numidr/docs/Operations.md

Basic Operations on Arrays

Constructing Arrays

The most important array constructor is array, which returns an array of the specified values:

array [[1, 2, 3], [4, 5, 6]]

Scalars, vectors and matrices have their own constructors, used in exactly the same way (scalar, vector, and matrix). These should be used instead of array whereever possible, as they provide more information to the type-checker.

There are also a few other, more basic constructors for convenience.

-- A 2x2x3 array filled with zeros
repeat 0 [2, 2, 3]

-- Same as previous
zeros [2, 2, 3]

-- A 2x2x3 array filled with ones
ones [2, 2, 3]

Accessing Array Properties

There are a few simple functions for accessing basic properties of arrays: shape and rank, which are self-explanatory, and size, which returns the total number of elements in the array.

The shape accessor is sufficient for most uses, but it can cause problems with the type-checker, as for an array arr : Array s a the type checker does not know that shape arr and s are equal. To solve this problem, a view for accessing the shape is provided:

example {s} arr with (viewShape arr)
  _ | Shape s = ...

This will fully placate the type-checker, as the s returned by the view is proven to be equal to the shape of the array.

Indexing Arrays

NumIdr provides multiple different indexing functions for different purposes. These functions can be grouped based on these categories:

Operation

• Access - Accesses and returns elements from the array.
• Update - Returns a new array with the specified element or range updated by a function. These are indicated by an Update suffix.
• Set - Returns a new array with the specified element or range set to new values. These are indicated by a Set suffix.

Range

• Default - Operates on a single array element.
• Range - Operates on multiple elements at once. These are indicated by a Range suffix.

Safety

• Safe - Guarantees through its type that the index is within range by requiring each index to be a Fin value.
• Non-Bounded - Does not guarantee through its type that the index is within range, and returns Nothing if the provided index is out of bounds. These are indicated by an NB suffix.
• Unsafe - Does not perform any bounds checks at all. These are indicated by an Unsafe suffix. Only use these if you really know what you are doing!

Not all combinations of these categories are defined by the library. Here are the currently provided indexing functions:

	Safe	Ranged Safe	Non-Bounded	Ranged Non-Bounded	Unsafe	Ranged Unsafe
Access	index, (!!)	indexRange, (!!..)	indexNB, (!?)	indexRangeNB, (!?..)	indexUnsafe, (!#)	indexRangeUnsafe, (!#..)
Update	indexUpdate	indexUpdateRange	indexUpdateNB
Set	indexSet	indexSetRange	indexSetNB

The accessor functions have operator forms for convenience, also specified within the table.

Specifying Coordinates

When indexing a single element, a coordinate is specified as a list of numbers, where each number is either a Fin value for safe indexing or a Nat value for non-bounded and unsafe indexing.

index [1, 0] arr

-- Equivalent operator form
arr !! [1, 0]

With ranged indexing, a sub-array of the original array is accessed or modified. This sub-array is given by a list of range specifiers, one for each axis, which can be one of the following:

• Bounds x y - Every index from x to y
• StartBound x - Every index from x to the end of the axis
• EndBound y - Every index from the start of the axis to y
• All - Every index in the axis
• Indices xs - A list of indices in the axis
• Filter p - A predicate specifying which indices to access or modify

There is also an extra specifier, One i, that selects exactly one index of the axis. It is only usable in safe indexing, and it is notable for being the only range specifier that decreases the rank of the sub-array. For example, the following matrix access returns the first column as a vector due to One being used:

matrix [[2, 0], [1, 2]] !!.. [All, One 0]
  == vector [2, 1]

Standard Interface Methods

The array type implements a number of standard interfaces. Most of these implementations are unremarkable, but a few have caveats that are worth noting.

Numeric Operations

Arrays implement the numeric interfaces (Num, Neg, Fractional), as well as Semigroup and Monoid, if their element type supports those operations. These functions are computed elementwise.

matrix [[1, 1], [2, 5]] + matrix [[2, 3], [-1, 3]]
 == matrix [[3, 4], [1, 8]]

This elementwise behavior holds for (+), (*), (-), (/), and (<+>). (*) is not matrix multiplication! For the generalized multiplication operator, which includes matrix multiplication, see the next chapter.

Note

Due to unfortunate restrictions in Idris's standard Num interface, the addition and multiplication operations can only be used when the array's shape is available at run-time. If this is not the case, you must use zipwith (+) or zipWith (*) instead.

Foldable and Traversable

When folding or traversing the elements of an array, these elements are ordered in row-major or "C-style" order, which corresponds to the order in which elements are written and displayed. This behavior should not be depended on, however, as it can change based on the internal array representation; use the elements function if you specifically want row-major order.

Other Common Operations

Concatenation and Stacking

Two arrays can be concatenated along an axis, so long as all other axes have the same dimensions. Two matrices being concatenated along the row axis requires that they must have the same number of columns.

-- 0 is the first axis i.e. the row axis
concat 0 (matrix [[1, 2], [3, 4]]) (matrix [[5, 6], [7, 8]])
  == matrix [[1, 2],
             [3, 4],
             [5, 6],
             [7, 8]]

Stacking is similar to concatenation, but slightly different. Stacking combines arrays with the exact same shape into a single array that is one rank higher. For example, vectors can be stacked along the row axis to obtain a matrix whose rows are the original vectors.

stack 0 [vector [1, 2], vector [3, 4]]
  == matrix [[1, 2],
             [3, 4]]

There are also specialized functions for operating on vectors and matrices: (++) for concatenating vectors; vconcat and hconcat for concatenating two matrices vertically (by row) and horizontally (by column) respectively, and vstack and hstack for stacking row vectors and column vectors respectively.

Reshaping and Resizing

An array can be "reshaped" into any other shape, so long as the total number of elements is the same. This reshaping is done by arranging the elements into a linear order before inserting them into the new array. As with folding, the default order is row-major.

reshape [3, 2] (vector [1, 2, 3, 4, 5, 6])
  == matrix [[1, 2],
             [3, 4],
             [5, 6]]

Arrays can also be resized, which changes their shape while keeping every element at the same index. A default element must be provided to fill any indices that did not exist in the original array.

resize [2, 4] 10 (matrix [[1, 2],
                          [3, 4],
                          [5, 6]])
  == matrix [[1, 2, 10, 10],
             [3, 4, 10, 10]]

Instead of the resize function, one can also use the resizeLTE function, which does not require a default element, but only works if the new array would be provably smaller than the original one.

Transpose

The transpose function reverses the axis order of an array. For matrices, this corresponds to the usual definition of switching rows and columns. There is also a postfix form (.T).

For more fine-grained control when rearranging arrays, there are the swapAxes and permuteAxes functions, where the first swaps only two axes and the second takes an arbitrary permutation. There are also swapInAxis and permuteInAxis, which permute inside an axis, e.g. swapping rows or columns in a matrix.

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