📐 A math framework for Swift. Includes: vectors, matrices, complex numbers, quaternions and polynomials.
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Carthage compatible

SwiftMath is a Swift framework providing some useful math constructs and functions, like complex numbers, vectors, matrices, quaternions, and polynomials.

⚠️ SwiftMath is work in progress, in alpha state. Master is currently targeting Swift 2.1.


SwiftMath requires iOS 8.0+ / OS X 10.9+.


SwiftMath can be installed with the dependency manager Carthage.

  • Add the following line to your project's Cartfile
github "madbat/SwiftMath"
  • In the terminal, run carthage update
  • Link your project target(s) with the built frameworks. Application targets should also ensure that the framework gets copied into their application bundle.



Vector3 — as the name suggests — represents a vector in the three-dimensional Euclidean space (aka R×R×R). Some of the most common uses of 3D vectors consist in encoding physical quantities like position, velocity, acceleration, force, and many others.

let v1 = Vector3(x: 1, y: 2, z: 3)
let v2 = Vector3(x: 5, y: 6, z: 7)

// vector sum
let v3 = v1 + v2 // Vector3(x: 6, y: 8, z: 10)

// length
v3.length // equals v3.norm

// zero vector
Vector3.zero() // Vector3(x: 0, y: 0, z: 0)

// unit-length vector
v3.unit() // divides v3 by its length


Pretty much like Vector3, but for 2D vectors.


Complex numbers extend real numbers in order to solve problems that cannot be solved with real numbers alone. For example, the roots of a polynomial equation of degree > 1 can always be expressed with complex numbers, but not with real numbers.

// the default constructor for Complex takes the real and imaginary parts as parameters
let c1 = Complex(1.0, 3.0)
c1.re // 1.0
c1.im // 3.0

// a complex can also be constructed by using the property i defined on Float and Double
let c2 = 5 + 1.i // Complex(5.0, 1.0)

// complex conjugate
c2.conj() // Complex(5.0, -1.0)

// polar form
let c3 = Complex(abs: 2.0, arg: -4.0)

let realComplex = Complex(10.0, 0.0)
realComplex.isReal // true


Quaternions extend complex numbers to 4 dimensions. They're handy to rotate three-dimensional vectors.

// rotating a vector by π/2 around its x axis
let original = Vector3(x: 3, y: 4, z: 0)
let rotation = Quaternion(axis: Vector3(x: 1, y: 0, z: 0), angle: Double.PI/2.0)
let rotated = original.rotate(rotation) // Vector3(x: 3, y: 0, z: 4.0)


Polynomial lets you represent – and find the roots of – a polynomial expression.

The following snippet shows how to express the polynomial x^2 + 4x + 8

let p = Polynomial(1, 4, 8)

Use Polynomial's roots() method to calculate its roots, represented as a (multi)set of complex numbers:

p.roots() // returns { (-2 - 2i), (-2 + 2i) }

For polynomials of degree <= 4, roots() defaults to using the analytic method, while for polynomials of higher degrees it uses the the Durand-Kerner method. It is possible to force the root finding process to use the numeric method also for polynomials of degree <= 4, using roots(preferClosedFormSolution: false).


Contributions in any form (especially pull requests) are very welcome!


SwiftMath is released under the MIT License. See the LICENSE file for more info.