Data.Vector.Fixed

Generic API for vectors with fixed length.

For encoding of vector size library uses Peano naturals defined in the library. At come point in the future it would make sense to switch to new GHC type level numerals.

Common pitfalls

Library provide instances for tuples. But there's a catch. Tuples are monomorphic in element type. Let consider 2-tuple (Int,Int). Vector type v is (,) Int and only allowed element type is Int. Because of that we cannot change element type and following code will fail:

>>> map (== 1) ((1,2) :: (Int,Int))

<interactive>:3:1:
    Couldn't match type `Int' with `Bool'
    In the expression: F.map (== 1) ((1, 2) :: (Int, Int))
    In an equation for `it': it = map (== 1) ((1, 2) :: (Int, Int))

To make it work we need to change vector type as well. Functions from module Data.Vector.Fixed.Generic provide this functionality.

>>> map (== 1) ((1,2) :: (Int,Int)) :: (Bool,Bool)
(True,False)

Vector type class

Vector size

type family Dim v

Size of vector expressed as type-level natural.

Instances

type Dim Complex = N2

type Dim Empty = Z

type Dim Only = S Z

type Dim ((,) a) = N2

type Dim (Proxy *) = Z

type Dim (ContVec n) = n

type Dim (VecList n) = n

type Dim (Vec n) = n

type Dim (Vec n) = n

type Dim (Vec n) = n

type Dim (Vec n) = n

type Dim ((,,) a b) = N3

type Dim ((,,,) a b c) = N4

type Dim ((,,,,) a b c d) = N5

type Dim ((,,,,,) a b c d e) = N6

type Dim ((,,,,,,) a b c d e f) = S N6

data Z

Type level zero

Instances

Arity Z

Typeable * Z

NatIso Z 0

Arity n => Index Z (S n)

type Add Z n = n

type Fn Z a b = b

data S n

Successor of n

Instances

Arity n => Index Z (S n)

Arity n => Arity (S n)

(NatIso k ((-) n 1), (~) * (ToPeano ((-) n 1)) k, (~) * (ToPeano n) (S k), (~) Nat n ((+) 1 ((-) n 1))) => NatIso (S k) n

Index k n => Index (S k) (S n)

Typeable (* -> *) S

type Add (S n) k = S (Add n k)

type Fn (S n) a b = a -> Fn n a b

Synonyms for small numerals

type N1 = S Z

type N2 = S N1

type N3 = S N2

type N4 = S N3

type N5 = S N4

type N6 = S N5

Type class

class Arity (Dim v) => Vector v a where

Type class for vectors with fixed length. Instance should provide two functions: one to create vector and another for vector deconstruction. They must obey following law:

inspect v construct = v

Minimal complete definition

construct, inspect

Methods

construct :: Fun (Dim v) a (v a)

N-ary function for creation of vectors.

inspect :: v a -> Fun (Dim v) a b -> b

Deconstruction of vector.

basicIndex :: v a -> Int -> a

Optional more efficient implementation of indexing. Shouldn't be used directly, use ! instead.

Instances

RealFloat a => Vector Complex a

Vector Empty a

Vector Only a

(~) * b a => Vector ((,) b) a

Note this instance (and other instances for tuples) is essentially monomorphic in element type. Vector type v of 2 element tuple (Int,Int) is (,) Int so it will only work with elements of type Int.

Vector (Proxy *) a

Arity n => Vector (ContVec n) a

Arity n => Vector (VecList n) a

Arity n => Vector (Vec n) a

(Arity n, Prim a) => Vector (Vec n) a

Unbox n a => Vector (Vec n) a

(Arity n, Storable a) => Vector (Vec n) a

((~) * b a, (~) * c a) => Vector ((,,) b c) a

((~) * b a, (~) * c a, (~) * d a) => Vector ((,,,) b c d) a

((~) * b a, (~) * c a, (~) * d a, (~) * e a) => Vector ((,,,,) b c d e) a

((~) * b a, (~) * c a, (~) * d a, (~) * e a, (~) * f a) => Vector ((,,,,,) b c d e f) a

((~) * b a, (~) * c a, (~) * d a, (~) * e a, (~) * f a, (~) * g a) => Vector ((,,,,,,) b c d e f g) a

class (Vector (v n) a, Dim (v n) ~ n) => VectorN v n a

Vector parametrized by length. In ideal world it should be:

forall n. (Arity n, Vector (v n) a, Dim (v n) ~ n) => VectorN v a

Alas polymorphic constraints aren't allowed in haskell.

