# Numeric Haskell: A Repa Tutorial

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Repa is a Haskell library for high performance, regular, multi-dimensional parallel arrays. All numeric data is stored unboxed and functions written with the Repa combinators are automatically parallel (provided you supply "+RTS -N" on the command line when running the program).

This document provides a tutorial on array programming in Haskell using the repa package.

Note: a companion tutorial to this is provided as the vector tutorial.

# 1 Quick Tour

Repa (REgular PArallel arrays) is an advanced, multi-dimensional parallel arrays library for Haskell, with a number of distinct capabilities:

• The arrays are "regular" (i.e. dense and rectangular); and
• Functions may be written that are polymorphic in the shape of the array;
• Many operations on arrays are accomplished by changing only the shape of the array (without copying elements);
• The library will automatically parallelize operations over arrays.

This is a quick start guide for the package. For further information, consult:

## 1.1 Importing the library

Download the repa package:

$cabal install repa and import it qualified: import qualified Data.Array.Repa as R The library needs to be imported qualified as it shares the same function names as list operations in the Prelude. Note: Operations that involve writing new index types for Repa arrays will require the '-XTypeOperators' language extension. For non-core functionality, a number of related packages are available: and example algorithms in: ## 1.2 Index types and shapes Before we can get started manipulating arrays, we need a grasp of repa's notion of array shape. Much like the classic 'array' library in Haskell, repa-based arrays are parameterized via a type which determines the dimension of the array, and the type of its index. However, while classic arrays take tuples to represent multiple dimensions, Repa arrays use a richer type language for describing multi-dimensional array indices and shapes. Index types consist of two parts: • a dimension component; and • an index type The most common dimensions are given by the shorthand names: type DIM0 = Z type DIM1 = DIM0 :. Int type DIM2 = DIM1 :. Int type DIM3 = DIM2 :. Int type DIM4 = DIM3 :. Int type DIM5 = DIM4 :. Int thus, Array DIM2 Double is the type of a two-dimensional array of doubles, indexed via Int keys, while Array Z Double is a zero-dimension object (i.e. a point) holding a Double. Many operations over arrays are polymorphic in the shape / dimension component. Others require operating on the shape itself, rather than the array. A typeclass, Shape, lets us operate uniformally over arrays with different shape. ## 1.3 Shapes To build values of shape type, we can use the Z and :. constructors: > Z -- the zero-dimension Z For arrays of non-zero dimension, we must give a size. Note: a common error is to leave off the type of the size. > :t Z :. 10 Z :. 10 :: Num head => Z :. head leading to annoying type errors about unresolved instances, such as: No instance for (Shape (Z :. head0)) To select the correct instance, you will need to annotate the size literals with their concrete type: > :t Z :. (10 :: Int) Z :. (10 :: Int) :: Z :. Int is the shape of 1D arrays of length 10, indexed via Ints. Given an array, you can always find its shape by calling extent. Additional convenience types for selecting particular parts of a shape are also provided (All, Any, Slice etc.) which are covered later in the tutorial. ## 1.4 Generating arrays New repa arrays ("arrays" from here on) can be generated in many ways, and we always begin by importing the Data.Array.Repa module:$ ghci
GHCi, version 7.0.3: http://www.haskell.org/ghc/  :? for help
Prelude> :m + Data.Array.Repa

They may be constructed from lists, for example. Here is a one dimensional array of length 10, here, given the shape (Z :. 10):

> let x = fromList (Z :. (10::Int)) [1..10]
> x
[1.0,2.0,3.0,4.0,5.0,6.0,7.0,8.0,9.0,10.0]

The type of x is inferred as:

> :t x
x :: Array (Z :. Int) Double

which we can read as "an array of dimension 1, indexed via Int keys, holding elements of type Double"

We could also have written the type as:

x :: Array DIM1 Double

The same data may also be treated as a two dimensional array, by changing the shape parameter:

