Factory.Data.Ring

AUTHOR

Dr. Alistair Ward

DESCRIPTION

• Describes a ring and operations on its members.
• https://en.wikipedia.org/wiki/Ring_%28mathematics%29.
• https://www.numericana.com/answer/rings.htm.

Type-classes

class Ring r where

• Define both the operations applicable to all members of the ring, and its mandatory members.
• Minimal definition; =+=, =*=, additiveInverse, multiplicativeIdentity, additiveIdentity.

Minimal complete definition

(=+=), (=*=), additiveInverse, multiplicativeIdentity, additiveIdentity

Methods

(=+=) infixl 6

Arguments

:: r

-> r

-> r

Addition of two members; required to be commutative; https://en.wikipedia.org/wiki/Commutativity.

(=*=) infixl 7

Arguments

:: r

-> r

-> r

Multiplication of two members.

additiveInverse

Arguments

:: r

-> r

The operand required to yield zero under addition; https://en.wikipedia.org/wiki/Additive_inverse.

multiplicativeIdentity

Arguments

:: r

The identity-member under multiplication; https://mathworld.wolfram.com/MultiplicativeIdentity.html.

additiveIdentity

Arguments

:: r

The identity-member under addition (AKA zero); https://en.wikipedia.org/wiki/Additive_identity.

(=-=) infixl 6

Arguments

:: r

-> r

-> r

Subtract the two specified ring-members.

square

Arguments

:: r

-> r

Square the ring.

Instances

(Eq c, Num c, Num e, Ord e) => Ring (Polynomial c e)

Makes Polynomial a Ring, over the field composed from all possible coefficients; https://en.wikipedia.org/wiki/Polynomial_ring.

Defined in Factory.Data.Polynomial

Methods

(=+=) :: Polynomial c e -> Polynomial c e -> Polynomial c e

(=*=) :: Polynomial c e -> Polynomial c e -> Polynomial c e

additiveInverse :: Polynomial c e -> Polynomial c e

multiplicativeIdentity :: Polynomial c e

additiveIdentity :: Polynomial c e

(=-=) :: Polynomial c e -> Polynomial c e -> Polynomial c e

square :: Polynomial c e -> Polynomial c e

(Eq c, Num c, Num e, Ord e, Show c, Show e) => Ring (MonicPolynomial c e)

Defined in Factory.Data.MonicPolynomial

Methods

(=+=) :: MonicPolynomial c e -> MonicPolynomial c e -> MonicPolynomial c e

(=*=) :: MonicPolynomial c e -> MonicPolynomial c e -> MonicPolynomial c e

additiveInverse :: MonicPolynomial c e -> MonicPolynomial c e

multiplicativeIdentity :: MonicPolynomial c e

additiveIdentity :: MonicPolynomial c e

(=-=) :: MonicPolynomial c e -> MonicPolynomial c e -> MonicPolynomial c e

square :: MonicPolynomial c e -> MonicPolynomial c e

Types

Data-types

Functions

product' :: Ring r => BisectionRatio -> MinLength -> [r] -> r

Returns the product of the list of ring-members.

sum' :: Ring r => BisectionRatio -> MinLength -> [r] -> r

Returns the sum of the list of ring-members.

Operators

(=^) :: (Eq r, Integral power, Ring r, Show power) => r -> power -> r infixr 8

• Raise a ring-member to the specified positive integral power.
• Exponentiation is implemented as a sequence of either squares of, or multiplications by, the ring-member; https://en.wikipedia.org/wiki/Exponentiation_by_squaring.