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Multiplicative.scala
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Multiplicative.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.{specialized => sp}
import scala.annotation.{nowarn, tailrec}
trait MultiplicativeSemigroup[@sp(Int, Long, Float, Double) A] extends Any with Serializable {
def multiplicative: Semigroup[A] = times(_, _)
def times(x: A, y: A): A
def pow(a: A, n: Int): A =
if (n > 0) positivePow(a, n)
else throw new IllegalArgumentException("Illegal non-positive exponent to pow: %s".format(n))
protected[this] def positivePow(a: A, n: Int): A = {
@tailrec def loop(b: A, k: Int, extra: A): A =
if (k == 1) times(b, extra)
else {
val x = if ((k & 1) == 1) times(b, extra) else extra
loop(times(b, b), k >>> 1, x)
}
if (n == 1) a else loop(a, n - 1, a)
}
/**
* Given a sequence of `as`, combine them and return the total.
*
* If the sequence is empty, returns None. Otherwise, returns Some(total).
*/
@nowarn("msg=deprecated")
def tryProduct(as: TraversableOnce[A]): Option[A] =
as.toIterator.reduceOption(times)
}
trait MultiplicativeCommutativeSemigroup[@sp(Int, Long, Float, Double) A] extends Any with MultiplicativeSemigroup[A] {
override def multiplicative: CommutativeSemigroup[A] = times(_, _)
}
trait MultiplicativeMonoid[@sp(Int, Long, Float, Double) A] extends Any with MultiplicativeSemigroup[A] {
override def multiplicative: Monoid[A] = new Monoid[A] {
def empty = one
def combine(x: A, y: A): A = times(x, y)
}
def one: A
/**
* Tests if `a` is one.
*/
def isOne(a: A)(implicit ev: Eq[A]): Boolean = ev.eqv(a, one)
override def pow(a: A, n: Int): A =
if (n > 0) positivePow(a, n)
else if (n == 0) one
else throw new IllegalArgumentException("Illegal negative exponent to pow: %s".format(n))
/**
* Given a sequence of `as`, compute the product.
*/
@nowarn("msg=deprecated")
def product(as: TraversableOnce[A]): A =
as.foldLeft(one)(times)
@nowarn("msg=deprecated")
override def tryProduct(as: TraversableOnce[A]): Option[A] =
if (as.isEmpty) None else Some(product(as))
}
trait MultiplicativeCommutativeMonoid[@sp(Int, Long, Float, Double) A]
extends Any
with MultiplicativeMonoid[A]
with MultiplicativeCommutativeSemigroup[A] {
override def multiplicative: CommutativeMonoid[A] = new CommutativeMonoid[A] {
def empty = one
def combine(x: A, y: A): A = times(x, y)
}
}
trait MultiplicativeGroup[@sp(Int, Long, Float, Double) A] extends Any with MultiplicativeMonoid[A] {
override def multiplicative: Group[A] = new Group[A] {
def empty = one
def combine(x: A, y: A): A = times(x, y)
override def remove(x: A, y: A): A = div(x, y)
def inverse(x: A): A = reciprocal(x)
}
def reciprocal(x: A): A = div(one, x)
def div(x: A, y: A): A
override def pow(a: A, n: Int): A =
if (n > 0) positivePow(a, n)
else if (n == 0) one
else if (n == Int.MinValue) positivePow(reciprocal(times(a, a)), 1073741824)
else positivePow(reciprocal(a), -n)
}
trait MultiplicativeCommutativeGroup[@sp(Int, Long, Float, Double) A]
extends Any
with MultiplicativeGroup[A]
with MultiplicativeCommutativeMonoid[A] {
override def multiplicative: CommutativeGroup[A] = new CommutativeGroup[A] {
def empty = one
def combine(x: A, y: A): A = times(x, y)
override def remove(x: A, y: A): A = div(x, y)
def inverse(x: A): A = reciprocal(x)
}
}
trait MultiplicativeSemigroupFunctions[S[T] <: MultiplicativeSemigroup[T]] {
def isMultiplicativeCommutative[A](implicit ev: S[A]): Boolean =
ev.isInstanceOf[MultiplicativeCommutativeSemigroup[A]]
def times[@sp(Int, Long, Float, Double) A](x: A, y: A)(implicit ev: S[A]): A =
ev.times(x, y)
def pow[@sp(Int, Long, Float, Double) A](a: A, n: Int)(implicit ev: S[A]): A =
ev.pow(a, n)
@nowarn("msg=deprecated")
def tryProduct[A](as: TraversableOnce[A])(implicit ev: S[A]): Option[A] =
ev.tryProduct(as)
}
trait MultiplicativeMonoidFunctions[M[T] <: MultiplicativeMonoid[T]] extends MultiplicativeSemigroupFunctions[M] {
def one[@sp(Int, Long, Float, Double) A](implicit ev: M[A]): A =
ev.one
def isOne[@sp(Int, Long, Float, Double) A](a: A)(implicit ev0: M[A], ev1: Eq[A]): Boolean =
ev0.isOne(a)
@nowarn("msg=deprecated")
def product[@sp(Int, Long, Float, Double) A](as: TraversableOnce[A])(implicit ev: M[A]): A =
ev.product(as)
}
trait MultiplicativeGroupFunctions[G[T] <: MultiplicativeGroup[T]] extends MultiplicativeMonoidFunctions[G] {
def reciprocal[@sp(Int, Long, Float, Double) A](x: A)(implicit ev: G[A]): A =
ev.reciprocal(x)
def div[@sp(Int, Long, Float, Double) A](x: A, y: A)(implicit ev: G[A]): A =
ev.div(x, y)
}
object MultiplicativeSemigroup extends MultiplicativeSemigroupFunctions[MultiplicativeSemigroup] {
@inline final def apply[A](implicit ev: MultiplicativeSemigroup[A]): MultiplicativeSemigroup[A] = ev
/**
* This method converts a multiplicative instance into a generic
* instance.
