numeric-prelude-0.4.3: An experimental alternative hierarchy of numeric type classes

Algebra.Ring

Contents

Synopsis

# 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)

Minimal complete definition

(*), (one | fromInteger)

Methods

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

one :: a Source #

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

 Source # Methods Source # Methods Source # Methods(*) :: Int -> Int -> Int Source #(^) :: Int -> Integer -> Int Source # Source # Methods(*) :: Int8 -> Int8 -> Int8 Source # Source # Methods Source # Methods Source # Methods Source # Methods Source # Methods(*) :: Word -> Word -> Word Source # Source # Methods Source # Methods Source # Methods Source # Methods Source # Methods(*) :: T -> T -> T Source #(^) :: T -> Integer -> T Source # Source # Methods(*) :: T -> T -> T Source #(^) :: T -> Integer -> T Source # Source # Methods(*) :: T -> T -> T Source #(^) :: T -> Integer -> T Source # Source # Methods(*) :: T -> T -> T Source #(^) :: T -> Integer -> T Source # Integral a => C (Ratio a) Source # Methods(*) :: Ratio a -> Ratio a -> Ratio a Source #(^) :: Ratio a -> Integer -> Ratio a Source # RealFloat a => C (Complex a) Source # Methods(*) :: Complex a -> Complex a -> Complex a Source #(^) :: Complex a -> Integer -> Complex a Source # C a => C (T a) Source # Methods(*) :: T a -> T a -> T a Source #one :: T a Source #(^) :: T a -> Integer -> T a Source # Num a => C (T a) Source # Methods(*) :: T a -> T a -> T a Source #one :: T a Source #(^) :: T a -> Integer -> T a Source # (Eq a, C a) => C (T a) Source # Methods(*) :: T a -> T a -> T a Source #one :: T a Source #(^) :: T a -> Integer -> T a Source # C a => C (T a) Source # Methods(*) :: T a -> T a -> T a Source #one :: T a Source #(^) :: T a -> Integer -> T a Source # (Eq a, C a) => C (T a) Source # Methods(*) :: T a -> T a -> T a Source #one :: T a Source #(^) :: T a -> Integer -> T a Source # C a => C (T a) Source # Methods(*) :: T a -> T a -> T a Source #one :: T a Source #(^) :: T a -> Integer -> T a Source # (C a, C a) => C (T a) Source # Methods(*) :: T a -> T a -> T a Source #one :: T a Source #(^) :: T a -> Integer -> T a Source # C a => C (T a) Source # Methods(*) :: T a -> T a -> T a Source #one :: T a Source #(^) :: T a -> Integer -> T a Source # C a => C (T a) Source # Methods(*) :: T a -> T a -> T a Source #one :: T a Source #(^) :: T a -> Integer -> T a Source # C a => C (T a) Source # Methods(*) :: T a -> T a -> T a Source #one :: T a Source #(^) :: T a -> Integer -> T a Source # C a => C (T a) Source # Methods(*) :: T a -> T a -> T a Source #one :: T a Source #(^) :: T a -> Integer -> T a Source # (C a, C a) => C (T a) Source # Methods(*) :: T a -> T a -> T a Source #one :: T a Source #(^) :: T a -> Integer -> T a Source # (C a, C a) => C (T a) Source # Methods(*) :: T a -> T a -> T a Source #one :: T a Source #(^) :: T a -> Integer -> T a Source # C a => C (T a) Source # Methods(*) :: T a -> T a -> T a Source #one :: T a Source #(^) :: T a -> Integer -> T a Source # C a => C (T a) Source # Methods(*) :: T a -> T a -> T a Source #one :: T a Source #(^) :: T a -> Integer -> T a Source # C a => C (T a) Source # Methods(*) :: T a -> T a -> T a Source #one :: T a Source #(^) :: T a -> Integer -> T a Source # C a => C (T a) Source # Methods(*) :: T a -> T a -> T a Source #one :: T a Source #(^) :: T a -> Integer -> T a Source # C a => C (T a) Source # Methods(*) :: T a -> T a -> T a Source #one :: T a Source #(^) :: T a -> Integer -> T a Source # (Ord a, C a, C b) => C (T a b) Source # Methods(*) :: T a b -> T a b -> T a b Source #one :: T a b Source #fromInteger :: Integer -> T a b Source #(^) :: T a b -> Integer -> T a b Source # C v => C (T a v) Source # Methods(*) :: T a v -> T a v -> T a v Source #one :: T a v Source #fromInteger :: Integer -> T a v Source #(^) :: T a v -> Integer -> T a v Source # (IsScalar u, C a) => C (T u a) Source # Methods(*) :: T u a -> T u a -> T u a Source #one :: T u a Source #fromInteger :: Integer -> T u a Source #(^) :: T u a -> Integer -> T u a Source # (Ord i, C a) => C (T i a) Source # Methods(*) :: T i a -> T i a -> T i a Source #one :: T i a Source #fromInteger :: Integer -> T i a Source #(^) :: T i a -> Integer -> T i a Source # C v => C (T a v) Source # Methods(*) :: T a v -> T a v -> T a v Source #one :: T a v Source #fromInteger :: Integer -> T a v Source #(^) :: T a v -> Integer -> T a v Source #

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

one :: C a => a Source #

(^) :: C a => a -> Integer -> a infixr 8 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 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 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 #

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.