-- Hoogle documentation, generated by Haddock
-- See Hoogle, http://www.haskell.org/hoogle/


-- | numeric classes
--   
--   A numeric class heirarchy.
@package numhask
@version 0.3.1


-- | Additive
module NumHask.Algebra.Abstract.Additive
class Additive a
(+) :: Additive a => a -> a -> a
zero :: Additive a => a
infixl 6 +
sum :: (Additive a, Foldable f) => f a -> a
class (Additive a) => Subtractive a
negate :: Subtractive a => a -> a
(-) :: Subtractive a => a -> a -> a
infixl 6 -
instance NumHask.Algebra.Abstract.Additive.Subtractive GHC.Types.Double
instance NumHask.Algebra.Abstract.Additive.Subtractive GHC.Types.Float
instance NumHask.Algebra.Abstract.Additive.Subtractive GHC.Types.Int
instance NumHask.Algebra.Abstract.Additive.Subtractive GHC.Integer.Type.Integer
instance NumHask.Algebra.Abstract.Additive.Subtractive GHC.Types.Bool
instance NumHask.Algebra.Abstract.Additive.Subtractive GHC.Natural.Natural
instance NumHask.Algebra.Abstract.Additive.Subtractive GHC.Int.Int8
instance NumHask.Algebra.Abstract.Additive.Subtractive GHC.Int.Int16
instance NumHask.Algebra.Abstract.Additive.Subtractive GHC.Int.Int32
instance NumHask.Algebra.Abstract.Additive.Subtractive GHC.Int.Int64
instance NumHask.Algebra.Abstract.Additive.Subtractive GHC.Types.Word
instance NumHask.Algebra.Abstract.Additive.Subtractive GHC.Word.Word8
instance NumHask.Algebra.Abstract.Additive.Subtractive GHC.Word.Word16
instance NumHask.Algebra.Abstract.Additive.Subtractive GHC.Word.Word32
instance NumHask.Algebra.Abstract.Additive.Subtractive GHC.Word.Word64
instance NumHask.Algebra.Abstract.Additive.Subtractive b => NumHask.Algebra.Abstract.Additive.Subtractive (a -> b)
instance NumHask.Algebra.Abstract.Additive.Additive GHC.Types.Double
instance NumHask.Algebra.Abstract.Additive.Additive GHC.Types.Float
instance NumHask.Algebra.Abstract.Additive.Additive GHC.Types.Int
instance NumHask.Algebra.Abstract.Additive.Additive GHC.Integer.Type.Integer
instance NumHask.Algebra.Abstract.Additive.Additive GHC.Types.Bool
instance NumHask.Algebra.Abstract.Additive.Additive GHC.Natural.Natural
instance NumHask.Algebra.Abstract.Additive.Additive GHC.Int.Int8
instance NumHask.Algebra.Abstract.Additive.Additive GHC.Int.Int16
instance NumHask.Algebra.Abstract.Additive.Additive GHC.Int.Int32
instance NumHask.Algebra.Abstract.Additive.Additive GHC.Int.Int64
instance NumHask.Algebra.Abstract.Additive.Additive GHC.Types.Word
instance NumHask.Algebra.Abstract.Additive.Additive GHC.Word.Word8
instance NumHask.Algebra.Abstract.Additive.Additive GHC.Word.Word16
instance NumHask.Algebra.Abstract.Additive.Additive GHC.Word.Word32
instance NumHask.Algebra.Abstract.Additive.Additive GHC.Word.Word64
instance NumHask.Algebra.Abstract.Additive.Additive b => NumHask.Algebra.Abstract.Additive.Additive (a -> b)


