Module

Data.EuclideanRing

#EuclideanRing 

class (CommutativeRing a) <= EuclideanRing a  where

The EuclideanRing class is for commutative rings that support division. The mathematical structure this class is based on is sometimes also called a Euclidean domain.

Instances must satisfy the following laws in addition to the Ring laws:

  • Integral domain: one /= zero, and if a and b are both nonzero then so is their product a * b
  • Euclidean function degree:
    • Nonnegativity: For all nonzero a, degree a >= 0
    • Quotient/remainder: For all a and b, where b is nonzero, let q = a / b and r = a `mod` b; then a = q*b + r, and also either r = zero or degree r < degree b
  • Submultiplicative euclidean function:
    • For all nonzero a and b, degree a <= degree (a * b)

The behaviour of division by zero is unconstrained by these laws, meaning that individual instances are free to choose how to behave in this case. Similarly, there are no restrictions on what the result of degree zero is; it doesn't make sense to ask for degree zero in the same way that it doesn't make sense to divide by zero, so again, individual instances may choose how to handle this case.

For any EuclideanRing which is also a Field, one valid choice for degree is simply const 1. In fact, unless there's a specific reason not to, Field types should normally use this definition of degree.

The EuclideanRing Int instance is one of the most commonly used EuclideanRing instances and deserves a little more discussion. In particular, there are a few different sensible law-abiding implementations to choose from, with slightly different behaviour in the presence of negative dividends or divisors. The most common definitions are "truncating" division, where the result of a / b is rounded towards 0, and "Knuthian" or "flooring" division, where the result of a / b is rounded towards negative infinity. A slightly less common, but arguably more useful, option is "Euclidean" division, which is defined so as to ensure that a `mod` b is always nonnegative. With Euclidean division, a / b rounds towards negative infinity if the divisor is positive, and towards positive infinity if the divisor is negative. Note that all three definitions are identical if we restrict our attention to nonnegative dividends and divisors.

In versions 1.x, 2.x, and 3.x of the Prelude, the EuclideanRing Int instance used truncating division. As of 4.x, the EuclideanRing Int instance uses Euclidean division. Additional functions quot and rem are supplied if truncating division is desired.

Members

Instances

#(/)

Operator alias for Data.EuclideanRing.div (left-associative / precedence 7)

#gcd 

gcd :: forall a. Eq a => EuclideanRing a => a -> a -> a

The greatest common divisor of two values.

#lcm 

lcm :: forall a. Eq a => EuclideanRing a => a -> a -> a

The least common multiple of two values.

Re-exports from Data.CommutativeRing

#CommutativeRing 

class (Ring a) <= CommutativeRing a

The CommutativeRing class is for rings where multiplication is commutative.

Instances must satisfy the following law in addition to the Ring laws:

  • Commutative multiplication: a * b = b * a

Instances

Re-exports from Data.Ring

#Ring 

class (Semiring a) <= Ring a  where

The Ring class is for types that support addition, multiplication, and subtraction operations.

Instances must satisfy the following laws in addition to the Semiring laws:

  • Additive inverse: a - a = zero
  • Compatibility of sub and negate: a - b = a + (zero - b)

Members

  • sub :: a -> a -> a

Instances

#(-)

Operator alias for Data.Ring.sub (left-associative / precedence 6)

Re-exports from Data.Semiring

#Semiring 

class Semiring a  where

The Semiring class is for types that support an addition and multiplication operation.

Instances must satisfy the following laws:

  • Commutative monoid under addition:
    • Associativity: (a + b) + c = a + (b + c)
    • Identity: zero + a = a + zero = a
    • Commutative: a + b = b + a
  • Monoid under multiplication:
    • Associativity: (a * b) * c = a * (b * c)
    • Identity: one * a = a * one = a
  • Multiplication distributes over addition:
    • Left distributivity: a * (b + c) = (a * b) + (a * c)
    • Right distributivity: (a + b) * c = (a * c) + (b * c)
  • Annihilation: zero * a = a * zero = zero

Note: The Number and Int types are not fully law abiding members of this class hierarchy due to the potential for arithmetic overflows, and in the case of Number, the presence of NaN and Infinity values. The behaviour is unspecified in these cases.

Members

Instances

#(+)

Operator alias for Data.Semiring.add (left-associative / precedence 6)

#(*)

Operator alias for Data.Semiring.mul (left-associative / precedence 7)

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