Safe Haskell	None
Language	Haskell98

Algebra.Field

Contents

Class
Properties

Synopsis

class C a => C a
(/) :: C a => a -> a -> a
recip :: C a => a -> a
fromRational' :: C a => Rational -> a
fromRational :: C a => Rational -> a
(^-) :: C a => a -> Integer -> a
propDivision :: (Eq a, C a, C a) => a -> a -> Property
propReciprocal :: (Eq a, C a, C a) => a -> Property

Class

class C a => C a Source #

Field again corresponds to a commutative ring. Division is partially defined and satisfies

   not (isZero b)  ==>  (a * b) / b === a
   not (isZero a)  ==>  a * recip a === one

when it is defined. To safely call division, the program must take type-specific action; e.g., the following is appropriate in many cases:

safeRecip :: (Integral a, Eq a, Field.C a) => a -> Maybe a
safeRecip x =
    let (q,r) = one `divMod` x
    in  toMaybe (isZero r) q

Typical examples include rationals, the real numbers, and rational functions (ratios of polynomial functions). An instance should be typically declared only if most elements are invertible.

Actually, we have also used this type class for non-fields containing lots of units, e.g. residue classes with respect to non-primes and power series. So the restriction not (isZero a) must be better isUnit a.

Minimal definition: recip or (/)

Minimal complete definition

recip | (/)