-- | The Group hierarchy
module NumHask.Algebra.Abstract.Group

-- | A <a>Magma</a> is a tuple (T,magma) consisting of
--   
--   <ul>
--   <li>a type a, and</li>
--   <li>a function (magma) :: T -&gt; T -&gt; T</li>
--   </ul>
--   
--   The mathematical laws for a magma are:
--   
--   <ul>
--   <li>magma is defined for all possible pairs of type T, and</li>
--   <li>magma is closed in the set of all possible values of type T</li>
--   </ul>
--   
--   or, more tersly,
--   
--   <pre>
--   ∀ a, b ∈ T: a magma b ∈ T
--   </pre>
--   
--   These laws are true by construction in haskell: the type signature of
--   <a>magma</a> and the above mathematical laws are synonyms.
class Magma a
magma :: Magma a => a -> a -> a

-- | A Unital Magma is a magma with an <a>identity element</a> (the unit).
--   
--   <pre>
--   unit magma a = a
--   a magma unit = a
--   </pre>
class Magma a => Unital a
unit :: Unital a => a

-- | An Associative Magma
--   
--   <pre>
--   (a magma b) magma c = a magma (b magma c)
--   </pre>
class Magma a => Associative a

-- | A Commutative Magma is a Magma where the binary operation is
--   <a>commutative</a>.
--   
--   <pre>
--   a magma b = b magma a
--   </pre>
class Magma a => Commutative a

-- | An Absorbing is a Magma with an <a>Absorbing Element</a>
--   
--   <pre>
--   a `times` absorb = absorb
--   </pre>
class Magma a => Absorbing a
absorb :: Absorbing a => a

-- | An Invertible Magma
--   
--   <pre>
--   ∀ a,b ∈ T: inv a `magma` (a `magma` b) = b = (b `magma` a) `magma` inv a
--   </pre>
class Magma a => Invertible a
inv :: Invertible a => a -> a

-- | An Idempotent Magma is a magma where every element is
--   <a>Idempotent</a>.
--   
--   <pre>
--   a magma a = a
--   </pre>
class Magma a => Idempotent a

-- | A <a>Group</a> is a Associative, Unital and Invertible Magma.
class (Associative a, Unital a, Invertible a) => Group a

-- | An <a>Abelian Group</a> is an Associative, Unital, Invertible and
--   Commutative Magma . In other words, it is a Commutative Group
class (Associative a, Unital a, Invertible a, Commutative a) => AbelianGroup a
instance (NumHask.Algebra.Abstract.Group.Associative a, NumHask.Algebra.Abstract.Group.Unital a, NumHask.Algebra.Abstract.Group.Invertible a, NumHask.Algebra.Abstract.Group.Commutative a) => NumHask.Algebra.Abstract.Group.AbelianGroup a
instance NumHask.Algebra.Abstract.Group.Idempotent b => NumHask.Algebra.Abstract.Group.Idempotent (a -> b)
instance NumHask.Algebra.Abstract.Group.Absorbing b => NumHask.Algebra.Abstract.Group.Absorbing (a -> b)
instance (NumHask.Algebra.Abstract.Group.Associative a, NumHask.Algebra.Abstract.Group.Unital a, NumHask.Algebra.Abstract.Group.Invertible a) => NumHask.Algebra.Abstract.Group.Group a
instance NumHask.Algebra.Abstract.Group.Invertible b => NumHask.Algebra.Abstract.Group.Invertible (a -> b)
instance NumHask.Algebra.Abstract.Group.Commutative b => NumHask.Algebra.Abstract.Group.Commutative (a -> b)
instance NumHask.Algebra.Abstract.Group.Associative b => NumHask.Algebra.Abstract.Group.Associative (a -> b)
instance NumHask.Algebra.Abstract.Group.Unital b => NumHask.Algebra.Abstract.Group.Unital (a -> b)
instance NumHask.Algebra.Abstract.Group.Magma b => NumHask.Algebra.Abstract.Group.Magma (a -> b)