Instances

Arity n => VectorN ContVec n a

Arity n => VectorN VecList n a

Arity n => VectorN Vec n a

(Arity n, Prim a) => VectorN Vec n a

Unbox n a => VectorN Vec n a

(Arity n, Storable a) => VectorN Vec n a

class Arity n

Type class for handling n-ary functions.

Minimal complete definition

accum, applyFun, applyFunM, arity, reverseF, gunfoldF, witSum

Instances

Arity Z

Arity n => Arity (S n)

newtype Fun n a b

Newtype wrapper which is used to make Fn injective. It's also a reader monad.

Constructors

Fun

Fields

unFun :: Fn n a b

Instances

Arity n => Monad (Fun n a)

Arity n => Functor (Fun n a)

Arity n => Applicative (Fun n a)

length :: forall v a. Arity (Dim v) => v a -> Int

Length of vector. Function doesn't evaluate its argument.

Constructors

There are several ways to construct fixed vectors except using their constructor if it's available. For small ones it's possible to use functions mk1, mk2, etc.

>>> mk3 'a' 'b' 'c' :: (Char,Char,Char)
('a','b','c')

Another option is to create tuple and convert it to desired vector type. For example:

v = convert (x,y,z)

It will work on if type of v is know from elsewhere. Same trick could be used to pattern match on the vector with opaque representation using view patterns

function :: Vec N3 Double -> ...
function (convert -> (x,y,z)) = ...

Third way is to use variadic function mkN. It works similarly to printf except it produces result of type ContVec which should be converted to vector of desired type by vector:

>>> vector $ mkN 'a' 'b' 'c' :: (Char,Char,Char)
('a','b','c')

Probably most generic way is to cons values to the ContVec and convert it vector of desired type using vector:

>>> vector $ 'a' <| 'b' <| 'c' <| empty :: (Char,Char,Char)
('a','b','c')

mk0 :: (Vector v a, Dim v ~ Z) => v a

mk1 :: (Vector v a, Dim v ~ N1) => a -> v a

mk2 :: (Vector v a, Dim v ~ N2) => a -> a -> v a

mk3 :: (Vector v a, Dim v ~ N3) => a -> a -> a -> v a

mk4 :: (Vector v a, Dim v ~ N4) => a -> a -> a -> a -> v a

mk5 :: (Vector v a, Dim v ~ N5) => a -> a -> a -> a -> a -> v a

Consing

data ContVec n a

Vector represented as continuation. Alternative wording: it's Church encoded N-element vector.

Instances

Arity n => VectorN ContVec n a

Arity n => Make n a (ContVec n a)

Arity n => Functor (ContVec n)

Arity n => Applicative (ContVec n)

Arity n => Foldable (ContVec n)

Arity n => Traversable (ContVec n)

Arity n => Vector (ContVec n) a

type Dim (ContVec n) = n

empty :: ContVec Z a

Create empty vector.

vector :: (Vector v a, Dim v ~ n) => ContVec n a -> v a

Convert continuation to the vector.

(<|) :: a -> ContVec n a -> ContVec (S n) a infixr 1

Cons value to continuation based vector.

Variadic function

class Make n a r

Type class for variadic vector constructors.

Minimal complete definition

make

Instances

Arity n => Make n a (ContVec n a)

((~) * a' a, Make (S n) a r) => Make n a' (a -> r)

mkN :: Make (S Z) a r => a -> r

Variadic vector constructor. Resulting vector should be converted from ContVec using vector function. For example:

>>> vector $ mkN 'a' 'b' 'c' :: (Char,Char,Char)
('a','b','c')

Functions

replicate :: Vector v a => a -> v a

Replicate value n times.

