dastrobu/NdArray 0.6.0

N dimensional array package for numeric computing in swift.

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.package(url: "https://github.com/dastrobu/NdArray.git", from: "0.6.0")

NdArray

N dimensional array package for numeric computing in Swift.

The package is inspired by NumPy, the well known python 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.

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.

Swift Package Manager

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. Here is an example:

let a = NdArray<Double>([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.

let a = NdArray<Double>.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

let a = NdArray<Double>.range(to: 10)
let _ = a[1] // does not work
let s1: NdArray<Double> = 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):

let a = NdArray<Double>.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 instead or use the Vector subtype which supports element indexing.

let a = NdArray<Double>.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

let a = NdArray<Double>.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.

let a = NdArray<Double>.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..<m and closed ranges n...m allow selecting certain sub arrays.

let a = NdArray<Double>.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 ...<m, ...m and n... define only one bound.

let a = NdArray<Double>.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.

let a = NdArray<Double>.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.

let a = NdArray<Double>.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

let a = NdArray<Double>.ones([4, 3])
for i in 0..<a.shape[0] {
    a[Slice(i), 0... ~ 2] *= Double(i)
}
print(a)
// [[0.0, 1.0, 0.0],
//  [1.0, 1.0, 1.0],
//  [2.0, 1.0, 2.0],
//  [3.0, 1.0, 3.0]]

Alternatively one can use classical loops and convert each row to a vector for efficient element indexing

let a = NdArray<Double>.ones([4, 3])
for i in 0..<a.shape[0] {
    let ai = Vector(a[Slice(i)])
    for j in stride(from: 0, to: a.shape[1], by: 2) {
        ai[j] *= Double(i)
    }
}
print(a)
// [[0.0, 1.0, 0.0],
//  [1.0, 1.0, 1.0],
//  [2.0, 1.0, 2.0],
//  [3.0, 1.0, 3.0]]

Reshaping

Like in NumPy, an array can be reshaped to any compatible shape without modifying data. That means the shape and strides are recomputed to re-interpret the data.

let a = NdArray<Double>.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,

let a = NdArray<Double>.ones([2, 2])
a *= 2
a /= 2
a += 2
a /= 2

or with implicit copies.

var b: NdArray<Double>
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.

let a = NdArray<Double>.ones([2, 2])
var b: NdArray<Double>

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.

let a = NdArray<Int>.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

let A = Matrix<Double>.ones([2, 2])
let x = Vector<Double>.ones(2)
print(A * x) // [2.0, 2.0]

Matrix Matrix Multiplication

let A = Matrix<Double>.ones([2, 2])
let x = Matrix<Double>.ones([2, 2])
print(A * x)
// [[2.0, 2.0],
//  [2.0, 2.0]]

Matrix Transpose

let A = Matrix<Double>(NdArray.range(to: 4).reshaped([2, 2]))
print(A.transposed())
// [[0.0,  2.0],
//  [1.0,  3.0]]

Matrix Inversion

let A = Matrix<Double>(NdArray.range(to: 4).reshaped([2, 2]))
print(try A.inverted())
// [[-1.5,  0.5],
//  [ 1.0,  0.0]]

LU Factorization

let A = Matrix<Double>(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)

let A = Matrix<Double>(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<Double>.zeros(A.shape)
let mn = A.shape.min()!
S[..<mn, ..<mn] = Sd
print(S)
// [[14.227407412633742,                0.0,                0.0,                0.0],
//  [               0.0, 1.2573298353791098,                0.0,                0.0]]
print(U * S * Vt)
// [[1.0000000000000004,                2.0, 3.0000000000000004, 3.9999999999999996],
//  [ 4.999999999999999,  6.000000000000001,  7.000000000000001,                8.0]]

Solve a Linear System of Equations

with single right-hand side

let A = Matrix<Double>(NdArray.range(to: 4).reshaped([2, 2]))
let x = Vector<Double>.ones(2)
print(try A.solve(x)) // [-1.0,  1.0]

with multiple right hand sides

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

print(NdArray<Double>.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<Double>.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<Double>.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.

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.

let v = Vector<Int>([1, 2, 3])
print(Array(v))
// [1, 2, 3]
let M = Matrix<Int>([
  [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.

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

let A = NdArray<Double>.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.

let a = NdArray<Double>.ones([2, 2])
let b = NdArray<Double>.zeros(2)
let A = Matrix<Double>(a) // matrix from array without copy
let x = Vector<Double>(b) // vector from array without copy
let Ax = A * x // matrix vector multiplication is defined
let _ = Vector<Double>(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

let A = Matrix<Double>(NdArray.range(to: 4).reshaped([2, 2]))
let x = Vector<Double>.ones(2)
print(try A.solve(x)) // [-1.0,  1.0]

Numerical Backend

Numerical operations are performed using BLAS, see also BLAS cheat sheet for an overview and LAPACK. The functions of these libraries are provided by the Accelerate Framework 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<Double>) $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.

let A = NdArray<Double>.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.

let A = NdArray<Double>.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.

let A = NdArray<Double>.ones([2, 2])
let B = NdArray<Double>.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

let A = NdArray<Double>.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<Complex>), just arithmetic operations will not be defined. These could of course be added inside application code.

Docs

Read the generated docs.

Swiftpack.co - dastrobu/NdArray as Swift Package