-- | The Homomorphism Hierarchy
module NumHask.Algebra.Abstract.Homomorphism

-- | A Homomorphism between two magmas law: forall a b. hom(a <a>magma</a>
--   b) = hom(a) <a>magma</a> hom(b)
class (Magma a, Magma b) => Hom a b
hom :: Hom a b => a -> b
class (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
invIso :: Iso a b => b -> a
class (Iso a a) => Automorphism a
instance NumHask.Algebra.Abstract.Homomorphism.Iso a a => NumHask.Algebra.Abstract.Homomorphism.Automorphism a
instance NumHask.Algebra.Abstract.Homomorphism.Iso b c => NumHask.Algebra.Abstract.Homomorphism.Iso (a -> b) (a -> c)
instance NumHask.Algebra.Abstract.Homomorphism.Hom a a => NumHask.Algebra.Abstract.Homomorphism.End a
instance NumHask.Algebra.Abstract.Homomorphism.Hom b c => NumHask.Algebra.Abstract.Homomorphism.Hom (a -> b) (a -> c)


-- | Multiplicative
module NumHask.Algebra.Abstract.Multiplicative
class Multiplicative a
(*) :: Multiplicative a => a -> a -> a
one :: Multiplicative a => a
infixl 7 *
product :: (Multiplicative a, Foldable f) => f a -> a
class (Multiplicative a) => Divisive a
recip :: Divisive a => a -> a
(/) :: Divisive a => a -> a -> a
infixl 7 /
instance NumHask.Algebra.Abstract.Multiplicative.Divisive GHC.Types.Double
instance NumHask.Algebra.Abstract.Multiplicative.Divisive GHC.Types.Float
instance NumHask.Algebra.Abstract.Multiplicative.Divisive b => NumHask.Algebra.Abstract.Multiplicative.Divisive (a -> b)
instance NumHask.Algebra.Abstract.Multiplicative.Multiplicative GHC.Types.Double
instance NumHask.Algebra.Abstract.Multiplicative.Multiplicative GHC.Types.Float
instance NumHask.Algebra.Abstract.Multiplicative.Multiplicative GHC.Types.Int
instance NumHask.Algebra.Abstract.Multiplicative.Multiplicative GHC.Integer.Type.Integer
instance NumHask.Algebra.Abstract.Multiplicative.Multiplicative GHC.Types.Bool
instance NumHask.Algebra.Abstract.Multiplicative.Multiplicative GHC.Natural.Natural
instance NumHask.Algebra.Abstract.Multiplicative.Multiplicative GHC.Int.Int8
instance NumHask.Algebra.Abstract.Multiplicative.Multiplicative GHC.Int.Int16
instance NumHask.Algebra.Abstract.Multiplicative.Multiplicative GHC.Int.Int32
instance NumHask.Algebra.Abstract.Multiplicative.Multiplicative GHC.Int.Int64
instance NumHask.Algebra.Abstract.Multiplicative.Multiplicative GHC.Types.Word
instance NumHask.Algebra.Abstract.Multiplicative.Multiplicative GHC.Word.Word8
instance NumHask.Algebra.Abstract.Multiplicative.Multiplicative GHC.Word.Word16
instance NumHask.Algebra.Abstract.Multiplicative.Multiplicative GHC.Word.Word32
instance NumHask.Algebra.Abstract.Multiplicative.Multiplicative GHC.Word.Word64
instance NumHask.Algebra.Abstract.Multiplicative.Multiplicative b => NumHask.Algebra.Abstract.Multiplicative.Multiplicative (a -> b)


-- | Action
module NumHask.Algebra.Abstract.Action

-- | a type class to represent an action on a higher-kinded number
type family Actor h
class (Additive (Actor h)) => AdditiveAction h
(.+) :: AdditiveAction h => h -> Actor h -> h
(+.) :: AdditiveAction h => Actor h -> h -> h
infixl 6 .+
infixl 6 +.
class (Subtractive (Actor h)) => SubtractiveAction h
(.-) :: SubtractiveAction h => h -> Actor h -> h
(-.) :: SubtractiveAction h => Actor h -> h -> h
infixl 6 .-
infixl 6 -.
class (Multiplicative (Actor h)) => MultiplicativeAction h
(.*) :: MultiplicativeAction h => h -> Actor h -> h
(*.) :: MultiplicativeAction h => Actor h -> h -> h
infixl 7 .*
infixl 7 *.
class (Divisive (Actor h)) => DivisiveAction h
(./) :: DivisiveAction h => h -> Actor h -> h
(/.) :: DivisiveAction h => Actor h -> h -> h
infixl 7 ./
infixl 7 /.


