SwiftyMath

The aim of this project is to understand Mathematics by realizing abstract concepts as codes. Mathematical axioms correspond to protocols, and objects satisfying some axioms correspond to structs.

Submodules

• SwiftyHomology
• SwiftyTopology
• SwiftyKnots

Getting Started

Using SwiftyMathREPL

Creating Your Own Project

1. Initialize a Package

$ mkdir YourProject
$ cd YourProject
$ swift package init --type executable

2. Edit Package.swift

 // swift-tools-version:5.1
 // The swift-tools-version declares the minimum version of Swift required to build this package.
 
 import PackageDescription
 
 let package = Package(
     name: "YourProject",
     dependencies: [
         // Dependencies declare other packages that this package depends on.
-        // .package(url: /* package url */, from: "1.0.0"),
+        .package(url: "https://github.com/taketo1024/SwiftyMath.git", from: "1.0.0"),
     ],
     targets: [
         // Targets are the basic building blocks of a package. A target can define a module or a test suite.
         // Targets can depend on other targets in this package, and on products in packages which this package depends on.
         .target(
             name: "YourProject",
-            dependencies: []),
+            dependencies: ["SwiftyMath"]),
     ]
 )

3. Edit Sources/YourProject/main.swift

import SwiftyMath

let a = RationalNumber(4, 5)  // 4/5
let b = RationalNumber(3, 2)  // 3/2

print(a + b)     // 23/10

4. Run

$ swift run

 23/10

Using Mathematical Symbols

We make use of mathematical symbols such as sets 𝐙, 𝐐, 𝐑, 𝐂 and operators ⊕, ⊗ etc. Copy the folder CodeSnippets to ~/Library/Developer/Xcode/UserData/ then you can quickly input these symbols by the completion of Xcode.

Examples

Rational Numbers

let a = 𝐐(4, 5)  // 4/5
let b = 𝐐(3, 2)  // 3/2

a + b  // 23/10
a * b  // 6/5
b / a  // 15/8

Matrices

typealias M = SquareMatrix<_2, 𝐙> // Matrix of integers with fixed size 2×2.

let a = M(1, 2, 3, 4)  // [1, 2; 3, 4]
let b = M(2, 1, 1, 2)  // [2, 1; 1, 2]

a + b  // [3, 3; 4, 6]
a * b  // [4, 5; 10, 11]

a + b == b + a  // true: addition is commutative
a * b == b * a  // false: multiplication is noncommutative

Permutation (Symmetric Group)

typealias S_5 = Permutation<_5>

let s = S_5(cyclic: 0, 1, 2) // cyclic notation
let t = S_5([0: 2, 1: 3, 2: 4, 3: 0, 4: 1]) // two-line notation

s[1]  // 2
t[2]  // 4

(s * t)[3]  // 3 -> 0 -> 1
(t * s)[3]  // 3 -> 3 -> 0

Polynomials

typealias P = Polynomial<𝐐>

let f = P(0, 2, -3, 1) // x^3 − 3x^2 + 2x
let g = P(6, -5, 1)    // x^2 − 5x + 6
    
f + g  // x^3 - 2x^2 - 3x + 6
f * g  // x^5 - 8x^4 + 23x^3 - 28x^2 + 12x
f % g  // 6x - 12
    
gcd(f, g) // 6x - 12

Integer Quotients, Finite Fields

typealias Z_4 = IntegerQuotientRing<_4>
Z_4.printAddTable()

+   |   0   1   2   3
----------------------
0   |   0   1   2   3
1   |   1   2   3   0
2   |   2   3   0   1
3   |   3   0   1   2

typealias F_5 = IntegerQuotientField<_5>
F_5.printMulTable()

*   |   0   1   2   3   4
--------------------------
0   |   0   0   0   0   0
1   |   0   1   2   3   4
2   |   0   2   4   1   3
3   |   0   3   1   4   2
4   |   0   4   3   2   1

Algebraic Extension

// Construct an algebraic extension over 𝐐:
// K = 𝐐(√2) = 𝐐[x]/(x^2 - 2).

struct p: _Polynomial {                            // p = x^2 - 2, as a struct
    typealias K = 𝐐
    static let value = Polynomial<𝐐>(-2, 0, 1)
}

typealias I = PolynomialIdeal<p>                   // I = (x^2 - 2)
typealias K = QuotientField<Polynomial<𝐐>, I>      // K = 𝐐[x]/I

let a = Polynomial<𝐐>(0, 1).asQuotient(in: K.self) // a = x mod I
a * a == 2                                         // true!

License

Swifty Math is released under MIT license.