Algebra.RealField

Documentation

class (C a, C a) => C a where

Minimal complete definition: splitFraction or floor

There are probably more laws, but some laws are

 (fromInteger.fst.splitFraction) a + (snd.splitFraction) a === a
    ceiling (toRational x) === ceiling x :: Integer
   truncate (toRational x) === truncate x :: Integer
      floor (toRational x) === floor x :: Integer

If there wouldn't be Real.C a and ToInteger.C b constraints, we could also use this class for splitting ratios of polynomials.

As an aside, let me note the similarities between splitFraction x and x divMod 1 (if that were defined). In particular, it might make sense to unify the rounding modes somehow.

IEEEFloat-specific calls are removed here (cf. Prelude.RealFloat) so probably nobody will actually use this default definition.

Henning: New function fraction doesn't return the integer part of the number. This also removes a type ambiguity if the integer part is not needed.

The new methods fraction and splitFraction differ from Prelude.properFraction semantics. They always round to floor. This means that the fraction is always non-negative and is always smaller than 1. This is more useful in practice and can be generalised to more than real numbers. Since every T denominator type supports divMod, every T can provide fraction and splitFraction, e.g. fractions of polynomials. However the ''integral'' part would not be of type class C.

Can there be a separate class for fraction, splitFraction, floor and ceiling since they do not need reals and their ordering?

Note: All of these methods can be defined exclusively with functions from Ord and Ring. We could write a power-of-two-algorithm like the one for finding the number of digits of an Integer in FixedPoint-fractions module. This would even be reasonably efficient. I think the module should be renamed to RealRing, and the superclass constraint should be lifted from Field to Ring.

We might also add a round method, that rounds 0.5 always up or always down. This is much more efficient in inner loops and is acceptable or even preferable for many applications.

The ToInteger constraint can be lifted to Ring.

Methods

splitFraction :: C b => a -> (b, a)

fraction :: a -> a

ceiling :: C b => a -> b

floor :: C b => a -> b

truncate :: C b => a -> b

round :: C b => a -> b

Instances

C Double

C Float

C T

C T

C a => C (T a)

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

fastSplitFraction :: (RealFrac a, C a, C b) => (a -> Int) -> (Int -> a) -> a -> (b, a)

fixSplitFraction :: (C a, C b, Ord a) => (b, a) -> (b, a)

fastFraction :: (RealFrac a, C a) => (a -> a) -> a -> a

preludeFraction :: (RealFrac a, C a) => a -> a

fixFraction :: (C a, Ord a) => a -> a

splitFractionInt :: (C a, Ord a) => (a -> Int) -> (Int -> a) -> a -> (Int, a)

floorInt :: (C a, Ord a) => (a -> Int) -> (Int -> a) -> a -> Int

ceilingInt :: (C a, Ord a) => (a -> Int) -> (Int -> a) -> a -> Int

roundInt :: (C a, Ord a) => (a -> Int) -> (Int -> a) -> a -> Int

approxRational :: (C a, C a) => a -> a -> Rational

TODO: Should be moved to a continued fraction module.

generic implementation of round functions

powersOfTwo :: C a => [a]

pairsOfPowersOfTwo :: (C a, C b) => [(a, b)]

genericFloor :: (Ord a, C a, C b) => a -> b

The generic rounding functions need a number of operations proportional to the number of binary digits of the integer portion. If operations like multiplication with two and comparison need time proportional to the number of binary digits, then the overall rounding requires quadratic time.

genericCeiling :: (Ord a, C a, C b) => a -> b

genericTruncate :: (Ord a, C a, C b) => a -> b

genericRound :: (Ord a, C a, C b) => a -> b

genericFraction :: (Ord a, C a) => a -> a

genericSplitFraction :: (Ord a, C a, C b) => a -> (b, a)

genericPosFloor :: (Ord a, C a, C b) => a -> b

genericPosCeiling :: (Ord a, C a, C b) => a -> b

genericHalfPosFloorDigits :: (Ord a, C a, C b) => a -> ((a, b), [Bool])

genericPosRound :: (Ord a, C a, C b) => a -> b

genericPosFraction :: (Ord a, C a) => a -> a

genericPosSplitFraction :: (Ord a, C a, C b) => a -> (b, a)