-- | Ring
module NumHask.Algebra.Abstract.Ring

-- | <a>Distributive</a> laws
--   
--   <pre>
--   a * (b + c) == a * b + a * c
--   (a * b) * c == a * c + b * c
--   </pre>
class (Additive a, Multiplicative a) => Distributive a

-- | A <a>Semiring</a> is a ring without, necessarily, negative elements.
--   
--   FIXME: rule zero' = zero. Is this somehow expressible in haskell?
class (Distributive a) => Semiring a

-- | A <a>Ring</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
class (Distributive a, Subtractive a) => Ring a

-- | A <a>Commutative Ring</a> is a ring with a Commutative Multiplication
--   operation. Recall that Addition is Commutative in all Rings
class (Distributive a, Subtractive a) => CommutativeRing a

-- | An <a>Integral Domain</a> 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) => IntegralDomain a

-- | A <a>StarSemiring</a> is a semiring with an additional unary operator
--   star satisfying:
--   
--   <pre>
--   star a = one + a `times` star a
--   </pre>
class (Distributive a) => StarSemiring a
star :: StarSemiring a => a -> a
plus :: StarSemiring a => a -> a

-- | A <a>Kleene Algebra</a> is a Star Semiring with idempotent addition
--   
--   <pre>
--   a `times` x + x = a ==&gt; star a `times` x + x = x
--   x `times` a + x = a ==&gt; x `times` star a + x = x
--   </pre>
class (StarSemiring a, Idempotent a) => KleeneAlgebra a

