# 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. 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 in vector tutorial.

# 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.

## 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:

* repa-bytestring
* repa-io
* repa-algorithms

and example algorithms in:

* repa-examples

## 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 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 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.

## 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. A common error is to leave off the type of the size,

> :t Z :. 10
Z :. 10 :: Num head => Z :. head

For arrays of non-zero dimension, we must give a size. 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.

## Generating arrays

New repa arrays ("arrays" from here on) can be generated in many ways:

\$ ghci
GHCi, version 7.0.3: 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.Array.Repa

They may be constructed from lists:

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:

> 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 would have the type:

x :: Array ((Z :. Int) :. Int) Double

or

x :: Array DIM2 Double

## 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

## Operations on arrays

### 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]