> let x = fromList (Z :. (5::Int) :. (2::Int)) [1..10]
> x
[1.0,2.0,3.0,4.0,5.0,6.0,7.0,8.0,9.0,10.0]

which has the type:

x :: Array ((Z :. Int) :. Int) Double

or, more simply:

x :: Array DIM2 Double

### 1.4.1 Building arrays from vectors

It is also possible to build arrays from unboxed vectors, from the 'vector' package:

fromVector :: Shape sh => sh -> Vector a -> Array sh a

New arrays are built by applying a shape to the vector. For example:

import Data.Vector.Unboxed

> let x = fromVector (Z :. (10::Int)) (enumFromN 0 10)
[0.0,1.0,2.0,3.0,4.0,5.0,6.0,7.0,8.0,9.0]

is a one-dimensional array of doubles. As usual, we can also impose other shapes:

> let x = fromVector (Z :. (3::Int) :. (3::Int)) (enumFromN 0 9)
> x
[0.0,1.0,2.0,3.0,4.0,5.0,6.0,7.0,8.0]
> :t x
x :: Array ((Z :. Int) :. Int) Double

to create a 3x3 array.

### 1.4.2 Reading arrays from files

Using the repa-io package, arrays may be written and read from files in a number of formats:

• as BMP files; and
• in a number of text formats.

with other formats rapidly appearing. For the special case of arrays of Word8 values, the repa-bytestring library supports generating bytestrings in memory.

An example: to write an 2D array to an ascii file:

writeMatrixToTextFile "/tmp/test.dat" x

This will result in a file containing:

MATRIX
2 5
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
9.0
10.0

To process .bmp files, use Data.Array.Repa.IO.BMP, as follows (currently reading only works for 24 bit .bmp):

as a 3D array of Word8, which can be further processed.

Note: at the time of writing, there are no binary instances for repa arrays

## 1.5 Indexing arrays

To access elements in repa arrays, you provide an array and a shape, to access the element:

(!) :: (Shape sh, Elt a) => Array sh a -> sh -> a

So:

> let x = fromList (Z :. (10::Int)) [1..10]
> x ! (Z :. 2)
3.0

Note that we can't, even for one-dimensional arrays, give just a bare literal as the shape:

> x ! 2
No instance for (Num (Z :. Int))
arising from the literal 2'

as the Z type isn't in the Num class.

What if the index is out of bounds, though?

> x ! (Z :. 11)
*** Exception: ./Data/Vector/Generic.hs:222 ((!)): index out of bounds (11,10)

an exception is thrown. An altnerative is to indexing functions that return a Maybe:

(!?) :: (Shape sh, Elt a) => Array sh a -> sh -> Maybe a

An example:

> x !? (Z :. 9)
Just 10.0
> x !? (Z :. 11)
Nothing

## 1.6 Operations on arrays

Besides indexing, there are many regular, list-like operations on arrays.

### 1.6.1 Maps, zips, filters and folds

We can map over multi-dimensional arrays:

> let x = fromList (Z :. (3::Int) :. (3::Int)) [1..9]
> x
[1.0,2.0,3.0,4.0,5.0,6.0,7.0,8.0,9.0]

since map` conflicts with the definition in the Prelude, we have to use it qualified:

> Data.Array.Repa.map (^2) x
[1.0,4.0,9.0,16.0,25.0,36.0,49.0,64.0,81.0]

Maps leave the dimension unchanged.

Folding reduces the inner dimension of the array.

fold :: (Shape sh, Elt a) => (a -> a -> a) -> a -> Array (sh :. Int) a -> Array sh a

So if 'x' is a 3D array:

> let x = fromList (Z :. (3::Int) :. (3::Int)) [1..9]
> x
[1.0,2.0,3.0,4.0,5.0,6.0,7.0,8.0,9.0]

We can sum each row, to yield a 2D array:

> fold (+) 0 x
[6.0,15.0,24.0]

Two arrays may be combined via zipWith:

zipWith :: (Shape sh, Elt b, Elt c, Elt a) =>
(a -> b -> c) -> Array sh a -> Array sh b -> Array sh c

an example:

> zipWith (*) x x
[1.0,4.0,9.0,16.0,25.0,36.0,49.0,64.0,81.0]