-- | Involutive Ring
--   
--   <pre>
--   adj (a + b) ==&gt; adj a + adj b
--   adj (a * b) ==&gt; adj a * adj b
--   adj one ==&gt; one
--   adj (adj a) ==&gt; a
--   </pre>
--   
--   Note: elements for which <tt>adj a == a</tt> are called
--   "self-adjoint".
class (Distributive a) => InvolutiveRing a
adj :: InvolutiveRing a => a -> a
two :: (Multiplicative a, Additive a) => a
instance NumHask.Algebra.Abstract.Ring.InvolutiveRing GHC.Types.Double
instance NumHask.Algebra.Abstract.Ring.InvolutiveRing GHC.Types.Float
instance NumHask.Algebra.Abstract.Ring.InvolutiveRing GHC.Integer.Type.Integer
instance NumHask.Algebra.Abstract.Ring.InvolutiveRing GHC.Types.Int
instance NumHask.Algebra.Abstract.Ring.InvolutiveRing GHC.Natural.Natural
instance NumHask.Algebra.Abstract.Ring.InvolutiveRing GHC.Int.Int8
instance NumHask.Algebra.Abstract.Ring.InvolutiveRing GHC.Int.Int16
instance NumHask.Algebra.Abstract.Ring.InvolutiveRing GHC.Int.Int32
instance NumHask.Algebra.Abstract.Ring.InvolutiveRing GHC.Int.Int64
instance NumHask.Algebra.Abstract.Ring.InvolutiveRing GHC.Types.Word
instance NumHask.Algebra.Abstract.Ring.InvolutiveRing GHC.Word.Word8
instance NumHask.Algebra.Abstract.Ring.InvolutiveRing GHC.Word.Word16
instance NumHask.Algebra.Abstract.Ring.InvolutiveRing GHC.Word.Word32
instance NumHask.Algebra.Abstract.Ring.InvolutiveRing GHC.Word.Word64
instance NumHask.Algebra.Abstract.Ring.InvolutiveRing b => NumHask.Algebra.Abstract.Ring.InvolutiveRing (a -> b)
instance NumHask.Algebra.Abstract.Ring.KleeneAlgebra b => NumHask.Algebra.Abstract.Ring.KleeneAlgebra (a -> b)
instance NumHask.Algebra.Abstract.Ring.StarSemiring b => NumHask.Algebra.Abstract.Ring.StarSemiring (a -> b)
instance NumHask.Algebra.Abstract.Ring.IntegralDomain GHC.Types.Double
instance NumHask.Algebra.Abstract.Ring.IntegralDomain GHC.Types.Float
instance NumHask.Algebra.Abstract.Ring.IntegralDomain b => NumHask.Algebra.Abstract.Ring.IntegralDomain (a -> b)
instance (NumHask.Algebra.Abstract.Ring.Distributive a, NumHask.Algebra.Abstract.Additive.Subtractive a) => NumHask.Algebra.Abstract.Ring.CommutativeRing a
instance (NumHask.Algebra.Abstract.Ring.Distributive a, NumHask.Algebra.Abstract.Additive.Subtractive a) => NumHask.Algebra.Abstract.Ring.Ring a
instance NumHask.Algebra.Abstract.Ring.Distributive a => NumHask.Algebra.Abstract.Ring.Semiring a
instance NumHask.Algebra.Abstract.Ring.Distributive GHC.Types.Double
instance NumHask.Algebra.Abstract.Ring.Distributive GHC.Types.Float
instance NumHask.Algebra.Abstract.Ring.Distributive GHC.Types.Int
instance NumHask.Algebra.Abstract.Ring.Distributive GHC.Integer.Type.Integer
instance NumHask.Algebra.Abstract.Ring.Distributive GHC.Natural.Natural
instance NumHask.Algebra.Abstract.Ring.Distributive GHC.Int.Int8
instance NumHask.Algebra.Abstract.Ring.Distributive GHC.Int.Int16
instance NumHask.Algebra.Abstract.Ring.Distributive GHC.Int.Int32
instance NumHask.Algebra.Abstract.Ring.Distributive GHC.Int.Int64
instance NumHask.Algebra.Abstract.Ring.Distributive GHC.Types.Word
instance NumHask.Algebra.Abstract.Ring.Distributive GHC.Word.Word8
instance NumHask.Algebra.Abstract.Ring.Distributive GHC.Word.Word16
instance NumHask.Algebra.Abstract.Ring.Distributive GHC.Word.Word32
instance NumHask.Algebra.Abstract.Ring.Distributive GHC.Word.Word64
instance NumHask.Algebra.Abstract.Ring.Distributive GHC.Types.Bool
instance NumHask.Algebra.Abstract.Ring.Distributive b => NumHask.Algebra.Abstract.Ring.Distributive (a -> b)


-- | Algebra for Modules
module NumHask.Algebra.Abstract.Module

-- | A <a>Module</a> over r a is a (Ring a), an abelian (Group r a) and an
--   scalar-mult. (.*, *.) with the laws:
--   
--   <pre>
--   a .* one == a
--   (a + b) .* c == (a .* c) + (b .* c)
--   c *. (a + b) == (c *. a) + (c *. b)
--   a .* zero == zero
--   a .* b == b *. a
--   </pre>
class (Distributive (Actor h), Divisive h, MultiplicativeAction h) => Module h


-- | TensorProduct
module NumHask.Algebra.Abstract.TensorProduct

-- | generalised outer product
--   
--   <pre>
--   a&gt;&lt;b + c&gt;&lt;b == (a+c) &gt;&lt; b
--   a&gt;&lt;b + a&gt;&lt;c == a &gt;&lt; (b+c)
--   a *. (b&gt;&lt;c) == (a&gt;&lt;b) .* c
--   (a&gt;&lt;b) .* c == a *. (b&gt;&lt;c)
--   </pre>
class TensorProduct a
(><) :: TensorProduct a => a -> a -> a >< a
outer :: TensorProduct a => a -> a -> a >< a
timesleft :: TensorProduct a => a -> (a >< a) -> a
timesright :: TensorProduct a => (a >< a) -> a -> a
infix 8 ><

