve [ n | n <- x, n `mod` p > 0 ] unfoldr sieve [2..] where sieve (p:x) = Just(p, [ n | n <- x, n `mod` p > 0 ]) -- or if you want to use the Sieve of Eratosthenes.. diff [] l = l diff l [] = l diff xl@(x:xs) yl@(y:ys) | x < y = x:diff xs yl | x > y = diff xl ys | otherwise = diff xs ys esieve [] = [] esieve (p:ps) = p:esieve (diff ps (iterate (+p) p)) eprimes = esieve [2..] -- enumerating the rationals (see [1]) rats :: [Rational] rats = iterate next 1 where next x = recip (fromInteger n+1-y) where (n,y) = properFraction x

[1] Gibbons, Lest, Bird - Enumerating the Rationals

## 3 Monad Magic

-- all combinations of a list of lists. -- these solutions are all pretty much equivalent in that they run -- in the List Monad. the "sequence" solution has the advantage of -- scaling to N sublists. -- "12" -> "45" -> ["14", "15", "24", "25"] sequence ["12", "45"] [[x,y] | x <- "12", y <- "45"] do { x <- "12"; y <- "45"; return [x,y] } "12" >>= \a -> "45" >>= \b -> return [a,b] -- all combinations of letters (inits . repeat) ['a'..'z'] >>= sequence -- apply a list of functions to an argument -- even -> odd -> 4 -> [True,False] map ($4) [even,odd] sequence [even,odd] 4 -- apply a function to two other function the same argument -- (lifting to the Function Monad (->)) -- even 4 && odd 4 -> False liftM2 (&&) even odd 4 liftM2 (>>) putStrLn return "hello" -- forward function concatenation (*3) >>> (+1) $ 2 foldl1 (flip (.)) [(+1),(*2)] 500 -- perform functions in/on a monad, lifting fmap (+2) (Just 2) liftM2 (+) (Just 4) (Just 2) -- [still to categorize] (id >>= (+) >>= (+) >>= (+)) 3 -- (3+3)+(3+3) = 12 (join . liftM2) (*) (+3) 5 -- 64 mapAccumL (\acc n -> (acc+n,acc+n)) 0 [1..10] -- interesting for fac, fib, ... do f <- [not, not]; d <- [True, False]; return (f d) -- [False,True,False,True] do { Just x <- [Nothing, Just 5, Nothing, Just 6, Just 7, Nothing]; return x }

## 4 Other

-- simulating lisp's cond case () of () | 1 > 2 -> True | 3 < 4 -> False | otherwise -> True -- match a constructor -- this is better than applying all the arguments, because this way the -- data type can be changed without touching the code (ideally). case a of Just{} -> True _ -> False {- TODO, IDEAS: more fun with monad, monadPlus (liftM, ap, guard, when) fun with arrows (second, first, &&&, ***) liftM, ap lazy search (searching as traversal of lazy structures) innovative data types (i.e. having fun with Maybe sequencing) LINKS: bananas, envelopes, ... (generic traversal) why functional fp matters (lazy search, ...) -}

### 4.1 Polynomials

In abstract algebra you learn that polynomials can be used the same way integers are used given the right assumptions about their coefficients and roots. Specifically, polynomials support addition, subtraction, multiplication and sometimes division. It also turns out that one way to think of polynomials is that they are just lists of numbers (their coefficients).

instance Num a => Num [a] where -- (1)

(f:fs) + (g:gs) = f+g : fs+gs -- (2) fs + [] = fs -- (3a) [] + gs = gs -- (3b)

(f:fs) * (g:gs) = f*g : [f]*gs + fs*(g:gs) -- (4) _ * _ = [] -- (5)

#### 4.1.1 Explanation

(1) puts lists into type class Num, the class to which operators + and * belong, provided the list elements are in class Num.

Lists are ordered by increasing powers. Thus `f:fs` means `f+x*fs` in algebraic notation. (2) and (4) follow from these algebraic identities:

(f+x*fs) + (g+x*gs) = f+g + x*(fs+gs) (f+x*fs) * (g+x*gs) = f*g + x*(f*gs + fs*(g+x*gs))

(3) and (5) handle list ends.

The bracketed `[f]` in (4) avoids mixed arithmetic, which Haskell doesn't support.

#### 4.1.2 Comments

The methods are qualitatively different from ordinary array-based methods; there is no vestige of subscripting or counting of terms.

The methods are suitable for on-line computation. Only
*n* terms of each input must be seen before the *n*-th term
of output is produced.

Thus the methods work on infinite series as well as polynomials.

Integer power comes for free. This example tests the cubing of (1+x):

[1, 1]^3 == [1, 3, 3, 1]

See also

- Pointfree
- NumericPrelude: Polynomials
- Add Polynomials
- Solve differential equations in terms of power series.