Data

Tools

Indexes

Applications

Experiments

Awesome

An open API service indexing awesome lists of open source software.

https://github.com/dastrobu/ndarray

N dimensional array package for numeric computing in swift.
https://github.com/dastrobu/ndarray

linear-algebra ndarray numerics swift

Last synced: 6 months ago
JSON representation

N dimensional array package for numeric computing in swift.

Awesome Lists containing this project

README 

          
# NdArray



[![Swift Version](https://img.shields.io/badge/swift-5.9-blue.svg)](https://swift.org)

![Platform](https://img.shields.io/badge/platform-macOS|iOS|tvOS|whatchOS-lightgray.svg)

![Build](https://github.com/dastrobu/NdArray/actions/workflows/ci.yaml/badge.svg)



N dimensional array package for numeric computing in [Swift](https://swift.org).



The package is inspired by [NumPy](https://www.numpy.org), the well known [python](https://python.org) package for

numerical computations. This Swift package is certainly far away from the maturity of NumPy but implements some key

features to enable fast and simple handling of multidimensional numeric data.



## Table of Contents



- [Installation](#installation)

  - [Swift Package Manager](#swift-package-manager)

- [Multiple Views on Underlying Data](#multiple-views-on-underlying-data)

- [Sliced and Strided Access](#sliced-and-strided-access)

  - [Slices and the Stride Operator `~`](#slices-and-the-stride-operator-)

  - [Single Slice](#single-slice)

  - [`UnboundedRange` Slices](#unboundedrange-slices)

  - [`Range` and `ClosedRange` Slices](#range-and-closedrange-slices)

  - [`PartialRangeFrom`, `PartialRangeUpTo` and `PartialRangeThrough` Slices](#partialrangefrom-partialrangeupto-and-partialrangethrough-slices)

- [Element Manipulation](#element-manipulation)

- [Reshaping](#reshaping)

- [Elementwise Operations](#elementwise-operations)

  - [Scalars](#scalars)

  - [Basic Functions](#basic-functions)

- [Linear Algebra Operations for `Double` and `Float` `NdArray`s.](#linear-algebra-operations-for-double-and-float-ndarrays)

  - [Matrix Vector Multiplication](#matrix-vector-multiplication)

  - [Matrix Matrix Multiplication](#matrix-matrix-multiplication)

  - [Matrix Transpose](#matrix-transpose)

  - [Matrix Inversion](#matrix-inversion)

  - [LU Factorization](#lu-factorization)

  - [Singular Value Decomposition (SVD)](#singular-value-decomposition-svd)

  - [Solve a Linear System of Equations](#solve-a-linear-system-of-equations)

- [Pretty Printing](#pretty-printing)

- [Interaction with Swift Arrays](#interaction-with-swift-arrays)

- [Raw Data Access](#raw-data-access)

- [Type Concept](#type-concept)

  - [Subtypes](#subtypes)

- [Numerical Backend](#numerical-backend)

- [Debugging](#debugging)

- [API Changes](#api-changes)

  - [TLDR](#tldr)

  - [Removal of `NdArraySlice`](#removal-of-ndarrayslice)

- [Not Implemented](#not-implemented)

- [Out of Scope](#out-of-scope)

- [Docs](#docs)



## Installation



The API is stable from versions up to `0.3.0`. Version `0.4.0` deprecates the old slicing API and introduces a more type

safe API. Version `0.5.0` will remove the old slicing API and thus contain breaking changes. It is recommended to fix

all compiler warnings on `0.4.0` before upgrading to `0.5.0`, see also [API Changes](#api-changes).



### Swift Package Manager



```swift

let package = Package(

    dependencies: [

        .package(url: "https://github.com/dastrobu/NdArray.git", from: "0.5.0"),

    ]

)

```



## Multiple Views on Underlying Data



Two arrays can easily point to the same data and data can be modified through both views. This is significantly

different from the Swift internal array object, which has copy on write semantics, meaning you cannot pass around

pointers to the same data. Whereas this behaviour is very nice for small amounts of data, since it reduces side effects.

