Swift で数学のススメ 〜 プログラミングと数学は同時に学べ

Swift で数学のススメ 〜 プログラミングと数学は同時に学べ

1. 1. protocol AdditiveGroup { // static var zero: Self { get } // static func + (a: Self, b: Self) -> Self // prefix static func - (x: Self) -> Self // } extension AdditiveGroup { static func -(a: Self, b: Self) -> Self { return (a + (-b)) // } }
2. 2. protocol AdditiveGroup { // … } protocol Ring: AdditiveGroup { // static var identity: Self { get } // 1 static func * (a: Self, b: Self) -> Self // var inverse: Self? { get } // (optional) }
3. 3. protocol AdditiveGroup { // … } protocol Ring: AdditiveGroup { // … } protocol Field: Ring {} // extension Field { static func / (a: Self, b: Self) -> Self { // return a * b.inverse! // 0 } }
4. 4. extension Int: Ring { // Int static var zero: Int { return 0 } static var identity: Int { return 1 } } // Int
5. 5. struct Rational: Field { // private let p, q: Int init(_ p: Int, _ q: Int) { (self.p, self.q) = (p, q) } static var zero: Int { return Rational(0, 1) } static var identity: Int { return Rational(1, 1) } var inverse: Rational? { // return (p != 0) ? Rational(q, p) : nil } …
6. 6. struct Rational: Field { … static func + (a: Rational, b: Rational) -> Rational { return Rational(a.p * b.q + a.q * b.p, a.q * b.q) } static prefix func - (a: Rational) -> Rational { return Rational(-a.p, a.q) } static func * (a: Rational, b: Rational) -> Rational { return Rational(a.p * b.p, a.q * b.q) } }
7. 7. protocol EuclideanRing: Ring { // static func eucDiv(_ a: Self, _ b: Self) -> (q: Self, r: Self) // } extension EuclideanRing { static func % (_ a: Self, b: Self) -> Self { // return Self.eucDiv(a, b).r } }
8. 8. extension Int: EuclideanRing { // Int EuclideanRing static func eucDiv(_ a: Int, _ b: Int) -> (q: Int, r: Int) { // let q = a / b return (q: q, r: a - q * b) } }
9. 9. struct Polynomial<K: Field>: EuclideanRing { public let coeffs: [K] public init(_ coeffs: K...) { self.coeffs = coeffs } public static func + (f: Polynomial<K>, g: Polynomial<K>) -> Polynomial<K> { return Polynomial<K>(degree: max(f.degree, g.degree)) { f.coeff($0) + g.coeff($0) } } public static prefix func - (f: Polynomial<K>) -> Polynomial<K> { return f.map { -$0 } } public static func * (f: Polynomial<K>, g: Polynomial<K>) -> Polynomial<K> { … } …
10. 10. struct Polynomial<K: Field>: EuclideanRing { … static func eucDiv<K: Field>(_ f: Polynomial<K>, _ g: Polynomial<K>) -> (q: Polynomial<K>, r: Polynomial<K>) { return (0 ... max(0, f.degree - g.degree)) .reversed() .reduce( (0, f) ) { (result: (Polynomial<K>, Polynomial<K>), degree: Int) in let (q, r) = result let m = eucDivMonomial(r, g) return (q + m.q, m.r) } } }
11. 11. public func gcd<R: EuclideanRing>(_ a: R, _ b: R) -> R { switch b { case 0: return a default: return gcd(b, a % b) } }
12. 12. let a = sqrt(2.0) // 1.41421356… a * a == 2.0 // false