numeric-prelude-0.3: An experimental alternative hierarchy of numeric type classes

Algebra.Ring

Contents

Synopsis

# Class

class C a => C a whereSource

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 -> aSource

one :: aSource

fromInteger :: Integer -> aSource

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

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 `Integer`s 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 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 (T a) (Eq a, C a) => C (T a) (Eq 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) Num 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 -> aSource

# Complex functions

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

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

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

# Properties

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

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

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

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

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

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

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

Commutativity need not be satisfied by all instances of `C`.