For numerical computation with huge arrays, it is preferable to let the programmer manage copies. The behaviour of the

NdArray is very similar to

[NumPy's ndarray object](https://numpy.org/doc/stable/reference/generated/numpy.ndarray.html). Here is an example:



```swift

let a = NdArray([9, 9, 0, 9])

let b = NdArray(a)

a[2] = 9.0

print(b) // [9.0, 9.0, 9.0, 9.0]

print(a.ownsData) // true

print(b.ownsData) // false

``` 



## Sliced and Strided Access



Like NumPy's ndarray, slices and strides can be created.



```swift

let a = NdArray.range(to: 10)

let b = NdArray(a[0... ~ 2]) // every second element

print(a) // [0.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0]

print(b) // [0.0, 2.0, 4.0, 6.0, 8.0]

print(b.strides) // [2]

b[0...].set(0)

print(a) // [0.0, 1.0, 0.0, 3.0, 0.0, 5.0, 0.0, 7.0, 0.0, 9.0]

print(b) // [0.0, 0.0, 0.0, 0.0, 0.0]

``` 



This creates an array first, then a strided view on the data, making it easy to set every second element to 0.



### Slices and the Stride Operator `~`



As shown in the previous example, strides can be defined via the stride operator `~`. The unbounded range slice `0...`

takes all elements along an axis. The stride `~ 2` selects only every second element. Here is a short comparison with

NumPy's syntax.



```

NdArray        NumPy

a[0...]        a[::]

a[0... ~ 2]    a[::2]

a[..<42 ~ 2]   a[:42:2]

a[3..<42 ~ 2]  a[3:42:2]

a[3...42 ~ 2]  a[3:41:2]

```



Alternatively, slice objects can be created programmatically. The following notations are equivalent:



```

 a[0...] ≡ a[Slice()]

 a[1...] ≡ a[Slice(lowerBound: 1)]

 a[..<42] ≡ a[Slice(upperBound: 42)]

 a[...42] ≡ a[Slice(upperBound: 43)]

 a[1..<42] ≡ a[Slice(lowerBound: 1, upperBound: 42)]

 a[1... ~ 2] ≡ a[Slice(lowerBound: 1, upperBound, stride: 2)]

 a[..<42 ~ 3] ≡ a[Slice(upperBound: 42, stride: 3)]

 a[1..<42 ~ 3] ≡ a[Slice(lowerBound: 1, upperBound: 42, stride: 3)]

```



Note, to avoid confusion with pure indexing, integer literals need to be converted to a slice explicitly. This means



```swift

let a = NdArray.range(to: 10)

let _ = a[1] // does not work

let s1: NdArray = a[Slice(1)] // selects slice at index one along zeroth dimension

let a1: Double = a[1] // selects first element

```



More detailed examples on each slice type are provided in the sections below.



### Single Slice



A single slice e.g. a row of a matrix is indexed by a so called index slice `Slice(_: Int)`:



```swift

let a = NdArray.ones([2, 2])

print(a)

// [[1.0, 1.0],

//  [1.0, 1.0]]

a[Slice(1)].set(0.0)

print(a)

// [[1.0, 1.0],

//  [0.0, 0.0]]

a[0..., 1].set(2.0)

print(a)

// [[1.0, 2.0],

//  [0.0, 2.0]]

``` 



Note, using element index on a one dimensional array will not access the element,

use [element indexing](#element-manipulation) instead or use the `Vector` subtype which supports element indexing.



```swift

let a = NdArray.range(to: 4)

print(a[Slice(0)]) // [0.0]

print(a[0]) // 0.0

let v = Vector(a)

print(v[0] as Double) // 0.0

print(v[0]) // 0.0

```



### `UnboundedRange` Slices



Unbounded ranges select all elements, this is helpful to access lower dimensions of a multidimensional array



```swift

let a = NdArray.ones([2, 2])

print(a)

// [[1.0, 1.0],

//  [1.0, 1.0]]

a[0..., 1].set(0.0)

print(a)

// [[1.0, 0.0],

//  [1.0, 0.0]]

