Numeric Haskell: A Vector Tutorial

Vector is a Haskell library for working with arrays, with an emphasis on raw performance, whilst retaining a rich interface. The main data types are boxed and unboxed arrays, and arrays may be immutable (pure), or mutable. Arrays are indexed by non-negative Int values.

The vector library has an API similar to the famous Haskell list library, with many of the same names.

This tutorial is modelled on the NumPy tutorial.

Quick Tour

Here is a quick overview to get you started.

Importing the library

Download the vector package:

$ cabal install vector

and import it as, for boxed arrays:

 import qualified Data.Vector as V

or:

 import qualified Data.Vector.Unboxed as V

for unboxed arrays. The library needs to be imported qualified as it shares the same function names as list operations in the Prelude.

Generating Vectors

New vectors can be generated in many ways:

$ ghci
GHCi, version 6.12.1: http://www.haskell.org/ghc/  :? for help
Loading package ghc-prim ... linking ... done.
Loading package integer-gmp ... linking ... done.
Loading package base ... linking ... done.
Loading package ffi-1.0 ... linking ... done.

Prelude> :m + Data.Vector

-- Generating a vector from a list:
Prelude Data.Vector> let a = fromList [10, 20, 30, 40]

Prelude Data.Vector> a
fromList [10,20,30,40] :: Data.Vector.Vector

-- Or filled from a sequence
Prelude Data.Vector> enumFromStepN 10 10 4
fromList [10,20,30,40] :: Data.Vector.Vector

-- A vector created from four consecutive values
Prelude Data.Vector> enumFromN 10 4
fromList [10,11,12,13] :: Data.Vector.Vector

You can also build vectors using operations similar to lists:

-- The empty vector
Prelude Data.Vector> empty
fromList [] :: Data.Vector.Vector

-- A vector of length one
Prelude Data.Vector> singleton 2
fromList [2] :: Data.Vector.Vector

-- A vector of length 10, filled with the value '2'
-- Note  that to disambiguate names,
-- and avoid a clash with the Prelude,
-- with use the full path to the Vector module
Prelude Data.Vector> Data.Vector.replicate 10 2
fromList [2,2,2,2,2,2,2,2,2,2] :: Data.Vector.Vector

In general, you may construct new vectors by applying a function to the index space:

Prelude Data.Vector> generate 10 (^2)
fromList [0,1,4,9,16,25,36,49,64,81] :: Data.Vector.Vector

Vectors may have more than one dimension:

-- Here we create a two dimensional vector, 10 columns,
-- each row filled with the row index.
Prelude Data.Vector> let x = generate 10 (\n -> Data.Vector.replicate 10 n)

-- The type is "Vector of Vector of Ints"
Prelude Data.Vector> :t x
x :: Vector (Vector Int)

Vectors may be grown or shrunk arbitrarily:

Prelude Data.Vector> let y = Data.Vector.enumFromTo 0 11
Prelude Data.Vector> y
fromList [0,1,2,3,4,5,6,7,8,9,10,11] :: Data.Vector.Vector

-- Take the first 3 elements as a new vector
Prelude Data.Vector> Data.Vector.take 3 y
fromList [0,1,2] :: Data.Vector.Vector

-- Duplicate and join the vector
Prelude Data.Vector> y Data.Vector.++ y
fromList [0,1,2,3,4,5,6,7,8,9,10,11,0,1,2,3,4,5,6,7,8,9,10,11] :: Data.Vector.Vector

Modifying vectors

Just as with lists, you can iterate (map) over arrays, reduce them (fold), filter them, or join them in various ways:

-- mapping a function over the elements of a vector
Prelude Data.Vector> Data.Vector.map (^2) y
fromList [0,1,4,9,16,25,36,49,64,81,100,121] :: Data.Vector.Vector

-- Extract only the odd elements from a vector
Prelude Data.Vector> Data.Vector.filter odd y
fromList [1,3,5,7,9,11] :: Data.Vector.Vector

-- Reduce a vector
Prelude Data.Vector> Data.Vector.foldl (+) 0 y
66

-- Take a scan (partial results from a reduction):
Prelude Data.Vector> Data.Vector.scanl (+) 0 y
fromList [0,0,1,3,6,10,15,21,28,36,45,55,66] :: Data.Vector.Vector

-- Zip two arrays pairwise, into an array of pairs
Prelude Data.Vector> Data.Vector.zip y y
fromList [(0,0),(1,1),(2,2),(3,3),(4,4),(5,5),(6,6),(7,7),(8,8),(9,9),(10,10),(11,11)] :: Data.Vector.Vector

Indexing vectors

And like all good arrays, you can index them in various ways:

-- Take the first element
Prelude Data.Vector> Data.Vector.head y
0

-- Take the last element
Prelude Data.Vector> Data.Vector.tail y
fromList [1,2,3,4,5,6,7,8,9,10,11] :: Data.Vector.Vector

-- Take an arbitrary element
Prelude Data.Vector> y ! 4
4

The Tutorial

The vector package provides a several types of array. The most general interface is via Data.Vector, which provides for boxed arrays, holding any type.