-- | generalised outer product
--   
--   <pre>
--   a&gt;&lt;b + c&gt;&lt;b == (a+c) &gt;&lt; b
--   a&gt;&lt;b + a&gt;&lt;c == a &gt;&lt; (b+c)
--   a *. (b&gt;&lt;c) == (a&gt;&lt;b) .* c
--   (a&gt;&lt;b) .* c == a *. (b&gt;&lt;c)
--   </pre>
class TensorProduct' a b
outer' :: TensorProduct' a b => a -> b -> a >< b
timesleft' :: TensorProduct' a b => a -> (a >< b) -> b
timesright' :: TensorProduct' a b => (a >< b) -> a -> b

-- | tensorial type
type family (><) (a :: k1) (b :: k2) :: *
infix 8 ><


-- | Element-by-element operations
module NumHask.Algebra.Linear.Hadamard

-- | element by element multiplication
--   
--   <pre>
--   (a .*. b) .*. c == a .*. (b .*. c)
--   singleton one .*. a = a
--   a .*. singleton one = a
--   a .*. b == b .*. a
--   </pre>
class (Multiplicative a) => HadamardMultiplication m a
(.*.) :: HadamardMultiplication m a => m a -> m a -> m a
infixl 7 .*.

-- | element by element division
--   
--   <pre>
--   a ./. a == singleton one
--   </pre>
class (Divisive a) => HadamardDivision m a
(./.) :: HadamardDivision m a => m a -> m a -> m a
infixl 7 ./.
class (HadamardMultiplication m a, HadamardDivision m a) => Hadamard m a
instance (NumHask.Algebra.Linear.Hadamard.HadamardMultiplication m a, NumHask.Algebra.Linear.Hadamard.HadamardDivision m a) => NumHask.Algebra.Linear.Hadamard.Hadamard m a


-- | Integral classes
module NumHask.Data.Integral

-- | Integral laws
--   
--   <pre>
--   b == zero || b * (a `div` b) + (a `mod` b) == a
--   </pre>
class (Distributive a) => Integral a
div :: Integral a => a -> a -> a
mod :: Integral a => a -> a -> a
divMod :: Integral a => a -> a -> (a, a)
quot :: Integral a => a -> a -> a
rem :: Integral a => a -> a -> a
quotRem :: Integral a => a -> a -> (a, a)
infixl 7 `div`
infixl 7 `mod`

-- | toIntegral is kept separate from Integral to help with compatability
--   issues. &gt; toIntegral a == a
class ToIntegral a b
toIntegral :: ToIntegral a b => a -> b
toIntegral :: (ToIntegral a b, a ~ b) => a -> b
type ToInteger a = ToIntegral a Integer
toInteger :: ToInteger a => a -> Integer

-- | fromIntegral abstracts the codomain type, compared with the preludes
--   Integral type. &gt; fromIntegral_ a == a
--   
--   fromIntegral is widely used as general coercion, hence the underscore
--   for the operator.
class FromIntegral a b
fromIntegral_ :: FromIntegral a b => b -> a
fromIntegral_ :: (FromIntegral a b, a ~ b) => b -> a

-- | ghc defaulting rules and, it seems, -XExtendedDefaultRules do not
--   permit multiple parameter typeclasses to be in the mix when types are
--   resolved, hence the simpler `type FromInteger a = FromIntegral a
--   Integer` does not suffice.
class FromInteger a
fromInteger :: FromInteger a => Integer -> a
fromInteger :: (FromInteger a, FromIntegral a Integer) => Integer -> a

-- | general coercion via Integer
fromIntegral :: (FromInteger b, ToInteger a) => a -> b
even :: (Eq a, Integral a) => a -> Bool
odd :: (Eq a, Integral a) => a -> Bool

