trait EuclideanRing[A] extends GCDRing[A]
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
- Self Type
- EuclideanRing[A]
- Source
- EuclideanRing.scala
- Alphabetic
- By Inheritance
- EuclideanRing
- GCDRing
- CommutativeRing
- CommutativeRng
- CommutativeRig
- MultiplicativeCommutativeMonoid
- CommutativeSemiring
- MultiplicativeCommutativeSemigroup
- Ring
- Rng
- AdditiveCommutativeGroup
- AdditiveGroup
- Rig
- MultiplicativeMonoid
- Semiring
- MultiplicativeSemigroup
- AdditiveCommutativeMonoid
- AdditiveCommutativeSemigroup
- AdditiveMonoid
- AdditiveSemigroup
- Serializable
- Any
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- Public
- Protected
Abstract Value Members
- abstract def emod(a: A, b: A): A
- abstract def equot(a: A, b: A): A
- abstract def euclideanFunction(a: A): BigInt
- abstract def getClass(): Class[_ <: AnyRef]
- Definition Classes
- Any
- abstract def negate(x: A): A
- Definition Classes
- AdditiveGroup
- abstract def one: A
- Definition Classes
- MultiplicativeMonoid
- abstract def plus(x: A, y: A): A
- Definition Classes
- AdditiveSemigroup
- abstract def times(x: A, y: A): A
- Definition Classes
- MultiplicativeSemigroup
- abstract def zero: A
- Definition Classes
- AdditiveMonoid
Concrete Value Members
- final def !=(arg0: Any): Boolean
- Definition Classes
- Any
- final def ##: Int
- Definition Classes
- Any
- final def ==(arg0: Any): Boolean
- Definition Classes
- Any
- def additive: CommutativeGroup[A]
- Definition Classes
- AdditiveCommutativeGroup → AdditiveCommutativeMonoid → AdditiveCommutativeSemigroup → AdditiveGroup → AdditiveMonoid → AdditiveSemigroup
- final def asInstanceOf[T0]: T0
- Definition Classes
- Any
- def equals(arg0: Any): Boolean
- Definition Classes
- Any
- def equotmod(a: A, b: A): (A, A)
- def fromBigInt(n: BigInt): A
Convert the given BigInt to an instance of A.
Convert the given BigInt to an instance of A.
This is equivalent to
nrepeated summations of this ring'sone, or-nsummations of-oneifnis negative.Most type class instances should consider overriding this method for performance reasons.
- Definition Classes
- Ring
- def fromInt(n: Int): A
Convert the given integer to an instance of A.
Convert the given integer to an instance of A.
Defined to be equivalent to
sumN(one, n).That is,
nrepeated summations of this ring'sone, or-nsummations of-oneifnis negative.Most type class instances should consider overriding this method for performance reasons.
- Definition Classes
- Ring
- def gcd(a: A, b: A)(implicit ev: Eq[A]): A
- Definition Classes
- EuclideanRing → GCDRing
- def hashCode(): Int
- Definition Classes
- Any
- final def isInstanceOf[T0]: Boolean
- Definition Classes
- Any
- def isOne(a: A)(implicit ev: Eq[A]): Boolean
Tests if
ais one.Tests if
ais one.- Definition Classes
- MultiplicativeMonoid
- def isZero(a: A)(implicit ev: Eq[A]): Boolean
Tests if
ais zero.Tests if
ais zero.- Definition Classes
- AdditiveMonoid
- def lcm(a: A, b: A)(implicit ev: Eq[A]): A
- Definition Classes
- EuclideanRing → GCDRing
- def minus(x: A, y: A): A
- Definition Classes
- AdditiveGroup
- def multiplicative: CommutativeMonoid[A]
- Definition Classes
- MultiplicativeCommutativeMonoid → MultiplicativeCommutativeSemigroup → MultiplicativeMonoid → MultiplicativeSemigroup
- def positivePow(a: A, n: Int): A
- Attributes
- protected[this]
- Definition Classes
- MultiplicativeSemigroup
- def positiveSumN(a: A, n: Int): A
- Attributes
- protected[this]
- Definition Classes
- AdditiveSemigroup
- def pow(a: A, n: Int): A
- Definition Classes
- MultiplicativeMonoid → MultiplicativeSemigroup
- def product(as: TraversableOnce[A]): A
Given a sequence of
as, compute the product.Given a sequence of
as, compute the product.- Definition Classes
- MultiplicativeMonoid
- Annotations
- @nowarn()
- def sum(as: TraversableOnce[A]): A
Given a sequence of
as, compute the sum.Given a sequence of
as, compute the sum.- Definition Classes
- AdditiveMonoid
- Annotations
- @nowarn()
- def sumN(a: A, n: Int): A
- Definition Classes
- AdditiveGroup → AdditiveMonoid → AdditiveSemigroup
- def toString(): String
- Definition Classes
- Any
- def tryProduct(as: TraversableOnce[A]): Option[A]
Given a sequence of
as, combine them and return the total.Given a sequence of
as, combine them and return the total.If the sequence is empty, returns None. Otherwise, returns Some(total).
- Definition Classes
- MultiplicativeMonoid → MultiplicativeSemigroup
- Annotations
- @nowarn()
- def trySum(as: TraversableOnce[A]): Option[A]
Given a sequence of
as, combine them and return the total.Given a sequence of
as, combine them and return the total.If the sequence is empty, returns None. Otherwise, returns Some(total).
- Definition Classes
- AdditiveMonoid → AdditiveSemigroup
- Annotations
- @nowarn()