### 1.6.2 Numeric operations: negation, addition, subtraction, multiplication

Repa arrays are instances of the Num. This means that operations on numerical elements are lifted automagically onto arrays of such elements. For example, (+) on two double values corresponds to element-wise addition, (+), of the two arrays of doubles:

> let x = fromList (Z :. (10::Int)) [1..10]
> x + x
[2.0,4.0,6.0,8.0,10.0,12.0,14.0,16.0,18.0,20.0]

Other operations from the Num class work just as well:

> -x
[-1.0,-2.0,-3.0,-4.0,-5.0,-6.0,-7.0,-8.0,-9.0,-10.0]
> x ^ 3
[1.0,8.0,27.0,64.0,125.0,216.0,343.0,512.0,729.0,1000.0]
> x * x
[1.0,4.0,9.0,16.0,25.0,36.0,49.0,64.0,81.0,100.0]

## 1.7 Changing the shape of an array

One of the main advantages of repa-style arrays over other arrays in Haskell is the ability to reshape data without copying. This is achieved via *index-space transformations*.

An example: transposing a 2D array (this example taken from the repa paper). First, the type of the transformation:

transpose2D :: Elt e => Array DIM2 e -> Array DIM2 e

Note that this transform will work on DIM2 arrays holding any elements. Now, to swap rows and columns, we have to modify the shape:

transpose2D a = backpermute (swap e) swap a
where
e = extent a
swap (Z :. i :. j) = Z :. j :. i

The swap function reorders the index space of the array. To do this, we extract the current shape of the array, and write a function that maps the index space from the old array to the new array. That index space function is then passed to backpermute which actually constructs the new array from the old one.

backpermute generates a new array from an old, when given the new shape, and a function that translates between the index space of each array (i.e. a shape transformer).

backpermute
:: (Shape sh, Shape sh', Elt a)
=> sh'
-> (sh' -> sh)
-> Array sh a
-> Array sh' a

Note that the array created is not actually evaluated (we only modified the index space of the array).

Transposition is such a common operation that it is provided by the library:

transpose :: (Shape sh, Elt a)
=> Array ((sh :. Int) :. Int) a -> Array ((sh :. Int) :. Int)

the type indicate that it works on the lowest two dimensions of the array.

Other operations on index spaces include taking slices and joining arrays into larger ones.

### 1.7.1 Example: matrix-matrix multiplication

A more advanced example from the Repa paper: matrix-matrix multiplication: the result of matrix multiplication is a matrix whose elements are found by multiplying the elements of each row from the first matrix by the associated elements of the same column from the second matrix and summing the result.

if $A=\begin{bmatrix}a&b\\c&d\end{bmatrix}$ and $B=\begin{bmatrix}e&f\\g&h\end{bmatrix}$

then

$AB=\begin{bmatrix}a&b\\c&d\end{bmatrix}\begin{bmatrix}e&f\\g&h\end{bmatrix}=\begin{bmatrix}ae+bg&af+bh\\ce+dg&cf+dh\end{bmatrix}$

So we take two, 2D arrays and generate a new array, using our transpose function from earlier:

mmMult :: (Num e, Elt e)
=> Array DIM2 e
-> Array DIM2 e
-> Array DIM2 e
mmMult a b = sum (zipWith (*) aRepl bRepl)
where
t   = transpose2D b
aRepl = extend (Z :.All :.colsB :.All) a
bRepl = extend (Z :.rowsA :.All :.All) t
(Z :.colsA :.rowsA) = extent a
(Z :.colsB :.rowsB) = extent b

The idea is to expand both 2D argument arrays into 3D arrays by replicating them across a new axis. The front face of the cuboid that results represents the array a, which we replicate as often as b has columns (colsB), producing aRepl.

The top face represents t (the transposed b), which we replicate as often as a has rows (rowsA), producing bRepl,. The two replicated arrays have the same extent, which corresponds to the index space of matrix multiplication

Optimized implementations of this function are available in the repa-algorithms package.