basis.math

CompleteField

trait CompleteField extends Field

A complete abstract field structure. Addition associates and commutes, and multiplication associates, commutes, and distributes over addition. Addition and multiplication both have an identity element, every element has an additive inverse, and every element except zero has a multiplicative inverse. Completeness implies that every Cauchy sequence of elements converges. To the extent practicable, the following axioms should hold.

Axioms for addition:

Axioms for multiplication:

The distributive law:

Completeness axiom:

Source
CompleteField.scala
Example: 

  1. // You can abstract over complete fields by parameterizing a class or
// function with a subtype of CompleteField with Singleton. Type elements
// with the #Element type projection of your CompleteField type parameter.
def testCompleteFieldOperations[F <: CompleteField with Singleton](a: F#Element, b: F#Element, c: F#Element): Unit = {
  assert(a + b == b + a, "commutativity of addition")
  assert((a + b) + c == a + (b + c), "associativity of addition")
  assert(a * b == b * a, "commutativity of multiplication")
  assert((a * b) * c == a * (b * c), "associativity of multiplication")
  assert(a * (b + c) == (a * b) + (a * c), "distributivity of multiplication over addition")
}

// Alternatively, functions can use path-dependent types of a CompleteField parameter.
def testCompleteFieldIdentities(F: CompleteField)(a: F.Element): Unit = {
  import F._
  assert(zero + a == a, "existence of additive identity")
  assert(a + (-a) == zero, "existence of additive inverse")
  assert(unit != zero && unit * a == a, "existence of multiplicative identity")
  assert(a * a.inverse == unit, "existence of multiplicative inverse")
}
Version

0.1

Since

0.0

Linear Supertypes
Field, Ring, AnyRef, Any
Known Subclasses
Real, RealField
Type Hierarchy Learn more about scaladoc diagrams
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Inherited
  1. CompleteField
  2. Field
  3. Ring
  4. AnyRef
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Visibility
  1. Public
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Type Members

  1. trait CompleteFieldElement extends FieldElement

  2. abstract type Element <: CompleteFieldElement

    The type of elements in this complete field.

    The type of elements in this complete field.

    Definition Classes
    CompleteField → Field → Ring

  3. trait FieldElement extends RingElement

    Definition Classes
    Field

  4. trait RingElement extends Any

    An element in this ring.

    An element in this ring.

    Definition Classes
    Ring

Abstract Value Members

  1. abstract def unit: Element

    Returns the multiplicative identity of this complete field.

    Returns the multiplicative identity of this complete field.

    Definition Classes
    CompleteField → Field → Ring

  2. abstract def zero: Element

    Returns the additive identity of this complete field.

    Returns the additive identity of this complete field.

    Definition Classes
    CompleteField → Field → Ring

Concrete Value Members

  1. final def !=(arg0: Any): Boolean

    Definition Classes
    AnyRef → Any

  2. final def ##(): Int

    Definition Classes
    AnyRef → Any

  3. final def ==(arg0: Any): Boolean

    Definition Classes
    AnyRef → Any

  4. final def asInstanceOf[T0]: T0

    Definition Classes
    Any

  5. def clone(): AnyRef

    Attributes
    protected[java.lang]
    Definition Classes
    AnyRef
    Annotations
    @throws( ... )

  6. final def eq(arg0: AnyRef): Boolean

    Definition Classes
    AnyRef

  7. def equals(arg0: Any): Boolean

    Definition Classes
    AnyRef → Any

  8. def finalize(): Unit

    Attributes
    protected[java.lang]
    Definition Classes
    AnyRef
    Annotations
    @throws( classOf[java.lang.Throwable] )

  9. final def getClass(): Class[_]

    Definition Classes
    AnyRef → Any

  10. def hashCode(): Int

    Definition Classes
    AnyRef → Any

  11. final def isInstanceOf[T0]: Boolean

    Definition Classes
    Any

  12. final def ne(arg0: AnyRef): Boolean

    Definition Classes
    AnyRef

  13. final def notify(): Unit

    Definition Classes
    AnyRef

  14. final def notifyAll(): Unit

    Definition Classes
    AnyRef

  15. final def synchronized[T0](arg0: ⇒ T0): T0

    Definition Classes
    AnyRef

  16. def toString(): String

    Definition Classes
    AnyRef → Any

  17. final def wait(): Unit

    Definition Classes
    AnyRef
    Annotations
    @throws( ... )

  18. final def wait(arg0: Long, arg1: Int): Unit

    Definition Classes
    AnyRef
    Annotations
    @throws( ... )

  19. final def wait(arg0: Long): Unit

    Definition Classes
    AnyRef
    Annotations
    @throws( ... )

Inherited from Field

Inherited from Ring

Inherited from AnyRef

Inherited from Any

Ungrouped