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EuclideanRing.scala
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EuclideanRing.scala
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/*
* Copyright (c) 2015 Typelevel
*
* Permission is hereby granted, free of charge, to any person obtaining a copy of
* this software and associated documentation files (the "Software"), to deal in
* the Software without restriction, including without limitation the rights to
* use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of
* the Software, and to permit persons to whom the Software is furnished to do so,
* subject to the following conditions:
*
* The above copyright notice and this permission notice shall be included in all
* copies or substantial portions of the Software.
*
* THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
* IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS
* FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR
* COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER
* IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN
* CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.
*/
package algebra
package ring
import scala.annotation.tailrec
import scala.{specialized => sp}
/**
* EuclideanRing implements a Euclidean domain.
*
* The formal definition says that every euclidean domain A has (at
* least one) euclidean function f: A -> N (the natural numbers) where:
*
* (for every x and non-zero y) x = yq + r, and r = 0 or f(r) < f(y).
*
* This generalizes the Euclidean division of integers, where f represents
* a measure of length (or absolute value), and the previous equation
* represents finding the quotient and remainder of x and y. So:
*
* quot(x, y) = q
* mod(x, y) = r
*/
trait EuclideanRing[@sp(Int, Long, Float, Double) A] extends Any with GCDRing[A] { self =>
def euclideanFunction(a: A): BigInt
def equot(a: A, b: A): A
def emod(a: A, b: A): A
def equotmod(a: A, b: A): (A, A) = (equot(a, b), emod(a, b))
def gcd(a: A, b: A)(implicit ev: Eq[A]): A =
EuclideanRing.euclid(a, b)(ev, self)
def lcm(a: A, b: A)(implicit ev: Eq[A]): A =
if (isZero(a) || isZero(b)) zero else times(equot(a, gcd(a, b)), b)
}
trait EuclideanRingFunctions[R[T] <: EuclideanRing[T]] extends GCDRingFunctions[R] {
def euclideanFunction[@sp(Int, Long, Float, Double) A](a: A)(implicit ev: R[A]): BigInt =
ev.euclideanFunction(a)
def equot[@sp(Int, Long, Float, Double) A](a: A, b: A)(implicit ev: R[A]): A =
ev.equot(a, b)
def emod[@sp(Int, Long, Float, Double) A](a: A, b: A)(implicit ev: R[A]): A =
ev.emod(a, b)
def equotmod[@sp(Int, Long, Float, Double) A](a: A, b: A)(implicit ev: R[A]): (A, A) =
ev.equotmod(a, b)
}
object EuclideanRing extends EuclideanRingFunctions[EuclideanRing] {
@inline final def apply[A](implicit e: EuclideanRing[A]): EuclideanRing[A] = e
/**
* Simple implementation of Euclid's algorithm for gcd
*/
@tailrec final def euclid[@sp(Int, Long, Float, Double) A: Eq: EuclideanRing](a: A, b: A): A = {
if (EuclideanRing[A].isZero(b)) a else euclid(b, EuclideanRing[A].emod(a, b))
}
}