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

Methods

(*) :: Double -> Double -> Double

one :: Double

fromInteger :: Integer -> Double

(^) :: Double -> Integer -> Double

C Float

Methods

(*) :: Float -> Float -> Float

one :: Float

fromInteger :: Integer -> Float

(^) :: Float -> Integer -> Float

C Int

Methods

(*) :: Int -> Int -> Int

one :: Int

fromInteger :: Integer -> Int

(^) :: Int -> Integer -> Int

C Int8

Methods

(*) :: Int8 -> Int8 -> Int8

one :: Int8

fromInteger :: Integer -> Int8

(^) :: Int8 -> Integer -> Int8

C Int16

Methods

(*) :: Int16 -> Int16 -> Int16

one :: Int16

fromInteger :: Integer -> Int16

(^) :: Int16 -> Integer -> Int16

C Int32

Methods

(*) :: Int32 -> Int32 -> Int32

one :: Int32

fromInteger :: Integer -> Int32

(^) :: Int32 -> Integer -> Int32

C Int64

Methods

(*) :: Int64 -> Int64 -> Int64

one :: Int64

fromInteger :: Integer -> Int64

(^) :: Int64 -> Integer -> Int64

C Integer

Methods

(*) :: Integer -> Integer -> Integer

one :: Integer

fromInteger :: Integer -> Integer

(^) :: Integer -> Integer -> Integer

C Word

Methods

(*) :: Word -> Word -> Word

one :: Word

fromInteger :: Integer -> Word

(^) :: Word -> Integer -> Word

C Word8

Methods

(*) :: Word8 -> Word8 -> Word8

one :: Word8

fromInteger :: Integer -> Word8

(^) :: Word8 -> Integer -> Word8

C Word16

Methods

(*) :: Word16 -> Word16 -> Word16

one :: Word16

fromInteger :: Integer -> Word16

(^) :: Word16 -> Integer -> Word16

C Word32

Methods

(*) :: Word32 -> Word32 -> Word32

one :: Word32

fromInteger :: Integer -> Word32

(^) :: Word32 -> Integer -> Word32

C Word64

Methods

(*) :: Word64 -> Word64 -> Word64

one :: Word64

fromInteger :: Integer -> Word64

(^) :: Word64 -> Integer -> Word64

C T

Methods

(*) :: T -> T -> T

one :: T

fromInteger :: Integer -> T

(^) :: T -> Integer -> T

C T

Methods

(*) :: T -> T -> T

one :: T

fromInteger :: Integer -> T

(^) :: T -> Integer -> T

C T

Methods

(*) :: T -> T -> T

one :: T

fromInteger :: Integer -> T

(^) :: T -> Integer -> T

C T

Methods

(*) :: T -> T -> T

one :: T

fromInteger :: Integer -> T

(^) :: T -> Integer -> T

Integral a => C (Ratio a)

Methods

(*) :: Ratio a -> Ratio a -> Ratio a

one :: Ratio a

fromInteger :: Integer -> Ratio a

(^) :: Ratio a -> Integer -> Ratio a

RealFloat a => C (Complex a)

Methods

(*) :: Complex a -> Complex a -> Complex a

one :: Complex a

fromInteger :: Integer -> Complex a

(^) :: Complex a -> Integer -> Complex a

C a => C (T a)

Methods

(*) :: T a -> T a -> T a

one :: T a

fromInteger :: Integer -> T a

(^) :: T a -> Integer -> T a

Num a => C (T a)

Methods

(*) :: T a -> T a -> T a

one :: T a

fromInteger :: Integer -> T a

(^) :: T a -> Integer -> T a

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

Methods

(*) :: T a -> T a -> T a

one :: T a

fromInteger :: Integer -> T a

(^) :: T a -> Integer -> T a

C a => C (T a)

Methods

(*) :: T a -> T a -> T a

one :: T a

fromInteger :: Integer -> T a

(^) :: T a -> Integer -> T a

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

Methods

(*) :: T a -> T a -> T a

one :: T a

fromInteger :: Integer -> T a

(^) :: T a -> Integer -> T a

C a => C (T a)

Methods

(*) :: T a -> T a -> T a

one :: T a

fromInteger :: Integer -> T a

(^) :: T a -> Integer -> T a

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

Methods

(*) :: T a -> T a -> T a

one :: T a

fromInteger :: Integer -> T a

(^) :: T a -> Integer -> T a

C a => C (T a)

Methods

(*) :: T a -> T a -> T a

one :: T a

fromInteger :: Integer -> T a

(^) :: T a -> Integer -> T a

C a => C (T a)

Methods

(*) :: T a -> T a -> T a

one :: T a

fromInteger :: Integer -> T a

(^) :: T a -> Integer -> T a

C a => C (T a)

Methods

(*) :: T a -> T a -> T a

one :: T a

fromInteger :: Integer -> T a

(^) :: T a -> Integer -> T a

C a => C (T a)

Methods

(*) :: T a -> T a -> T a

one :: T a

fromInteger :: Integer -> T a

(^) :: T a -> Integer -> T a

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

Methods

(*) :: T a -> T a -> T a

one :: T a

fromInteger :: Integer -> T a

(^) :: T a -> Integer -> T a

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

Methods

(*) :: T a -> T a -> T a

one :: T a

fromInteger :: Integer -> T a

(^) :: T a -> Integer -> T a

C a => C (T a)

Methods

(*) :: T a -> T a -> T a

one :: T a

fromInteger :: Integer -> T a

(^) :: T a -> Integer -> T a

C a => C (T a)

Methods

(*) :: T a -> T a -> T a

one :: T a

fromInteger :: Integer -> T a

(^) :: T a -> Integer -> T a

C a => C (T a)

Methods

(*) :: T a -> T a -> T a

one :: T a

fromInteger :: Integer -> T a

(^) :: T a -> Integer -> T a

C a => C (T a)

Methods

(*) :: T a -> T a -> T a

one :: T a

fromInteger :: Integer -> T a

(^) :: T a -> Integer -> T a

C a => C (T a)

Methods

(*) :: T a -> T a -> T a

one :: T a

fromInteger :: Integer -> T a

(^) :: T a -> Integer -> T a

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

Methods

(*) :: T a b -> T a b -> T a b

one :: T a b

fromInteger :: Integer -> T a b

(^) :: T a b -> Integer -> T a b

C v => C (T a v)

Methods

(*) :: T a v -> T a v -> T a v

one :: T a v

fromInteger :: Integer -> T a v

(^) :: T a v -> Integer -> T a v

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

Methods

(*) :: T u a -> T u a -> T u a

one :: T u a

fromInteger :: Integer -> T u a

(^) :: T u a -> Integer -> T u a

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

Methods

(*) :: T i a -> T i a -> T i a

one :: T i a

fromInteger :: Integer -> T i a

(^) :: T i a -> Integer -> T i a

C v => C (T a v)

Methods

(*) :: T a v -> T a v -> T a v

one :: T a v

fromInteger :: Integer -> T a v

(^) :: T a v -> Integer -> T a v

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

one :: C a => a

fromInteger :: C a => Integer -> a

(^) :: C 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.

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.