diff --git a/numhask.cabal b/numhask.cabal
--- a/numhask.cabal
+++ b/numhask.cabal
@@ -1,5 +1,5 @@
name: numhask
-version: 0.3.1
+version: 0.4.0
synopsis:
numeric classes
description:
@@ -56,14 +56,11 @@
NumHask.Algebra.Abstract.Additive
NumHask.Algebra.Abstract.Field
NumHask.Algebra.Abstract.Group
- NumHask.Algebra.Abstract.Homomorphism
NumHask.Algebra.Abstract.Lattice
NumHask.Algebra.Abstract.Module
NumHask.Algebra.Abstract.Multiplicative
NumHask.Algebra.Abstract.Ring
- NumHask.Algebra.Abstract.TensorProduct
NumHask.Algebra.Linear.Hadamard
- NumHask.Analysis.Banach
NumHask.Analysis.Metric
NumHask.Data.Complex
NumHask.Data.Integral
diff --git a/other/field.svg b/other/field.svg
new file mode 100644
--- /dev/null
+++ b/other/field.svg
@@ -0,0 +1,3 @@
+
+
diff --git a/other/ring.svg b/other/ring.svg
deleted file mode 100644
--- a/other/ring.svg
+++ /dev/null
@@ -1,3 +0,0 @@
-
-
diff --git a/src/NumHask/Algebra/Abstract.hs b/src/NumHask/Algebra/Abstract.hs
--- a/src/NumHask/Algebra/Abstract.hs
+++ b/src/NumHask/Algebra/Abstract.hs
@@ -14,7 +14,6 @@
, module NumHask.Algebra.Abstract.Module
, module NumHask.Algebra.Abstract.Action
, module NumHask.Algebra.Abstract.Lattice
- , module NumHask.Algebra.Abstract.Homomorphism
)
where
@@ -26,7 +25,6 @@
import NumHask.Algebra.Abstract.Module
import NumHask.Algebra.Abstract.Action
import NumHask.Algebra.Abstract.Lattice
-import NumHask.Algebra.Abstract.Homomorphism
-- $numMap
--
diff --git a/src/NumHask/Algebra/Abstract/Additive.hs b/src/NumHask/Algebra/Abstract/Additive.hs
--- a/src/NumHask/Algebra/Abstract/Additive.hs
+++ b/src/NumHask/Algebra/Abstract/Additive.hs
@@ -14,15 +14,29 @@
import Prelude (Int, Integer, Float, Double, Bool)
import qualified Prelude as P
+-- | For practical reasons, 'Additive' has no super classes. Using `Associative` and 'Unital' from this library, or using 'Semigroup' and 'Monoid' from base tends to complexify the interface once you start having to disinguish between (say) monoidal addition and monoidal multiplication.
+--
+-- > zero + a == a
+-- > a + zero == a
+-- > (a + b) + c == a + (b + c)
+-- > a + b == b + a
+--
+-- By convention, (+) is regarded as commutative, but this is not universal, and the introduction of another symbol which means non-commutative multiplication seems a bit dogmatic.
class Additive a where
infixl 6 +
(+) :: a -> a -> a
zero :: a
+-- | Compute the sum of a 'Foldable'.
sum :: (Additive a, P.Foldable f) => f a -> a
sum = P.foldr (+) zero
+-- |
+-- > a - a = zero
+-- > negate a = zero - a
+-- > negate a + a = zero
+-- > a + negate a = zero
class (Additive a) => Subtractive a where
negate :: a -> a
diff --git a/src/NumHask/Algebra/Abstract/Field.hs b/src/NumHask/Algebra/Abstract/Field.hs
--- a/src/NumHask/Algebra/Abstract/Field.hs
+++ b/src/NumHask/Algebra/Abstract/Field.hs
@@ -27,11 +27,10 @@
import qualified Prelude as P
import Prelude ((.))
--- | A is an
--- Integral domain in which every non-zero element has a multiplicative
--- inverse.
+-- | A is a set
+-- on which addition, subtraction, multiplication, and division are defined. It is also assumed that multiplication is distributive over addition.