There are also more specialized array types:

• Unboxed
• Storable

which provide unboxed arrays (i.e. no closures) and storable arrays (data that is pinned, and may be passed to and from C via a Ptr).

In all cases, the operations are subject to loop fusion. That is, if you compose two functions,

map f . map g

the compiler will rewrite it into a single traversal:

map (f . g)

saving time and space.

Simple example

You can create the arrays in many ways, for example, from a regular Haskell list:

 let a = fromList [2,3,4]

Prelude Data.Vector> a
fromList [2,3,4] :: Data.Vector.Vector

Prelude Data.Vector> :t a
a :: Vector Integer

GHCi will print the contents of the vector as executable code.

To create a multidimensional array, you can use a nested list generator to fill it:

Prelude Data.Vector> let x = fromList [ fromList [1 .. x] | x <- [1..10] ]

Prelude Data.Vector> :t x
x :: Vector (Vector Integer)

-- XXX TODO need a better printing function for multidimensional arrays.

You can also just create arrays filled with zeroes:

Prelude Data.Vector> Data.Vector.replicate 10 0
fromList [0,0,0,0,0,0,0,0,0,0] :: Data.Vector.Vector

And you can fill arrays from a sequence generator:

Prelude Data.Vector> enumFromN 1 10
fromList [1,2,3,4,5,6,7,8,9,10] :: Data.Vector.Vector

Prelude Data.Vector> enumFromStepN 0 10 10
fromList [0,10,20,30,40,50,60,70,80,90] :: Data.Vector.Vector

Array Types

The vector package provides several array types, with an identical interface. They have different flexibility with respect to the types of values that may be stored in them, and different performance characteristics.

Boxed Arrays: Data.Vector

The most flexible type is Data.Vector.Vector, which provides *boxed* arrays: arrays of pointers to Haskell values.

• Data.Vector.Vector's are fully polymorphic: they can hold any valid Haskell type

These arrays are suitable for storing complex Haskell types (sum types, or algebraic data types), but a better choice for simple data types is Data.Vector.Unboxed.

Unboxed Arrays: Data.Vector.Unboxed

Simple, atomic types, and pair types can be stored in a more efficient manner: consecutive memory slots without pointers. The Data.Array.Unboxed.Vector type provides unboxed arrays of types that are members of the Unbox class, including:

• Bool
• ()
• Char
• Double
• Float
• Int
• Int8, 16, 32, 64
• Word
• Word8, 16, 32, 64
• Complex a's, where 'a' is in Unbox
• Tuple types, where the elements are unboxable

Unboxed arrays should be preferred when you have unboxable elements, as they are generally more efficient.

Storable Arrays: passing data to C

Storable arrays (Data.Vector.Storable.Vector) are vectors of any type in the Storable class.

These arrays are pinned, and may be converted to and from pointers, that may be passed to C functions, using a number of functions:

unsafeFromForeignPtr
    :: Storable a
    => ForeignPtr a
    -> Int
    -> Int	
    -> Vector a	

-- Create a vector from a ForeignPtr with an offset and a length. The data may --- not be modified through the ForeignPtr afterwards.

unsafeToForeignPtr
    :: Storable a
    => Vector a
    -> (ForeignPtr a, Int, Int)

-- Yield the underlying ForeignPtr together with the offset to the data and its -- length. The data may not be modified through the ForeignPtr.

unsafeWith
    :: Storable a
    => Vector a
    -> (Ptr a -> IO b)
    -> IO b

-- Pass a pointer to the vector's data to the IO action. The data may not be -- -- modified through the 'Ptr.

