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.

Table of Contents

• Installation
  • Swift Package Manager
• Multiple Views on Underlying Data
• Sliced and Strided Access
  • Single Slice
  • UnboundedRange Slices
  • Range and ClosedRange Slices
  • PartialRangeFrom, PartialRangeUpTo and PartialRangeThrough Slices
• Element Manipulation
• Reshaping
• Elementwise Operations
  • Scalars
  • Basic Functions
• Linear Algebra Operations for Double and Float NdArrays.
  • Matrix Vector Multiplication
  • Matrix Matrix Multiplication
  • Matrix Inversion
  • Solve a Linear System of Equations
• Pretty Printing
• Type Concept
  • Subtypes
• Numerical Backend
• Not Implemented
• Out of Scope
• Docs

Installation

Swift Package Manager

let package = Package(
    dependencies: [
        .package(url: "https://github.com/dastrobu/NdArray.git", from: "0.3.0"),
    ]
)

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[..., 2])
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[...].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.

Single Slice

A single slice e.g. a row of a matrix is indexed by simple integer

let a = NdArray<Double>.ones([2, 2])
print(a)
// [[1.0, 1.0],
//  [1.0, 1.0]]
a[1].set(0.0)
print(a)
// [[1.0, 1.0],
//  [0.0, 0.0]]
a[...][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[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

Unbound 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[...][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[..., 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]]

Range and ClosedRange Slices

Ranges n..<m and closed ranges n...m allow to select 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

The syntax for indexing individual elements is by passing an (Swift) array as index. Passing indices individually cannot be implemented, since Swift does not support varargs on subscript.

let a = NdArray<Double>.range(to: 12).reshaped([2, 2, 3])
a[[0, 1, 2]]
a[0, 1, 2]  // does not work with Swift

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[i][..., 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[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 NdArrays.

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 Inversion

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

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

Multi dimensional 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]]

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[..., 2]) // no copy, but B will not be contiguous
B = NdArray(A[..., 2], order: .C) // copy, because otherwise new array will not have C ordering

When using slices on an NdArray it returns 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[...][..., 2] // NdArraySlice with sliced = 2, i.e. one dimension has been sliced
B = A[...][..., 2][..<1] // NdArraySlice with sliced = 3, i.e. one dimension has been sliced
B = A[...][..., 2][..<1][...] // Precondition failed: Cannot slice array with ndim 3 more than 3 times.

So it is 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[...][..., 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] = B[...]
print(A)
// [[0.0, 1.0],
//  [0.0, 1.0]]

Subtypes

To be able to define operators for matrix vector multiplication and matrix matrix multiplication, sub types 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 sub type.

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.

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 numbers (currently support for complex numbers is not planned)

Docs

Read the generated docs.