--
--- A summary of the rules inherited from super-classes of Field
+-- A summary of the rules thus inherited from super-classes of Field
--
-- > zero + a == a
-- > a + zero == a
@@ -53,7 +52,7 @@
-- > recip a = one / a
-- > recip a * a = one
-- > a * recip a = one
-class (IntegralDomain a) =>
+class (Distributive a, Subtractive a, Divisive a) =>
Field a
instance Field P.Double
@@ -156,7 +155,7 @@
-- > zero / zero != nan
--
-- Note the tricky law that, although nan is assigned to zero/zero, they are never-the-less not equal. A committee decided this.
-class (IntegralDomain a) =>
+class (Field a) =>
UpperBoundedField a where
infinity :: a
@@ -244,5 +243,6 @@
acosh f = acosh . f
atanh f = atanh . f
+-- | A 'half' is a 'Field' because it requires addition, multiplication and division to be computed.
half :: (Field a) => a
half = one / two
diff --git a/src/NumHask/Algebra/Abstract/Homomorphism.hs b/src/NumHask/Algebra/Abstract/Homomorphism.hs
deleted file mode 100644
--- a/src/NumHask/Algebra/Abstract/Homomorphism.hs
+++ /dev/null
@@ -1,43 +0,0 @@
-{-# LANGUAGE FlexibleInstances #-}
-{-# LANGUAGE MultiParamTypeClasses #-}
-{-# LANGUAGE UndecidableInstances #-}
-{-# OPTIONS_GHC -Wall #-}
-
--- | The Homomorphism Hierarchy
-module NumHask.Algebra.Abstract.Homomorphism
- ( Hom(..)
- , End
- , Iso
- , iso
- , invIso
- , Automorphism
- )
-where
-
-import NumHask.Algebra.Abstract.Group
-
--- | A Homomorphism between two magmas
--- law: forall a b. hom(a `magma` b) = hom(a) `magma` hom(b)
-class (Magma a, Magma b) => Hom a b where
- hom :: a -> b
-
-instance Hom b c => Hom (a -> b) (a -> c) where
- hom f = hom . f
-
-class (Hom a a) => End a
-instance (Hom a a) => End a
-
--- | A Isomorphism between two magmas
--- an Isomorphism is a bijective Homomorphism
-class (Hom a b, Hom b a) => Iso a b
-
-iso :: Iso a b => a -> b
-iso = hom
-
-invIso :: Iso a b => b -> a
-invIso = hom
-
-instance Iso b c => Iso (a -> b) (a -> c)
-
-class (Iso a a) => Automorphism a
-instance (Iso a a) => Automorphism a
diff --git a/src/NumHask/Algebra/Abstract/Module.hs b/src/NumHask/Algebra/Abstract/Module.hs
--- a/src/NumHask/Algebra/Abstract/Module.hs
+++ b/src/NumHask/Algebra/Abstract/Module.hs
@@ -12,7 +12,7 @@
import NumHask.Algebra.Abstract.Action
-- | A over r a is
--- a (Ring a), an abelian (Group r a) and an scalar-mult. (.*, *.) with the
+-- a (Ring a), an abelian (Group r a) and a scalar multiplier (.*, *.) with the
-- laws:
--
-- > a .* one == a
diff --git a/src/NumHask/Algebra/Abstract/Multiplicative.hs b/src/NumHask/Algebra/Abstract/Multiplicative.hs
--- a/src/NumHask/Algebra/Abstract/Multiplicative.hs
+++ b/src/NumHask/Algebra/Abstract/Multiplicative.hs
@@ -14,6 +14,14 @@
import Prelude (Int, Integer, Float, Double)
import qualified Prelude as P
+-- | For practical reasons, 'Multiplicative' has no super classes. Using 'Associative' and 'Unital' from this library, or using 'Semigroup' and 'Monoid' from base tends to complexify the interface once you start having to disinguish between (say) monoidal addition and monoidal multiplication.
