A real/complex vector library in Swift.

Compatible with Swift 6.0.

VecLab is a Swift Package for real and complex vector operations with NumPy and MATLAB-style functions.

• Real and complex scalars and vectors.
• Overloaded arithmetic operators.
• Basic MATLAB-style functions.
• Vectorized using Accelerate and vDSP.

Full documentation can be found on the Swift Package Index.

The library includes an FFT function using Accelerate, but here is an example of creating a complex FFT using a recursive algorithm and its NumPy and MATLAB equivalents:

// FFTX Fast Finite Fourier Transform.
public func fftx(_ x: ComplexArray) -> ComplexArray {
    let n = length(x) 
    let omega = exp(-2 * Real.pi * Real.i / Real(n))
    if rem(n, 2) == 0 {
        // Recursive divide and conquer.
        let k = vector(0 ... (n / 2 - 1))
        let w = omega ** k
        let u = fftx(slice(x, 0 ..< n - 1, 2))
        let v = w * fftx(slice(x, 1 ..< n, 2))
        return cat(u + v, u - v)
    } else {
        return x
    }
}

Here's a breakdown of the real and complex vector operations in the function:

1. x is the complex input array.
2. omega is a complex exponential number. Real.i is the imaginary unit i.
3. Tests if the input x length is divisible by 2.
4. k is a real vector from 0, 1, 2,... n/2-1.
5. w is the complex vector of omega to the power of vector k.
6. u is the complex result of a recursive call with even x.
7. v is the complex result of a recursive call with odd x.
8. The result is the concatenation of the complex vector addition and subtraction of u and v.
9. The recursion ends when there is one element in the input array. The FFT of a single element is itself.

import numpy as np

def fftx(x):
    """
    FFTX Fast Finite Fourier Transform.
    """
    n = len(x)
    omega = np.exp(-2j * np.pi / n)
    
    if n % 2 == 0:
        # Recursive divide and conquer
        k = np.arange(n//2)
        w = omega ** k
        u = fftx(x[::2])
        v = w * fftx(x[1::2])
        y = np.concatenate([u + v, u - v])
    else:
        y = x
    
    return y

function y = fftx(x)
% FFTX Fast Finite Fourier Transform.
n = length(x);
omega = exp(-2*pi*i/n);
if rem(n,2) == 0
    % Recursive divide and conquer.
    k = (0:n/2-1)';
    w = omega.^k;
    u = fftx(x(1:2:n-1));
    v = w.*fftx(x(2:2:n));
    y = [u+v; u-v];
else
    y = x
end

The library works with existing Swift types, using only arrays and tuples. For convenience, these have been given type aliases for the underlying native types. Only the Real need be defined, the others are all derived from this type.

public typealias Real = Double
public typealias RealArray = [Real]
public typealias Complex = (Real, Real)
public typealias ComplexArray = ([Real], [Real])

Real numbers are Double types.

Real arrays are just a normal Swift Array of Double.

Complex numbers are defined as a tuple of two real numbers, representing the real and imaginary parts of the number.

let c = (10.0, 2.0)

A complex array consists of a tuple of two real arrays. This arrangement is sometimes known as split complex.

let realArray = [1.0, 2.0, 3.0, 4.0]
let imagArray = [1.0, 2.0, 3.0, 4.0]
let complexArray = (realArray, imagArray)

The imaginary unit, i, is defined as an extension to Real, similar to other constants such as pi.

let c = 10 + Real.i

It can be used in any expression. This a a complex exponential:

let phi = 100.0
let c = exp(Real.i * 2 * Real.pi * phi)

Ranges can be defined using the Swift Range or ClosedRange types but with the addition of an optional by value. This has been implemented as an extension to the Array type.

Swift style:

let t = [Double](0...<100)
let s = [Double](1...100, 2)

VecLab style using the vector function:

let t = vector(0..<100)
let s = vector(1...100, 2)

Overloaded operators for scalar and vectors.