Pure Arrays

Impure Arrays

Some examples

The most important attributes of an array are available in O(1) time, such as the size (length),

-- how big is the array?
Prelude Data.Vector> let a = fromList [1,2,3,4,5,6,7,8,9,10]
Prelude Data.Vector> Data.Vector.length a
10

-- is the array empty?
Prelude Data.Vector> Data.Vector.null a
False

Array Creation

An example: filling a vector from a file

We often want to populate a vector using a external data file. The easiest way to do this is with bytestring IO, and Data.Vector.unfoldr (or the equivalent functions in Data.Vector.Unboxed or Data.Vector.Storable:

{-# LANGUAGE BangPatterns #-}

import qualified Data.ByteString.Lazy.Char8 as L
import qualified Data.Vector                as U
import System.Environment

main = do
    [f] <- getArgs
    s   <- L.readFile f
    print . U.sum . parse $ s

-- Fill a new vector from a file containing a list of numbers.
parse = U.unfoldr step
  where
     step !s = case L.readInt s of
        Nothing       -> Nothing
        Just (!k, !t) -> Just (k, L.tail t)

Note the use of bang patterns to ensure the parsing accumulated state is produced strictly.

Create a data file filled with 1 million integers:

   $ seq 1 1000000 >  data

Compile with -Odph (enables special optimizations to help fusion):

   $ ghc -Odph --make vector.hs

Run:

   $ time ./vector data   
   500000500000
   ./vector data  0.08s user 0.01s system 98% cpu 0.088 total

Random numbers

If we can parse from a file, we can also fill a vector with random numbers. We'll use the mersenne-random package:

$ cabal install mersenne-random

We can then use MTRandom class to generate random vectors of different types:

import qualified Data.Vector.Unboxed as U
import System.Random.Mersenne
import Control.Monad

main = do
    -- create a new source of randomness
    -- andan infinite list of randoms
    g   <- newMTGen Nothing
    rs  <- randoms g

    -- fill a vector with the first 10 random Ints
    let a = U.fromList (take 10 rs) :: U.Vector Int

    -- print the sum
    print (U.sum a)

    -- print each element
    forM_ (U.toList a) print

Running it:

 $ runhaskell B.hs
-56426044567146682
-2144043065897064806
-2361203915295681429
1023751035988638668
-5145147152582103336
6545758323081548799
-7630294342751488332
-5861937811333342436
-3198510304070719259
8949914511577398116
-8681457396993884283

Transformations on Vectors

A primary operation on arrays are the zip class of functions.

> let a = fromList [20,30,40,50]

> let b = enumFromN 0 4

> a 
fromList [20,30,40,50]

> b
fromList [0,1,2,3]

> Data.Vector.zipWith (-) a b
fromList [20,29,38,47]

We can also, of course, apply a function to each element of a vector (map):

> Data.Vector.map (^2) b
fromList [0,1,4,9]

> Data.Vector.map (\e -> 10 * sin (fromIntegral e)) a
fromList [9.129452507276277,-9.880316240928618,7.451131604793488,-2.6237485370392877]

> Data.Vector.map (< 35) a
fromList [True,True,False,False]

Folds: Sums, Products, Min, Max

Many special purpose folds (reductions) on vectors are available:

> let a = enumFromN 1 100

> Data.Vector.sum a
5050

> Data.Vector.product a
93326215443944152681699238856266700490715968264381621468592963895217599993229915608941463976156518286253697920827223758251185210916864000000000000000000000000

> Data.Vector.minimum a
1

> Data.Vector.maximum a
100

Indexing, Slicing and Iterating

One dimensional arrays can be indexed, sliced and iterated over pretty much like lists.

Because Haskell values are by default immutable, all slice operations are zero-copying.

> let a = enumFromN 0 10

> a
fromList [0,1,2,3,4,5,6,7,8,9]

> let b = Data.Vector.map (^3) a

> b ! 2
8

> slice 2 3 b
fromList [8,27,64]

slice takes 3 arguments: the initial index to slice from, the number of elements to slice, and the vector to operate on.

A number of special-purpose slices are also available:

> Data.Vector.init b
fromList [0,1,8,27,64,125,216,343,512]

> Data.Vector.tail b
fromList [1,8,27,64,125,216,343,512,729]

> Data.Vector.take 3 b
fromList [0,1,8] :: Data.Vector.Vector

> Data.Vector.drop 3 b
fromList [27,64,125,216,343,512,729] :: Data.Vector.Vector

Bulk operations

Stacking together different arrays

Splitting one array into several smaller ones

Copies and Views

No Copy at All

Indexing with Arrays of Indices

Indexing with Boolean Arrays

Permutations

Randoms

IO

References