+--
+-- > one * a == a
+-- > a * one == a
+-- > (a * b) * c == a * (b * c)
+-- > a * b == b * a
+--
+-- By convention, (*) is regarded as commutative, but this is not universal, and the introduction of another symbol which means non-commutative multiplication seems a bit dogmatic.
class Multiplicative a where
infixl 7 *
@@ -21,9 +29,16 @@
one :: a
+-- | Compute the product of a 'Foldable'.
product :: (Multiplicative a, P.Foldable f) => f a -> a
product = P.foldr (*) one
+-- |
+--
+-- > a / a = one
+-- > recip a = one / a
+-- > recip a * a = one
+-- > a * recip a = one
class (Multiplicative a) => Divisive a where
recip :: a -> a
@@ -102,4 +117,3 @@
instance Divisive b => Divisive (a -> b) where
recip f = recip P.. f
-
diff --git a/src/NumHask/Algebra/Abstract/Ring.hs b/src/NumHask/Algebra/Abstract/Ring.hs
--- a/src/NumHask/Algebra/Abstract/Ring.hs
+++ b/src/NumHask/Algebra/Abstract/Ring.hs
@@ -28,6 +28,10 @@
--
-- > a * (b + c) == a * b + a * c
-- > (a * b) * c == a * c + b * c
+-- > zero * a == zero
+-- > a * zero == zero
+--
+-- The sneaking in of the laws here glosses over the possibility that the multiplicative zero element does not have to correspond with the additive unital zero.
class (Additive a, Multiplicative a) =>
Distributive a
@@ -48,27 +52,22 @@
instance Distributive P.Bool
instance Distributive b => Distributive (a -> b)
--- | A is a ring without,
--- necessarily, negative elements.
---
--- FIXME: rule zero' = zero. Is this somehow expressible in haskell?
+-- | A is commutative monoidal under addition, has a monoidal multiplication operator (not necessarily commutative), and where multiplication distributes over addition.
class (Distributive a) =>
Semiring a where
instance (Distributive a) =>
Semiring a
-- | A is an abelian
--- group under addition and monoid under multiplication where multiplication
--- distributes over addition. Alternatively, a ring is semiring where additive
--- inverses exist
+-- group under addition and monoidal under multiplication, and where multiplication
+-- distributes over addition.
class (Distributive a, Subtractive a) =>
Ring a
instance (Distributive a, Subtractive a) =>
Ring a
-- | A is a
--- ring with a Commutative Multiplication operation. Recall that Addition is
--- Commutative in all Rings
+-- ring with commutative multiplication.
class (Distributive a, Subtractive a) =>
CommutativeRing a
instance (Distributive a, Subtractive a) =>
@@ -78,9 +77,8 @@
-- generalizes a ring of integers by requiring the product of any two nonzero
-- elements to be nonzero. This means that if a ≠ 0, an equality ab = ac
-- implies b = c.
--- FIXME: write a rule for this
--
-class (Distributive a, Divisive a) =>
+class (Distributive a) =>
IntegralDomain a
instance IntegralDomain P.Double
@@ -142,5 +140,6 @@
instance InvolutiveRing Word64
instance InvolutiveRing b => InvolutiveRing (a -> b)
+-- | Defining 'two' requires adding the multiplicative unital to itself.
two :: (Multiplicative a, Additive a) => a
two = one + one
diff --git a/src/NumHask/Algebra/Abstract/TensorProduct.hs b/src/NumHask/Algebra/Abstract/TensorProduct.hs
deleted file mode 100644
--- a/src/NumHask/Algebra/Abstract/TensorProduct.hs
+++ /dev/null
@@ -1,81 +0,0 @@
-{-# LANGUAGE PolyKinds #-}
-{-# LANGUAGE TypeFamilies #-}
-{-# LANGUAGE TypeOperators #-}
-{-# LANGUAGE MultiParamTypeClasses #-}
-{-# OPTIONS_GHC -Wall #-}
-
--- | TensorProduct
-module NumHask.Algebra.Abstract.TensorProduct
- ( TensorProduct(..)
- , TensorProduct'(..)
