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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 data.
Table of Contents
(generated with DocToc)
- Installation
- Multiple Views on Underlying Data
- Sliced and Strided Access
- Element Manipulation
- Reshaping
- Linear Algebra Operations for
DoubleandFloatNdArrays. - Pretty Printing
- Type Concept
- Numerical Backend
- Not Implemented
- Out of Scope
Installation
Swift Package Manager
dependencies: [
.package(url: "https://github.com/dastrobu/NdArrays.git", from: "0.1.2"),
],
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) // falseSliced 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]]) // 0.0UnboundedRange 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 SwiftFor 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
}) // 12.0Scaling 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].apply {
$0 * 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, and array can be reshaped to any compatible shape without modifying the 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.
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 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 orderingWhen 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][...] // Assertion 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) // Assertion 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.
- Trigonometric functions
- Elementwise multiplication of Double and Float arrays. Planned as
multiply(elementwiseBy, divide(elementwiseBy)employingvDSP_vmulDNote that this can be done with help ofmapcurrently.
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.