Algebra.Ring
 Contents Class Complex functions Properties
Synopsis
class C a => C a where
 (*) :: a -> a -> a one :: a fromInteger :: Integer -> a (^) :: a -> Integer -> a
sqr :: C a => a -> a
product :: C a => [a] -> a
product1 :: C a => [a] -> a
scalarProduct :: C a => [a] -> [a] -> a
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
Class
 class C a => C a where Source

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)

Methods
 (*) :: a -> a -> a Source
 one :: a Source
 fromInteger :: Integer -> a Source
 (^) :: a -> Integer -> a Source

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 Algebra.ToInteger.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 Algebra.Core.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 Integral a => C (Ratio 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) 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 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) 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 Source
Complex functions
 product :: C a => [a] -> a Source
 product1 :: C a => [a] -> a Source
 scalarProduct :: C a => [a] -> [a] -> a Source
Properties
 propAssociative :: (Eq a, C a) => a -> a -> a -> Bool Source
 propLeftDistributive :: (Eq a, C a) => a -> a -> a -> Bool Source
 propRightDistributive :: (Eq a, C a) => a -> a -> a -> Bool Source
 propLeftIdentity :: (Eq a, C a) => a -> Bool Source
 propRightIdentity :: (Eq a, C a) => a -> Bool Source
 propPowerCascade :: (Eq a, C a) => a -> Integer -> Integer -> Property Source
 propPowerProduct :: (Eq a, C a) => a -> Integer -> Integer -> Property Source
 propPowerDistributive :: (Eq a, C a) => Integer -> a -> a -> Property Source
 propCommutative :: (Eq a, C a) => a -> a -> Bool Source
Commutativity need not be satisfied by all instances of C.