GitHub - taketo1024/SwiftyMath: Pure Math in Pure Swift.
source link: https://github.com/taketo1024/SwiftyMath
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README.md
SwiftyMath
The aim of this project is to understand Mathematics by realizing abstract concepts as codes. Mathematical axioms correspond to protocol
s, and objects satisfying some axioms correspond to struct
s.
Getting Started
Swift REPL
With Xcode installed, you can run SwiftyMath on the Swift-REPL by:
$ swift build
$ swift -I .build/debug/ -L .build/debug/ -ldSwiftyMath
Try something like:
:set set print-decls false // suppress print-decl import SwiftyMath typealias F5 = IntegerQuotientRing<_5> F5.printAddTable() F5.printMulTable()
Creating Your Own Project
1. Initialize a Package
$ mkdir YourProject
$ cd YourProject
$ swift package init --type executable
2. Edit Package.swift
// swift-tools-version:4.0 // 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: "0.1.0"), ], 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", "SwiftyTopology"]), ] )
3. Edit Sources/YourProject/main.swift
import SwiftyMath let a = ?(4, 5) // 4/5 let b = ?(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 ?, ?, ?, ? and operators ⊕, ⊗ etc. Copy the folder CodeSnippets
to ~/Library/Xcode/UserData/
then you can quickly input these symbols by the completion of Xcode.
Examples
Rational Numbers
let a = ?(4, 5) // 4/5 let b = ?(3, 2) // 3/2 a + b // 23/10 a * b // 6/5 b / a // 15/8
Matrices (type safe)
typealias M = Matrix<_2, _2, ?> // 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<?> 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 ?: // K = ?(√2) = ?[x]/(x^2 - 2). struct p: _Polynomial { // p = x^2 - 2, as a struct typealias K = ? static let value = Polynomial<?>(-2, 0, 1) } typealias I = PolynomialIdeal<p> // I = (x^2 - 2) typealias K = QuotientField<Polynomial<?>, I> // K = ?[x]/I let a = Polynomial<?>(0, 1).asQuotient(in: K.self) // a = x mod I a * a == 2 // true!
Homology, Cohomology
import SwiftyMath import SwiftyTopology let S2 = SimplicialComplex.sphere(dim: 2) let H = Homology(S2, ?.self) print("H(S^2; ?) =", H.detailDescription, "\n")
H(S^2; ?) = {
0 : ?, [(v1)],
1 : 0, [],
2 : ?, [-1(v0, v2, v3) + -1(v0, v1, v2) + (v1, v2, v3) + (v0, v1, v3)]
}
let RP2 = SimplicialComplex.realProjectiveSpace(dim: 2) let H = Homology(RP2, ?₂.self) print("H(RP^2; ?₂) =", H.detailDescription, "\n")
H(RP^2; ?₂) = {
0 : ?₂, [(v1)],
1 : ?₂, [(v0, v1) + (v1, v2) + (v0, v3) + (v2, v3)],
2 : ?₂, [(v0, v2, v3) + (v3, v4, v5) + (v2, v3, v5) + (v1, v2, v5) + (v0, v4, v5) + (v1, v3, v4) + (v0, v1, v5) + (v1, v2, v4) + (v0, v2, v4) + (v0, v1, v3)]
}
References
License
Swifty Math is licensed under CC0 1.0 Universal.
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