- , type (><)
- ) where
-
-import Data.Int (Int8, Int16, Int32, Int64)
-import Data.Word (Word, Word8, Word16, Word32, Word64)
-import GHC.Natural
-import Prelude (Double, Float, Int, Integer)
-
--- | tensorial type
-type family (><) (a :: k1) (b :: k2) :: *
-
-type instance Int >< Int = Int
-
-type instance Integer >< Integer = Integer
-
-type instance Double >< Double = Double
-
-type instance Float >< Float = Float
-
-type instance Natural >< Natural = Natural
-
-type instance Int8 >< Int8 = Int8
-
-type instance Int16 >< Int16 = Int16
-
-type instance Int32 >< Int32 = Int32
-
-type instance Int64 >< Int64 = Int64
-
-type instance Word >< Word = Word
-
-type instance Word8 >< Word8 = Word8
-
-type instance Word16 >< Word16 = Word16
-
-type instance Word32 >< Word32 = Word32
-
-type instance Word64 >< Word64 = Word64
-
--- | representation synthesis
-type family TensorRep k1 k2 where
- TensorRep (r a) (r a) = r (r a)
- TensorRep (r a) (s a) = r (s a)
- TensorRep (r a) a = r a
-
-type instance r a >< b = TensorRep (r a) b
-
--- | generalised outer product
---
--- > a>< b
--- > a>< (b+c)
--- > a *. (b> (a><
- (><) :: a -> a -> (a >< a)
- outer :: a -> a -> (a >< a)
- outer = (><)
- timesleft :: a -> (a >< a) -> a
- timesright :: (a >< a) -> a -> a
-
--- | generalised outer product
---
--- > a>< b
--- > a>< (b+c)
--- > a *. (b> (a> b -> (a >< b)
- timesleft' :: a -> (a >< b) -> b
- timesright' :: (a >< b) -> a -> b
diff --git a/src/NumHask/Algebra/Linear/Hadamard.hs b/src/NumHask/Algebra/Linear/Hadamard.hs
--- a/src/NumHask/Algebra/Linear/Hadamard.hs
+++ b/src/NumHask/Algebra/Linear/Hadamard.hs
@@ -1,7 +1,6 @@
{-# LANGUAGE FlexibleInstances #-}
{-# LANGUAGE MultiParamTypeClasses #-}
{-# LANGUAGE UndecidableInstances #-}
-{-# LANGUAGE TypeApplications #-}
{-# LANGUAGE ScopedTypeVariables #-}
{-# OPTIONS_GHC -Wall #-}
@@ -14,7 +13,6 @@
where
import NumHask.Algebra.Abstract.Multiplicative
-import Data.Coerce
-- | element by element multiplication
--
@@ -26,7 +24,6 @@
HadamardMultiplication m a where
infixl 7 .*.
(.*.) :: m a -> m a -> m a
- -- (.*.) = coerce ((.*.) @m @a)
-- | element by element division
@@ -36,7 +33,6 @@
HadamardDivision m a where
infixl 7 ./.
(./.) :: m a -> m a -> m a
- (./.) = coerce ((./.) @m @a)
class (HadamardMultiplication m a, HadamardDivision m a) => Hadamard m a
instance (HadamardMultiplication m a, HadamardDivision m a) => Hadamard m a
diff --git a/src/NumHask/Analysis/Banach.hs b/src/NumHask/Analysis/Banach.hs
deleted file mode 100644
--- a/src/NumHask/Analysis/Banach.hs
+++ /dev/null
@@ -1,38 +0,0 @@
-{-# LANGUAGE FlexibleContexts #-}
-{-# LANGUAGE RankNTypes #-}
-{-# LANGUAGE MultiParamTypeClasses #-}
-{-# OPTIONS_GHC -Wall #-}
-
--- | Metric classes
-module NumHask.Analysis.Banach
- ( Banach(..)
- , Hilbert(..)
- )
-where
-
-import NumHask.Algebra.Abstract.Action
-import NumHask.Algebra.Abstract.Ring
-import NumHask.Algebra.Abstract.Field
-import NumHask.Analysis.Metric
-
--- | Banach (with Norm) laws form rules around size and direction of a number, with a potential crossing into another codomain.