``` 



or with a stride, selecting every nth element.



```swift

let a = NdArray.range(to: 10).reshaped([5, 2])

print(a)

// [[0.0, 1.0],

//  [2.0, 3.0],

//  [4.0, 5.0],

//  [6.0, 7.0],

//  [8.0, 9.0]]

a[0... ~ 2].set(0.0)

print(a)

// [[0.0, 0.0],

//  [2.0, 3.0],

//  [0.0, 0.0],

//  [6.0, 7.0],

//  [0.0, 0.0]]

``` 



Due to a limitation in the type system, the true unbounded range operator `...` cannot be used. Instead, the

idiom `0...`

should be preferred to specify an unbound range.



### `Range` and `ClosedRange` Slices



Ranges `n...range(to: 10)

print(a[2..<4]) // [2.0, 3.0]

print(a[2...4]) // [2.0, 3.0, 4.0]

print(a[2...4 ~ 2]) // [2.0, 4.0]

``` 



### `PartialRangeFrom`, `PartialRangeUpTo` and `PartialRangeThrough` Slices



Partial ranges `....range(to: 10)

print(a[..<4]) // [0.0, 1.0, 2.0, 3.0]

print(a[...4]) // [0.0, 1.0, 2.0, 3.0, 4.0]

print(a[4...]) // [4.0, 5.0, 6.0, 7.0, 8.0, 9.0]

print(a[4... ~ 2]) // [4.0, 6.0, 8.0]

``` 



## Element Manipulation



Individual elements can be indexed by passing a (Swift) array as index or varargs.



```swift

let a = NdArray.range(to: 12).reshaped([2, 2, 3])

a[[0, 1, 2]]

a[0, 1, 2]

```



For efficient iteration of all indices consider using e.g. `apply`, `map` or `reduce`.



```swift

let a = NdArray.ones(4).reshaped([2, 2])

let b = a.map {

    $0 * 2

} // map to new array

print(b)

// [[2.0, 2.0],

//  [2.0, 2.0]]

a.apply {

    $0 * 3

} // in place

print(a)

// [[3.0, 3.0],

//  [3.0, 3.0]]

print(a.reduce(0) {

    $0 + $1

}) // 12.0

```



Scaling every second element in a matrix by its row index could be done in the following way



```swift

let a = NdArray.ones([4, 3])

for i in 0...ones([4, 3])

for i in 0...range(to: 12)

print(a.reshaped([2, 6]))

// [[ 0.0,  1.0,  2.0,  3.0,  4.0,  5.0],

//  [ 6.0,  7.0,  8.0,  9.0, 10.0, 11.0]]

print(a.reshaped([2, 6], order: .F))

// [[ 0.0,  2.0,  4.0,  6.0,  8.0, 10.0],

//  [ 1.0,  3.0,  5.0,  7.0,  9.0, 11.0]]

print(a.reshaped([3, 4]))

// [[ 0.0,  1.0,  2.0,  3.0],

//  [ 4.0,  5.0,  6.0,  7.0],

//  [ 8.0,  9.0, 10.0, 11.0]]

print(a.reshaped([4, 3]))

// [[ 0.0,  1.0,  2.0],

//  [ 3.0,  4.0,  5.0],

//  [ 6.0,  7.0,  8.0],

//  [ 9.0, 10.0, 11.0]]

print(a.reshaped([2, 2, 3]))

// [[[ 0.0,  1.0,  2.0],

//   [ 3.0,  4.0,  5.0]],

//

//  [[ 6.0,  7.0,  8.0],

//   [ 9.0, 10.0, 11.0]]]

```



A copy will only be made if required to create an array with the specified order.