Examples:

>>> import Data.Vector.Fixed.Boxed (Vec2)
>>> replicate 1 :: Vec2 Int
fromList [1,1]

>>> replicate 2 :: (Double,Double,Double)
(2.0,2.0,2.0)

>>> import Data.Vector.Fixed.Boxed (Vec4)
>>> replicate "foo" :: Vec4 String
fromList ["foo","foo","foo","foo"]

replicateM :: (Vector v a, Monad m) => m a -> m (v a)

Execute monadic action for every element of vector.

Examples:

>>> import Data.Vector.Fixed.Boxed (Vec2,Vec3)
>>> replicateM (Just 3) :: Maybe (Vec3 Int)
Just fromList [3,3,3]
>>> replicateM (putStrLn "Hi!") :: IO (Vec2 ())
Hi!
Hi!
fromList [(),()]

generate :: Vector v a => (Int -> a) -> v a

Generate vector from function which maps element's index to its value.

Examples:

>>> import Data.Vector.Fixed.Unboxed (Vec4)
>>> generate (^2) :: Vec4 Int
fromList [0,1,4,9]

generateM :: (Monad m, Vector v a) => (Int -> m a) -> m (v a)

Generate vector from monadic function which maps element's index to its value.

unfoldr :: Vector v a => (b -> (a, b)) -> b -> v a

Unfold vector.

basis :: (Vector v a, Num a) => Int -> v a

Unit vector along Nth axis. If index is larger than vector dimensions returns zero vector.

Examples:

>>> import Data.Vector.Fixed.Boxed (Vec3)
>>> basis 0 :: Vec3 Int
fromList [1,0,0]
>>> basis 1 :: Vec3 Int
fromList [0,1,0]
>>> basis 3 :: Vec3 Int
fromList [0,0,0]

Modifying vectors

Transformations

head :: (Vector v a, Dim v ~ S n) => v a -> a

First element of vector.

Examples:

>>> import Data.Vector.Fixed.Boxed (Vec3)
>>> let x = mk3 1 2 3 :: Vec3 Int
>>> head x
1

tail :: (Vector v a, Vector w a, Dim v ~ S (Dim w)) => v a -> w a

Tail of vector.

Examples:

>>> import Data.Complex
>>> tail (1,2,3) :: Complex Double
2.0 :+ 3.0

cons :: (Vector v a, Vector w a, S (Dim v) ~ Dim w) => a -> v a -> w a

Cons element to the vector

snoc :: (Vector v a, Vector w a, S (Dim v) ~ Dim w) => a -> v a -> w a

Append element to the vector

concat :: (Vector v a, Vector u a, Vector w a, Add (Dim v) (Dim u) ~ Dim w) => v a -> u a -> w a

reverse :: Vector v a => v a -> v a

Reverse order of elements in the vector

Indexing & lenses

class Index k n

Type class for indexing of vector when index value is known at compile time.

Minimal complete definition

getF, putF, lensF

Instances

Arity n => Index Z (S n)

Index k n => Index (S k) (S n)

(!) :: Vector v a => v a -> Int -> a

Retrieve vector's element at index. Generic implementation is O(n) but more efficient one is used when possible.

index :: (Vector v a, Index k (Dim v)) => v a -> k -> a

Get element from vector at statically known index

set :: (Vector v a, Index k (Dim v)) => k -> a -> v a -> v a

Set n'th element in the vector

element :: (Vector v a, Functor f) => Int -> (a -> f a) -> v a -> f (v a)

Twan van Laarhoven's lens for element of vector

elementTy :: (Vector v a, Index k (Dim v), Functor f) => k -> (a -> f a) -> v a -> f (v a)

Twan van Laarhoven's lens for element of vector with statically known index.

Comparison

eq :: (Vector v a, Eq a) => v a -> v a -> Bool

Test two vectors for equality.