*
* Given an implicit `MultiplicativeSemigroup[A]`, this method returns
* a `Semigroup[A]`.
*/
@inline final def multiplicative[A](implicit ev: MultiplicativeSemigroup[A]): Semigroup[A] =
ev.multiplicative
}
object MultiplicativeCommutativeSemigroup extends MultiplicativeSemigroupFunctions[MultiplicativeCommutativeSemigroup] {
@inline final def apply[A](implicit
ev: MultiplicativeCommutativeSemigroup[A]
): MultiplicativeCommutativeSemigroup[A] = ev
/**
* This method converts a multiplicative instance into a generic
* instance.
*
* Given an implicit `MultiplicativeCommutativeSemigroup[A]`, this method returns
* a `CommutativeSemigroup[A]`.
*/
@inline final def multiplicative[A](implicit ev: MultiplicativeCommutativeSemigroup[A]): CommutativeSemigroup[A] =
ev.multiplicative
}
object MultiplicativeMonoid extends MultiplicativeMonoidFunctions[MultiplicativeMonoid] {
@inline final def apply[A](implicit ev: MultiplicativeMonoid[A]): MultiplicativeMonoid[A] = ev
/**
* This method converts a multiplicative instance into a generic
* instance.
*
* Given an implicit `MultiplicativeMonoid[A]`, this method returns
* a `Monoid[A]`.
*/
@inline final def multiplicative[A](implicit ev: MultiplicativeMonoid[A]): Monoid[A] =
ev.multiplicative
}
object MultiplicativeCommutativeMonoid extends MultiplicativeMonoidFunctions[MultiplicativeCommutativeMonoid] {
@inline final def apply[A](implicit ev: MultiplicativeCommutativeMonoid[A]): MultiplicativeCommutativeMonoid[A] = ev
/**
* This method converts a multiplicative instance into a generic
* instance.
*
* Given an implicit `MultiplicativeCommutativeMonoid[A]`, this method returns
* a `CommutativeMonoid[A]`.
*/
@inline final def multiplicative[A](implicit ev: MultiplicativeCommutativeMonoid[A]): CommutativeMonoid[A] =
ev.multiplicative
}
object MultiplicativeGroup extends MultiplicativeGroupFunctions[MultiplicativeGroup] {
@inline final def apply[A](implicit ev: MultiplicativeGroup[A]): MultiplicativeGroup[A] = ev
/**
* This method converts a multiplicative instance into a generic
* instance.
*
* Given an implicit `MultiplicativeGroup[A]`, this method returns
* a `Group[A]`.
*/
@inline final def multiplicative[A](implicit ev: MultiplicativeGroup[A]): Group[A] =
ev.multiplicative
}
object MultiplicativeCommutativeGroup extends MultiplicativeGroupFunctions[MultiplicativeCommutativeGroup] {
@inline final def apply[A](implicit ev: MultiplicativeCommutativeGroup[A]): MultiplicativeCommutativeGroup[A] = ev
/**
* This method converts a multiplicative instance into a generic
* instance.
*
* Given an implicit `MultiplicativeCommutativeGroup[A]`, this method returns
* a `CommutativeGroup[A]`.
*/
@inline final def multiplicative[A](implicit ev: MultiplicativeCommutativeGroup[A]): CommutativeGroup[A] =
ev.multiplicative
}