Factory.Data.Ring

AUTHOR

Dr. Alistair Ward

DESCRIPTION

• Describes a ring and operations on its members.
• http://en.wikipedia.org/wiki/Ring_%28mathematics%29.
• http://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; http://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; http://en.wikipedia.org/wiki/Additive_inverse.

multiplicativeIdentity

Arguments

:: r

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

additiveIdentity

Arguments

:: r

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

(Eq c, Num c, Num e, Ord e, Show c, Show e) => Ring (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; http://en.wikipedia.org/wiki/Exponentiation_by_squaring.