Examples:

>>> import Data.Vector.Fixed.Boxed (Vec2)
>>> let v0 = basis 0 :: Vec2 Int
>>> let v1 = basis 1 :: Vec2 Int
>>> v0 `eq` v0
True
>>> v0 `eq` v1
False

ord :: (Vector v a, Ord a) => v a -> v a -> Ordering

Lexicographic ordering of two vectors.

Maps

map :: (Vector v a, Vector v b) => (a -> b) -> v a -> v b

Map over vector

mapM :: (Vector v a, Vector v b, Monad m) => (a -> m b) -> v a -> m (v b)

Monadic map over vector.

mapM_ :: (Vector v a, Monad m) => (a -> m b) -> v a -> m ()

Apply monadic action to each element of vector and ignore result.

imap :: (Vector v a, Vector v b) => (Int -> a -> b) -> v a -> v b

Apply function to every element of the vector and its index.

imapM :: (Vector v a, Vector v b, Monad m) => (Int -> a -> m b) -> v a -> m (v b)

Apply monadic function to every element of the vector and its index.

imapM_ :: (Vector v a, Monad m) => (Int -> a -> m b) -> v a -> m ()

Apply monadic function to every element of the vector and its index and discard result.

scanl :: (Vector v a, Vector w b, Dim w ~ S (Dim v)) => (b -> a -> b) -> b -> v a -> w b

Left scan over vector

scanl1 :: Vector v a => (a -> a -> a) -> v a -> v a

Left scan over vector

sequence :: (Vector v a, Vector v (m a), Monad m) => v (m a) -> m (v a)

Evaluate every action in the vector from left to right.

sequence_ :: (Vector v (m a), Monad m) => v (m a) -> m ()

Evaluate every action in the vector from left to right and ignore result

sequenceA :: (Vector v a, Vector v (f a), Applicative f) => v (f a) -> f (v a)

Analog of sequenceA from Traversable.

traverse :: (Vector v a, Vector v b, Applicative f) => (a -> f b) -> v a -> f (v b)

Analog of traverse from Traversable.

distribute :: (Vector v a, Vector v (f a), Functor f) => f (v a) -> v (f a)

collect :: (Vector v a, Vector v b, Vector v (f b), Functor f) => (a -> v b) -> f a -> v (f b)

distributeM :: (Vector v a, Vector v (m a), Monad m) => m (v a) -> v (m a)

collectM :: (Vector v a, Vector v b, Vector v (m b), Monad m) => (a -> v b) -> m a -> v (m b)

Folding

foldl :: Vector v a => (b -> a -> b) -> b -> v a -> b

Left fold over vector

foldr :: Vector v a => (a -> b -> b) -> b -> v a -> b

Right fold over vector

foldl1 :: (Vector v a, Dim v ~ S n) => (a -> a -> a) -> v a -> a

Left fold over vector

fold :: (Vector v m, Monoid m) => v m -> m

Combine the elements of a structure using a monoid. Similar to fold

foldMap :: (Vector v a, Monoid m) => (a -> m) -> v a -> m

Map each element of the structure to a monoid, and combine the results. Similar to foldMap

ifoldl :: Vector v a => (b -> Int -> a -> b) -> b -> v a -> b

Left fold over vector. Function is applied to each element and its index.

ifoldr :: Vector v a => (Int -> a -> b -> b) -> b -> v a -> b

Right fold over vector

foldM :: (Vector v a, Monad m) => (b -> a -> m b) -> b -> v a -> m b

Monadic fold over vector.

ifoldM :: (Vector v a, Monad m) => (b -> Int -> a -> m b) -> b -> v a -> m b

Left monadic fold over vector. Function is applied to each element and its index.

Special folds

sum :: (Vector v a, Num a) => v a -> a

Sum all elements in the vector.

maximum :: (Vector v a, Dim v ~ S n, Ord a) => v a -> a

Maximal element of vector.

Examples:

>>> import Data.Vector.Fixed.Boxed (Vec3)
>>> let x = mk3 1 2 3 :: Vec3 Int
>>> maximum x
3

minimum :: (Vector v a, Dim v ~ S n, Ord a) => v a -> a

Minimal element of vector.