## Elementwise Operations



### Scalars



Arithmetic operations with scalars work in-place,



```swift

let a = NdArray.ones([2, 2])

a *= 2

a /= 2

a += 2

a /= 2

```



or with implicit copies.



```swift

var b: NdArray

b = a * 2

b = a / 2

b = a + 2

b = a - 2

```



### Basic Functions



The following basic functions can be applied to any `Float` or `Double` array.



```swift

let a = NdArray.ones([2, 2])

var b: NdArray



b = abs(a)



b = acos(a)

b = asin(a)

b = atan(a)



b = cos(a)

b = sin(a)

b = tan(a)



b = cosh(a)

b = sinh(a)

b = tanh(a)



b = exp(a)

b = exp2(a)



b = log(a)

b = log10(a)

b = log1p(a)

b = log2(a)

b = logb(a)

```



The `abs` function is also defined for `SignedNumeric`, such as `Int` arrays.



```swift

let a = NdArray.range(from: -2, to: 2)

print(a) // [-2, -1,  0,  1]

print(abs(a)) // [2, 1, 0, 1]

```



## Linear Algebra Operations for `Double` and `Float` `NdArray`s.



Linear algebra support is currently very basic.



### Matrix Vector Multiplication



```swift

let A = Matrix.ones([2, 2])

let x = Vector.ones(2)

print(A * x) // [2.0, 2.0]

```



### Matrix Matrix Multiplication



```swift

let A = Matrix.ones([2, 2])

let x = Matrix.ones([2, 2])

print(A * x)

// [[2.0, 2.0],

//  [2.0, 2.0]]

```



### Matrix Transpose



```swift

let A = Matrix(NdArray.range(to: 4).reshaped([2, 2]))

print(A.transposed())

// [[0.0,  2.0],

//  [1.0,  3.0]]

```



### Matrix Inversion



```swift

let A = Matrix(NdArray.range(to: 4).reshaped([2, 2]))

print(try A.inverted())

// [[-1.5,  0.5],

//  [ 1.0,  0.0]]

```



### LU Factorization



```swift

let A = Matrix(NdArray.range(to: 4).reshaped([2, 2]))

let (P, L, U) = try A.lu()

print(P)

// [[0.0, 1.0],

//  [1.0, 0.0]]

print(L)

// [[1.0, 0.0],

//  [0.0, 1.0]]

print(U)

// [[2.0, 3.0],

//  [0.0, 1.0]]

print(P * L * U)

// [[0.0, 1.0],

//  [2.0, 3.0]]

```



See also `luInPlace()` for more advanced use cases that avoid creating full matrices.



### Singular Value Decomposition (SVD)



```swift

let A = Matrix(NdArray.range(from: 1, to: 9).reshaped([2, 4]))

let (U, s, Vt) = try A.svd()

print(U)

// [[-0.3761682344281408, -0.9265513797988838],

//  [-0.9265513797988838,  0.3761682344281408]]

print(s)

// [14.227407412633742, 1.2573298353791098]

print(Vt)

// [[ -0.3520616924890126, -0.44362578258952023,  -0.5351898726900277,  -0.6267539627905352],

//  [  0.7589812676751458,  0.32124159914593237,  -0.1164980693832819,   -0.554237737912496],

//  [ -0.4000874340557387,  0.25463292200666415,   0.6909964581538871,  -0.5455419461048127],

//  [ -0.3740722458438949,   0.7969705609558909,   -0.471724384380099,  0.04882606926810252]]

let Sd = Matrix(diag: s)

let S = Matrix.zeros(A.shape)

let mn = A.shape.min()!

S[..(NdArray.range(to: 4).reshaped([2, 2]))

let x = Vector.ones(2)

print(try A.solve(x)) // [-1.0,  1.0]

```



with multiple right hand sides



```swift

let A = Matrix(NdArray.range(to: 4).reshaped([2, 2]))

let x = Matrix.ones([2, 2])

print(try A.solve(x))

// [[-1.0, -1.0],

//  [ 1.0,  1.0]]