Instances

C Double Source #
Instance details Defined in Algebra.Field Methods (/) :: Double -> Double -> Double Source # recip :: Double -> Double Source # fromRational' :: Rational -> Double Source # (^-) :: Double -> Integer -> Double Source #
C Float Source #
Instance details Defined in Algebra.Field Methods (/) :: Float -> Float -> Float Source # recip :: Float -> Float Source # fromRational' :: Rational -> Float Source # (^-) :: Float -> Integer -> Float Source #
C T Source #
Instance details Defined in Number.GaloisField2p32m5 Methods (/) :: T -> T -> T Source # recip :: T -> T Source # fromRational' :: Rational -> T Source # (^-) :: T -> Integer -> T Source #
C T Source #
Instance details Defined in Number.FixedPoint.Check Methods (/) :: T -> T -> T Source # recip :: T -> T Source # fromRational' :: Rational -> T Source # (^-) :: T -> Integer -> T Source #
C T Source #
Instance details Defined in Number.Positional.Check Methods (/) :: T -> T -> T Source # recip :: T -> T Source # fromRational' :: Rational -> T Source # (^-) :: T -> Integer -> T Source #
Integral a => C (Ratio a) Source #
Instance details Defined in Algebra.Field Methods (/) :: Ratio a -> Ratio a -> Ratio a Source # recip :: Ratio a -> Ratio a Source # fromRational' :: Rational -> Ratio a Source # (^-) :: Ratio a -> Integer -> Ratio a Source #
RealFloat a => C (Complex a) Source #
Instance details Defined in Algebra.Field Methods (/) :: Complex a -> Complex a -> Complex a Source # recip :: Complex a -> Complex a Source # fromRational' :: Rational -> Complex a Source # (^-) :: Complex a -> Integer -> Complex a Source #
(Ord a, C a) => C (T a) Source #
Instance details Defined in Number.NonNegative Methods (/) :: T a -> T a -> T a Source # recip :: T a -> T a Source # fromRational' :: Rational -> T a Source # (^-) :: T a -> Integer -> T a Source #
C a => C (T a) Source #
Instance details Defined in Algebra.Field Methods (/) :: T a -> T a -> T a Source # recip :: T a -> T a Source # fromRational' :: Rational -> T a Source # (^-) :: T a -> Integer -> T a Source #
Fractional a => C (T a) Source #
Instance details Defined in MathObj.Wrapper.Haskell98 Methods (/) :: T a -> T a -> T a Source # recip :: T a -> T a Source # fromRational' :: Rational -> T a Source # (^-) :: T a -> Integer -> T a Source #
C a => C (T a) Source #
Instance details Defined in Number.ResidueClass.Func Methods (/) :: T a -> T a -> T a Source # recip :: T a -> T a Source # fromRational' :: Rational -> T a Source # (^-) :: T a -> Integer -> T a Source #
(Eq a, C a) => C (T a) Source #
Instance details Defined in Number.ResidueClass.Check Methods (/) :: T a -> T a -> T a Source # recip :: T a -> T a Source # fromRational' :: Rational -> T a Source # (^-) :: T a -> Integer -> T a Source #
C a => C (T a) Source #
Instance details Defined in Number.PartiallyTranscendental Methods (/) :: T a -> T a -> T a Source # recip :: T a -> T a Source # fromRational' :: Rational -> T a Source # (^-) :: T a -> Integer -> T a Source #
C a => C (T a) Source #
Instance details Defined in MathObj.PowerSeries Methods (/) :: T a -> T a -> T a Source # recip :: T a -> T a Source # fromRational' :: Rational -> T a Source # (^-) :: T a -> Integer -> T a Source #
C a => C (T a) Source #
Instance details Defined in MathObj.PowerSeries2 Methods (/) :: T a -> T a -> T a Source # recip :: T a -> T a Source # fromRational' :: Rational -> T a Source # (^-) :: T a -> Integer -> T a Source #
(C a, C a) => C (T a) Source #
Instance details Defined in MathObj.PowerSum Methods (/) :: T a -> T a -> T a Source # recip :: T a -> T a Source # fromRational' :: Rational -> T a Source # (^-) :: T a -> Integer -> T a Source #
(C a, C a) => C (T a) Source #
Instance details Defined in MathObj.RootSet Methods (/) :: T a -> T a -> T a Source # recip :: T a -> T a Source # fromRational' :: Rational -> T a Source # (^-) :: T a -> Integer -> T a Source #
C a => C (T a) Source #
Instance details Defined in Number.Complex Methods (/) :: T a -> T a -> T a Source # recip :: T a -> T a Source # fromRational' :: Rational -> T a Source # (^-) :: T a -> Integer -> T a Source #
C a => C (T a) Source #
Instance details Defined in Number.Quaternion Methods (/) :: T a -> T a -> T a Source # recip :: T a -> T a Source # fromRational' :: Rational -> T a Source # (^-) :: T a -> Integer -> T a Source #
(C a, C a) => C (T a) Source #
Instance details Defined in MathObj.LaurentPolynomial Methods (/) :: T a -> T a -> T a Source # recip :: T a -> T a Source # fromRational' :: Rational -> T a Source # (^-) :: T a -> Integer -> T a Source #
C a => C (T a) Source #
Instance details Defined in MathObj.Wrapper.NumericPrelude Methods (/) :: T a -> T a -> T a Source # recip :: T a -> T a Source # fromRational' :: Rational -> T a Source # (^-) :: T a -> Integer -> T a Source #
C v => C (T a v) Source #
Instance details Defined in Number.OccasionallyScalarExpression Methods (/) :: T a v -> T a v -> T a v Source # recip :: T a v -> T a v Source # fromRational' :: Rational -> T a v Source # (^-) :: T a v -> Integer -> T a v Source #
(IsScalar u, C a) => C (T u a) Source #
Instance details Defined in Number.DimensionTerm Methods (/) :: T u a -> T u a -> T u a Source # recip :: T u a -> T u a Source # fromRational' :: Rational -> T u a Source # (^-) :: T u a -> Integer -> T u a Source #
(Ord i, C a) => C (T i a) Source #
Instance details Defined in Number.Physical Methods (/) :: T i a -> T i a -> T i a Source # recip :: T i a -> T i a Source # fromRational' :: Rational -> T i a Source # (^-) :: T i a -> Integer -> T i a Source #
C v => C (T a v) Source #
Instance details Defined in Number.SI Methods (/) :: T a v -> T a v -> T a v Source # recip :: T a v -> T a v Source # fromRational' :: Rational -> T a v Source # (^-) :: T a v -> Integer -> T a v Source #

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

recip :: C a => a -> a Source #

fromRational' :: C a => Rational -> a Source #

fromRational :: C a => Rational -> a Source #

Needed to work around shortcomings in GHC.

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

Properties

propDivision :: (Eq a, C a, C a) => a -> a -> Property Source #

the restriction on the divisor should be isUnit a instead of not (isZero a)

propReciprocal :: (Eq a, C a, C a) => a -> Property Source #