---
--- > a == singleton zero || normalizeL2 a *. normL2 a == a
-class (ExpField (Actor h), Normed h (Actor h), DivisiveAction h) =>
- Banach h where
- normalizeL1 :: h -> h
- normalizeL1 a = a ./ normL1 a
-
- normalizeL2 :: h -> h
- normalizeL2 a = a ./ normL2 a
-
--- | the inner product: a distributive fold
---
--- > a <.> b == b <.> a
--- > a <.> (b +c) == a <.> b + a <.> c
--- > a <.> (s *. b + c) == s * (a <.> b) + a <.> c
--- (s0 *. a) <.> (s1 *. b) == s0 * s1 * (a <.> b)
-class (Distributive (Actor h)) =>
- Hilbert h where
- infix 8 <.>
- (<.>) :: h -> h -> Actor h
diff --git a/src/NumHask/Analysis/Metric.hs b/src/NumHask/Analysis/Metric.hs
--- a/src/NumHask/Analysis/Metric.hs
+++ b/src/NumHask/Analysis/Metric.hs
@@ -27,7 +27,7 @@
import NumHask.Algebra.Abstract.Multiplicative
import NumHask.Algebra.Abstract.Lattice
--- | 'signum' from base is not an operator replicated in numhask, being such a very silly name, and preferred is the much more obvious 'sign'. Compare with 'Norm' and 'Banach' where there is a change in codomain
+-- | 'signum' from base is not an operator replicated in numhask, being such a very silly name, and preferred is the much more obvious 'sign'. Compare with 'Norm' where there is a change in codomain
--
-- > abs a * sign a == a
--
diff --git a/src/NumHask/Data/LogField.hs b/src/NumHask/Data/LogField.hs
--- a/src/NumHask/Data/LogField.hs
+++ b/src/NumHask/Data/LogField.hs
@@ -160,7 +160,7 @@
{-# INLINE [0] log1p #-}
log1p x = log (one + x)
-expm1 :: (ExpField a, Subtractive a) => a -> a
+expm1 :: (ExpField a) => a -> a
{-# INLINE [0] expm1 #-}
expm1 x = exp x - one
@@ -230,7 +230,7 @@
nearZero (LogField x) = nearZero $ exp x
aboutEqual (LogField x) (LogField y) = aboutEqual (exp x) (exp y)
-instance (Ord a, ExpField a, LowerBoundedField a) => Field (LogField a)
+instance (Ord a, ExpField a, LowerBoundedField a, UpperBoundedField a) => Field (LogField a)
instance (Ord a, ExpField a, LowerBoundedField a, UpperBoundedField a) =>
LowerBoundedField (LogField a)
@@ -281,7 +281,7 @@
-- it is not amenable to list fusion, and hence will use a lot of
-- memory when summing long lists.
{-# INLINE accurateSum #-}
-accurateSum :: (ExpField a, Subtractive a, Foldable f, Ord a) => f (LogField a) -> LogField a
+accurateSum :: (ExpField a, Foldable f, Ord a) => f (LogField a) -> LogField a
accurateSum xs = LogField (theMax + log theSum)
where
LogField theMax = maximum xs
@@ -295,7 +295,7 @@
--
-- > LogField . accurateProduct == accurateProduct . map LogField
{-# INLINE accurateProduct #-}
-accurateProduct :: (ExpField a, Subtractive a, Foldable f) => f (LogField a) -> LogField a
+accurateProduct :: (ExpField a, Foldable f) => f (LogField a) -> LogField a
accurateProduct = LogField . fst . F.foldr kahanPlus (zero, zero)
where
kahanPlus (LogField x) (t, c) =
diff --git a/src/NumHask/Data/Positive.hs b/src/NumHask/Data/Positive.hs
--- a/src/NumHask/Data/Positive.hs
+++ b/src/NumHask/Data/Positive.hs
@@ -63,11 +63,11 @@
QuotientField (Positive a) (Positive P.Integer) where
properFraction (Positive a) = let (i,r) = properFraction a in (Positive i, Positive r)
-instance (UpperBoundedField a) =>
+instance (P.Ord a, UpperBoundedField a) =>
UpperBoundedField (Positive a) where
infinity = Positive infinity
-instance (UpperBoundedField a) => P.Bounded (Positive a) where
+instance (P.Ord a, UpperBoundedField a) => P.Bounded (Positive a) where
minBound = zero
maxBound = infinity