```



## Pretty Printing



Multidimensional arrays can be printed in a human friendly way.



```swift

print(NdArray.ones([2, 3, 4]))

// [[[1.0, 1.0, 1.0, 1.0],

//  [1.0, 1.0, 1.0, 1.0],

//  [1.0, 1.0, 1.0, 1.0]],

//

// [[1.0, 1.0, 1.0, 1.0],

//  [1.0, 1.0, 1.0, 1.0],

//  [1.0, 1.0, 1.0, 1.0]]]

print("this is a 2d array in one line \(NdArray.zeros([2, 2]), style: .singleLine)")

// this is a 2d array in one line [[0.0, 0.0], [0.0, 0.0]]

print("this is a 2d array in multi line format line \n\(NdArray.zeros([2, 2]), style: .multiLine)")

// this is a 2d array in multi line format line

// [[0.0, 0.0],

//  [0.0, 0.0]]

```



## Interaction with Swift Arrays



Normal Swift arrays can be converted to a NdArray and back as follows.



```swift

let a = [1, 2, 3]

let b = NdArray(a)

let c = b.dataArray

print(c)

// [1, 2, 3]

```



It should be noted that the conversion requires copying data. This is usually quite fast, but if a numeric algorithm

would convert very small array back and forth, it could slow down the algorithm unnecessarily.

Multidimensional arrays will be represented as flat arrays, to convert a vector or a matrix to nested arrays, make use

of the sequence protocols as shown below.



```swift

let v = Vector([1, 2, 3])

print(Array(v))

// [1, 2, 3]

let M = Matrix([

  [1, 2, 3],

  [3, 2, 1],

])

let a = Array(M).map({ Array($0) })

print(a)

// [[1, 2, 3], [3, 2, 1]]

```



## Raw Data Access



Instead of converting to another type, sometimes it can be helpful to access raw data. Especially, when passing data to

another low level numeric library.

Raw data can be accessed via the `data` property.



```swift

let a = NdArray([1, 2, 3])

let aData = a.data

print(aData)

// UnsafeMutableBufferPointer(start: 0x0000600002796760, count: 3)

```



Note that strides and dimensions must be taken care of manually.



## Type Concept



The idea is to have basic `NdArray` type, which keeps a pointer to data and stores shape and stride information. Since

there can be multiple `NdArray` objects referring to the same data, ownership is tracked explicitly. If an array owns

its data is stored in the `ownsData` flag (similar to NumPy's ndarray)

When creating a new array from an existing one, no copy is made unless necessary. Here are a few examples



```swift

let A = NdArray.ones(5)

var B = NdArray(A) // no copy

B = NdArray(copy: A) // copy explicitly required

B = NdArray(A[0... ~ 2]) // no copy, but B will not be contiguous

B = NdArray(A[0... ~ 2], order: .C) // copy, because otherwise new array will not have C ordering

```



### Subtypes



To be able to define operators for matrix vector multiplication and matrix matrix multiplication, subtypes like

`Matrix` and `Vector` are defined. Since no data is copied when creating a matrix or vector from an array, they can be

converted anytime, thereby making sure the shapes match requirements of the subtype.



```swift

let a = NdArray.ones([2, 2])

let b = NdArray.zeros(2)

let A = Matrix(a) // matrix from array without copy

let x = Vector(b) // vector from array without copy

let Ax = A * x // matrix vector multiplication is defined

let _ = Vector(a) // Precondition failed: Cannot create vector with shape [2, 2]. Vector must have one dimension.

````



Furthermore, algorithms specific for subtypes like a matrix will be defined as method on the subtype, e.g. `solve`



```swift

let A = Matrix(NdArray.range(to: 4).reshaped([2, 2]))

let x = Vector.ones(2)

print(try A.solve(x)) // [-1.0,  1.0]

```



## Numerical Backend



Numerical operations are performed using [BLAS](http://www.netlib.org/blas), see also

[BLAS cheat sheet](http://www.netlib.org/blas/blasqr.pdf) for an overview and [LAPACK](http://www.netlib.org/lapack).

The functions of these libraries are provided by the

[Accelerate Framework](https://developer.apple.com/documentation/accelerate) and are available on most Apple platforms.



