# Blow your mind

### From HaskellWiki

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

Line 212: | Line 212: | ||

=== 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). Here is one way to use lists to model polynomials. Since polynomials can support the same operations as integers, we model polynomials by making a list of numbers an instance of the Num type class. |
+ | 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). |

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

− | -- First we tell Haskell that we want to make lists (or [a]) an instance of Num. |
||

− | -- We refer to this instance of the Num type class as Num [a]. |
||

− | -- If you tried to use just: |
||

− | -- "instance Num [a] where" |
||

− | -- You'd get errors because the element type a is too general, too unconstrainted |
||

− | -- for what we need. |
||

− | -- So we add constraints to "a" by saying "Num a", this means whatever "a" is, it |
||

− | -- must be in the Num type class. |
||

− | instance Num a => Num [a] where |
+ | (f:fs) + (g:gs) = f+g : fs+gs -- (2) |

− | -- Next, we have to implement all the operations that instances of Num support. |
+ | fs + [] = fs -- (3a) |

− | -- A minimal set of operations is +, *, negate, abs, signum and fromInteger. |
+ | [] + gs = gs -- (3b) |

− | xs + ys = zipWith' (+) xs ys |
||

− | where |
||

− | zipWith' f [] ys = ys |
||

− | zipWith' f xs [] = xs |
||

− | zipWith' f (x:xs) (y:ys) = f x y : zipWith' f xs ys |
||

− | -- We define a new version of zipWith that returns a list as long as the longest |
||

− | -- of the two lists it is given. If we did not do this then when we add polynomials |
||

− | -- the result would be truncated to the length of the shorter polynomial. |
||

− | xs * ys = foldl1 (+) (padZeros partialProducts) |
||

− | where |
||

− | partialProducts = map (\x -> [x*y | y <- ys]) xs |
||

− | padZeros = map (\(z,zs) -> replicate z 0 ++ zs) . (zip [0..]) |
||

− | -- This function is sort of hard to explain.... basically [1,2,3] should correspond |
||

− | -- to the polynomial 1 + 2x + 3x^2. partialProducts does the steps of the multiplication |
||

− | -- just like you would by hand when multiplying polynomials. |
||

− | -- padZeros takes a list of polynomials and creates tuples of the form |
||

− | -- (offset, poly). If you notice when you add the partial products by hand |
||

− | -- that you have to shift the partial products to the left on each new line. |
||

− | -- we accomplish this by padding by zeros at the beginning of the partial product. |
||

− | -- Finally we use foldl1 to sum the partial products. Since they are polynomials |
||

− | -- They are added by the definition of plus we already gave. |
||

− | negate xs = map negate xs |
||

− | abs xs = map abs xs -- is this reasonable? |
||

− | signum xs = fromIntegral ((length xs)-1) |
||

− | -- signum isn't really defined for polynomials, but polynomials do have a concept |
||

− | -- of degree. We might as well reuse signum as the degree of the |
||

− | -- the polynomial. Notice that constants have degree zero. |
||

− | fromInteger x = [fromInteger x] |
||

− | -- This definition of fromInteger seems cyclical, it is left |
||

− | -- as an exercise to the reader to figure out why it is correct :) |
||

− | </haskell> |
||

− | The reader is encouraged to write a simple pretty printer that takes into account the many special cases of displaying a polynomial. For example, [1,3,-2, 0, 1,-1,0] should display as: <tt>-x^5 + x^4 - 2x^2 + 3x + 1</tt> |
||

− | Other execrises for the reader include writing <haskell>polyApply :: (Num a) => [a] -> a -> a </haskell> which evaluates the polynomial at a specific value or writing a differentiation function. |
+ | (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. |
||

+ | |||

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

+ | |||

+ | ====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 |
||

+ | <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.