Algebra.Ring

Class

class C a => C a

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)

Instances

C Double

Defined in Algebra.Ring

Methods

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

one :: Double

fromInteger :: Integer -> Double

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

C Float

Defined in Algebra.Ring

Methods

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

one :: Float

fromInteger :: Integer -> Float

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

C Int

Defined in Algebra.Ring

Methods

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

one :: Int

fromInteger :: Integer -> Int

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

C Int8

Defined in Algebra.Ring

Methods

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

one :: Int8

fromInteger :: Integer -> Int8

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

C Int16

Defined in Algebra.Ring

Methods

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

one :: Int16

fromInteger :: Integer -> Int16

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

C Int32

Defined in Algebra.Ring

Methods

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

one :: Int32

fromInteger :: Integer -> Int32

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

C Int64

Defined in Algebra.Ring

Methods

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

one :: Int64

fromInteger :: Integer -> Int64

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

C Integer

Defined in Algebra.Ring

Methods

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

one :: Integer

fromInteger :: Integer -> Integer

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

C Word

Defined in Algebra.Ring

Methods

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

one :: Word

fromInteger :: Integer -> Word

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

C Word8

Defined in Algebra.Ring

Methods

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

one :: Word8

fromInteger :: Integer -> Word8

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

C Word16

Defined in Algebra.Ring

Methods

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

one :: Word16

fromInteger :: Integer -> Word16

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

C Word32

Defined in Algebra.Ring

Methods

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

one :: Word32

fromInteger :: Integer -> Word32

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

C Word64

Defined in Algebra.Ring

Methods

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

one :: Word64

fromInteger :: Integer -> Word64

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

C T

Defined in Number.Peano

Methods

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

one :: T

fromInteger :: Integer -> T

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

C T

Defined in Number.GaloisField2p32m5

Methods

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

one :: T

fromInteger :: Integer -> T

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

C T

Defined in Number.FixedPoint.Check

Methods

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

one :: T

fromInteger :: Integer -> T

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

C T

Defined in Number.Positional.Check

Methods

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

one :: T

fromInteger :: Integer -> T

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

Integral a => C (Ratio a)

Defined in Algebra.Ring

Methods

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

one :: Ratio a

fromInteger :: Integer -> Ratio a

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

RealFloat a => C (Complex a)

Defined in Algebra.Ring

Methods

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

one :: Complex a

fromInteger :: Integer -> Complex a

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

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

Defined in Number.NonNegative

Methods

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

one :: T a

fromInteger :: Integer -> T a

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

C a => C (T a)

Defined in Number.Ratio

Methods

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

one :: T a

fromInteger :: Integer -> T a

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

Num a => C (T a)

Defined in MathObj.Wrapper.Haskell98

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)

Defined in Number.ResidueClass.Maybe

Methods

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

one :: T a

fromInteger :: Integer -> T a

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

C a => C (T a)

Defined in Number.ResidueClass.Func

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)

Defined in Number.ResidueClass.Check

Methods

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

one :: T a

fromInteger :: Integer -> T a

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

C a => C (T a)

Defined in Number.PartiallyTranscendental

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)

Defined in Number.NonNegativeChunky

Methods

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

one :: T a

fromInteger :: Integer -> T a

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

C a => C (T a)

QC.choose (1,10) /\ \expon (QC.Positive x) xs -> let xt = x:xs in  equalTrunc 15 (PS.pow (const x) (1 % expon) (PST.coeffs (PST.fromCoeffs xt ^ expon)) ++ repeat zero) (xt ++ repeat zero)

Defined in MathObj.PowerSeries

Methods

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

one :: T a

fromInteger :: Integer -> T a

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

C a => C (T a)

Defined in MathObj.PowerSeries2

Methods

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

one :: T a

fromInteger :: Integer -> T a

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

C a => C (T a)

Defined in MathObj.Polynomial

Methods

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

one :: T a

fromInteger :: Integer -> T a

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

C a => C (T a)

Defined in MathObj.PowerSum

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)

Defined in MathObj.RootSet

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)

Defined in MathObj.PartialFraction

Methods

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

one :: T a

fromInteger :: Integer -> T a

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

C a => C (T a)

genIntMatrix /\ \a -> Laws.leftIdentity  (*) (Matrix.one (Matrix.numRows a)) a

genIntMatrix /\ \a -> Laws.rightIdentity (*) (Matrix.one (Matrix.numColumns a)) a

genIntMatrix /\ \a -> genFactorMatrix a /\ \b -> Laws.homomorphism Matrix.transpose (*) (flip (*)) a b

genIntMatrix /\ \a -> genFactorMatrix a /\ \b -> genFactorMatrix b /\ \c -> Laws.associative (*) a b c

genIntMatrix /\ \b -> genSameMatrix b /\ \c -> genFactorMatrix b /\ \a -> Laws.leftDistributive (*) (+) a b c

genIntMatrix /\ \a -> genFactorMatrix a /\ \b -> genSameMatrix b /\ \c -> Laws.rightDistributive (*) (+) a b c

QC.choose (0,10) /\ \k -> genDimension /\ \n -> genMatrixFor n n /\ \a -> a^k == nest (fromInteger k) ((a::Matrix.T Integer)*) (Matrix.one n)

Defined in MathObj.Matrix

Methods

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

one :: T a

fromInteger :: Integer -> T a

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

C a => C (T a)

Defined in Number.Complex

Methods

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

one :: T a

fromInteger :: Integer -> T a

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

C a => C (T a)

Defined in Number.Quaternion

Methods

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

one :: T a

fromInteger :: Integer -> T a

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

C a => C (T a)

Defined in MathObj.LaurentPolynomial

Methods

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

one :: T a

fromInteger :: Integer -> T a

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

C a => C (T a)

Defined in MathObj.Wrapper.NumericPrelude

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)

Defined in MathObj.Algebra

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)

Defined in Number.OccasionallyScalarExpression

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)

Defined in Number.DimensionTerm

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)

Defined in Number.Physical

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)

Defined in Number.SI

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.