Algebra.Ring

Class

class C a => C a where

Ring encapsulates the mathematical structure of a (not necessarily commutative) ring, with the laws

  a * (b * c) === (a * b) * c
      one * a === a
      a * one === a
  a * (b + c) === a * b + a * c

Typical examples include integers, polynomials, matrices, and quaternions.

Minimal definition: *, (one or fromInteger)

Minimal complete definition

(*), (one | fromInteger)

Methods

(*) :: a -> a -> a infixl 7

one :: a

fromInteger :: Integer -> a

(^) :: a -> Integer -> a infixr 8

The exponent has fixed type Integer in order to avoid an arbitrarily limitted range of exponents, but to reduce the need for the compiler to guess the type (default type). In practice the exponent is most oftenly fixed, and is most oftenly 2. Fixed exponents can be optimized away and thus the expensive computation of Integers doesn't matter. The previous solution used a C constrained type and the exponent was converted to Integer before computation. So the current solution is not less efficient.

A variant of ^ with more flexibility is provided by ringPower.

Instances

C Double

C Float

C Int

C Int8

C Int16

C Int32

C Int64

C Integer

C Word

C Word8

C Word16

C Word32

C Word64

C T

C T

C T

C T

Integral a => C (Ratio a)

RealFloat a => C (Complex a)

(Ord a, C a) => C (T a)

C a => C (T a)

C a => C (T a)

(C a, C a) => C (T a)

C a => C (T a)

C a => C (T a)

C a => C (T a)

(Eq a, C a) => C (T a)

(Eq a, C a) => C (T a)

Num a => C (T a)

C a => C (T a)

C a => C (T a)

(C a, C a) => C (T a)

C a => C (T a)

C a => C (T a)

C a => C (T a)

C a => C (T a)

C a => C (T a)

(C a, C a) => C (T a)

(Ord a, C a, C b) => C (T a b)

(IsScalar u, C a) => C (T u a)

C v => C (T a v)

(Ord i, C a) => C (T i a)

C v => C (T a v)

sqr :: C a => a -> a

Complex functions

product :: C a => [a] -> a

product1 :: C a => [a] -> a

scalarProduct :: C a => [a] -> [a] -> a

Properties

propAssociative :: (Eq a, C a) => a -> a -> a -> Bool

propLeftDistributive :: (Eq a, C a) => a -> a -> a -> Bool

propRightDistributive :: (Eq a, C a) => a -> a -> a -> Bool

propLeftIdentity :: (Eq a, C a) => a -> Bool

propRightIdentity :: (Eq a, C a) => a -> Bool

propPowerCascade :: (Eq a, C a) => a -> Integer -> Integer -> Property

propPowerProduct :: (Eq a, C a) => a -> Integer -> Integer -> Property

propPowerDistributive :: (Eq a, C a) => Integer -> a -> a -> Property

propCommutative :: (Eq a, C a) => a -> a -> Bool

Commutativity need not be satisfied by all instances of C.