Examples:

>>> import Data.Vector.Fixed.Boxed (Vec3)
>>> let x = mk3 1 2 3 :: Vec3 Int
>>> minimum x
1

and :: Vector v Bool => v Bool -> Bool

Conjunction of all elements of a vector.

or :: Vector v Bool => v Bool -> Bool

Disjunction of all elements of a vector.

all :: Vector v a => (a -> Bool) -> v a -> Bool

Determines whether all elements of vector satisfy predicate.

any :: Vector v a => (a -> Bool) -> v a -> Bool

Determines whether any of element of vector satisfy predicate.

Zips

zipWith :: (Vector v a, Vector v b, Vector v c) => (a -> b -> c) -> v a -> v b -> v c

Zip two vector together using function.

Examples:

>>> import Data.Vector.Fixed.Boxed (Vec3)
>>> let b0 = basis 0 :: Vec3 Int
>>> let b1 = basis 1 :: Vec3 Int
>>> let b2 = basis 2 :: Vec3 Int
>>> let vplus x y = zipWith (+) x y
>>> vplus b0 b1
fromList [1,1,0]
>>> vplus b0 b2
fromList [1,0,1]
>>> vplus b1 b2
fromList [0,1,1]

zipWithM :: (Vector v a, Vector v b, Vector v c, Monad m) => (a -> b -> m c) -> v a -> v b -> m (v c)

Zip two vector together using monadic function.

izipWith :: (Vector v a, Vector v b, Vector v c) => (Int -> a -> b -> c) -> v a -> v b -> v c

Zip two vector together using function which takes element index as well.

izipWithM :: (Vector v a, Vector v b, Vector v c, Monad m) => (Int -> a -> b -> m c) -> v a -> v b -> m (v c)

Zip two vector together using monadic function which takes element index as well..

Conversion

convert :: (Vector v a, Vector w a, Dim v ~ Dim w) => v a -> w a

Convert between different vector types

toList :: Vector v a => v a -> [a]

Convert vector to the list

fromList :: Vector v a => [a] -> v a

Create vector form list. Will throw error if list is shorter than resulting vector.

fromList' :: Vector v a => [a] -> v a

Create vector form list. Will throw error if list has different length from resulting vector.

fromListM :: Vector v a => [a] -> Maybe (v a)

Create vector form list. Will return Nothing if list has different length from resulting vector.

fromFoldable :: (Vector v a, Foldable f) => f a -> Maybe (v a)

Create vector from Foldable data type. Will return Nothing if data type different number of elements that resulting vector.

Data types

data VecList n a where

Vector based on the lists. Not very useful by itself but is necessary for implementation.

Constructors

Nil :: VecList Z a

Cons :: a -> VecList n a -> VecList (S n) a

Instances

Arity n => VectorN VecList n a

Arity n => Functor (VecList n)

Arity n => Applicative (VecList n)

Arity n => Foldable (VecList n)

Arity n => Traversable (VecList n)

Arity n => Vector (VecList n) a

(Eq a, Arity n) => Eq (VecList n a)

(Ord a, Arity n) => Ord (VecList n a)

(Show a, Arity n) => Show (VecList n a)

(Arity n, Monoid a) => Monoid (VecList n a)

Typeable (* -> * -> *) VecList

type Dim (VecList n) = n

newtype Only a

Single-element tuple.

Constructors

Only a

Instances

Functor Only

Foldable Only

Traversable Only

Vector Only a

Eq a => Eq (Only a)

Data a => Data (Only a)

Ord a => Ord (Only a)

Show a => Show (Only a)

Monoid a => Monoid (Only a)

Typeable (* -> *) Only

type Dim Only = S Z

data Empty a

Empty tuple.

Constructors

Empty

Instances

Functor Empty

Foldable Empty

Traversable Empty

Vector Empty a

Data a => Data (Empty a)

Typeable (* -> *) Empty

type Dim Empty = Z

Tuple synonyms

type Tuple2 a = (a, a)

type Tuple3 a = (a, a, a)

type Tuple4 a = (a, a, a, a)

type Tuple5 a = (a, a, a, a, a)