## Debugging



When debugging some code, sometimes it can be helpful to look at the raw data in the debugger. This can be done with

help of the `data` property, which is a typed `UnsafeMutableBufferPointer` pointing to the raw data.



Here is an example in lldb:



```

(lldb) p a.data

(UnsafeMutableBufferPointer) $R1 = 6 values (0x113d05000) {

  [0] = 1

  [1] = 2

  [2] = 3

  [3] = 4

  [4] = 5

  [5] = 6

}

```



## API Changes



### TLDR



To migrate from `<=0.3.0` to `0.4.0` upgrade to `0.4.0` first and fix all compile warnings. Do not skip `0.4.0`, since

this can result in undesired behaviour (`a[0..., 2]` will be interpreted as "take slice along zeroth and first

dimension" from `0.5.0` instead of "take slice along zeroth dimension with stride 2" `<=0.3.0`).



Here are a few rules of thumb to fix compile warnings after upgrading to `0.4.0`:



```

a[...] => a[[0...]] // UnboundedRange is now expresed by 0...

a[..., 2] => a[[0... ~ 2]] // strides are now expressed by the stride operator ~

a[...][3] => a[[0..., Slice(3)]] // multi dimensional slices are now created within one subscript call [] not many [][][]

```



### Removal of `NdArraySlice`



Prior to version `0.4.0` using slices on an `NdArray` returned a `NdArraySlice` object. This slice object is similar to

an array but keeps track how deeply it is sliced.



```swift

let A = NdArray.ones([2, 2, 2])

var B = A[...] // NdArraySlice with sliced = 1, i.e. one dimension has been sliced

B = A[0...][0... ~ 2] // NdArraySlice with sliced = 2, i.e. one dimension has been sliced

B = A[0...][0... ~ 2][..<1] // NdArraySlice with sliced = 3, i.e. one dimension has been sliced

B = A[0...][0... ~ 2][..<1][0...] // Precondition failed: Cannot slice array with ndim 3 more than 3 times.

```



So it was recommended to convert to an `NdArray` after slicing before continuing to work with the data.



```swift

let A = NdArray.ones([2, 2, 2])

var B = NdArray(A[...]) // B has shape [2, 2, 2]

B = NdArray(A[...][..., 2]) // B has shape [2, 1, 2]

B = NdArray(A[0...][0..., 2][..<1]) // B has shape [2, 1, 1]

```



When using slices to assign data, no type conversion is required.



```swift

let A = NdArray.ones([2, 2])

let B = NdArray.zeros(2)

A[0..., 0] = B[0...]

print(A)

// [[0.0, 1.0],

//  [0.0, 1.0]]

```



These conversions are not necessary anymore, starting from version `0.4.0`. With the new slice API, based on the `Slice`

object, slices are obtained by



```swift

let A = NdArray.ones([2, 2, 2])

var B = A[0...] // NdArray with sliced = 1, i.e. one dimension has been sliced

B = A[0..., 0... ~ 2] // NdArray with sliced = 2, i.e. one dimension has been sliced

B = A[0..., 0... ~ 2, ..<1] // NdArray with sliced = 3, i.e. one dimension has been sliced

B = A[0..., 0... ~ 2, ..<1, 0...] // Precondition failed: Cannot slice array with ndim 3 more than 3 times.

```



With this API, there is no subtypes returned when slicing, requiring to remember how many times the array was already

sliced. The old slice API is deprecated and will be removed in `0.5.0`.



## Not Implemented



Some features are not implemented yet, but are planned for the near future.



* Elementwise multiplication of Double and Float arrays. Planned as `multiply(elementwiseBy), divide(elementwiseBy)`

  employing `vDSP_vmulD`

  Note that this can be done with help of `map` currently.



## Out of Scope



Some features would be nice to have at some time but currently out of scope.



* Complex number arithmetic (explicit support for complex numbers is not planned). One can create arrays for any type

  though (`NdArray`), just arithmetic operations will not be defined. These could of course be added inside

  application code.



## Docs



Read the generated [docs](https://dastrobu.github.io/NdArray/documentation/ndarray).