-- | raise a number to a non-negative integral power
(^) :: (Ord b, Multiplicative a, Integral b) => a -> b -> a
(^^) :: (Divisive a, Subtractive b, Integral b, Ord b) => a -> b -> a
instance NumHask.Data.Integral.FromInteger GHC.Integer.Type.Integer
instance NumHask.Data.Integral.FromInteger GHC.Types.Int
instance NumHask.Data.Integral.FromInteger GHC.Types.Double
instance NumHask.Data.Integral.FromInteger GHC.Types.Float
instance NumHask.Data.Integral.FromInteger GHC.Natural.Natural
instance NumHask.Data.Integral.FromInteger GHC.Int.Int8
instance NumHask.Data.Integral.FromInteger GHC.Int.Int16
instance NumHask.Data.Integral.FromInteger GHC.Int.Int32
instance NumHask.Data.Integral.FromInteger GHC.Int.Int64
instance NumHask.Data.Integral.FromInteger GHC.Types.Word
instance NumHask.Data.Integral.FromInteger GHC.Word.Word8
instance NumHask.Data.Integral.FromInteger GHC.Word.Word16
instance NumHask.Data.Integral.FromInteger GHC.Word.Word32
instance NumHask.Data.Integral.FromInteger GHC.Word.Word64
instance NumHask.Data.Integral.FromIntegral a b => NumHask.Data.Integral.FromIntegral (c -> a) b
instance NumHask.Data.Integral.FromIntegral GHC.Types.Double GHC.Integer.Type.Integer
instance NumHask.Data.Integral.FromIntegral GHC.Types.Float GHC.Integer.Type.Integer
instance NumHask.Data.Integral.FromIntegral GHC.Types.Int GHC.Integer.Type.Integer
instance NumHask.Data.Integral.FromIntegral GHC.Integer.Type.Integer GHC.Integer.Type.Integer
instance NumHask.Data.Integral.FromIntegral GHC.Natural.Natural GHC.Integer.Type.Integer
instance NumHask.Data.Integral.FromIntegral GHC.Int.Int8 GHC.Integer.Type.Integer
instance NumHask.Data.Integral.FromIntegral GHC.Int.Int16 GHC.Integer.Type.Integer
instance NumHask.Data.Integral.FromIntegral GHC.Int.Int32 GHC.Integer.Type.Integer
instance NumHask.Data.Integral.FromIntegral GHC.Int.Int64 GHC.Integer.Type.Integer
instance NumHask.Data.Integral.FromIntegral GHC.Types.Word GHC.Integer.Type.Integer
instance NumHask.Data.Integral.FromIntegral GHC.Word.Word8 GHC.Integer.Type.Integer
instance NumHask.Data.Integral.FromIntegral GHC.Word.Word16 GHC.Integer.Type.Integer
instance NumHask.Data.Integral.FromIntegral GHC.Word.Word32 GHC.Integer.Type.Integer
instance NumHask.Data.Integral.FromIntegral GHC.Word.Word64 GHC.Integer.Type.Integer
instance NumHask.Data.Integral.FromIntegral GHC.Types.Int GHC.Types.Int
instance NumHask.Data.Integral.FromIntegral GHC.Natural.Natural GHC.Natural.Natural
instance NumHask.Data.Integral.FromIntegral GHC.Int.Int8 GHC.Int.Int8
instance NumHask.Data.Integral.FromIntegral GHC.Int.Int16 GHC.Int.Int16
instance NumHask.Data.Integral.FromIntegral GHC.Int.Int32 GHC.Int.Int32
instance NumHask.Data.Integral.FromIntegral GHC.Int.Int64 GHC.Int.Int64
instance NumHask.Data.Integral.FromIntegral GHC.Types.Word GHC.Types.Word
instance NumHask.Data.Integral.FromIntegral GHC.Word.Word8 GHC.Word.Word8
instance NumHask.Data.Integral.FromIntegral GHC.Word.Word16 GHC.Word.Word16
instance NumHask.Data.Integral.FromIntegral GHC.Word.Word32 GHC.Word.Word32
instance NumHask.Data.Integral.FromIntegral GHC.Word.Word64 GHC.Word.Word64
instance NumHask.Data.Integral.ToIntegral GHC.Integer.Type.Integer GHC.Integer.Type.Integer
instance NumHask.Data.Integral.ToIntegral GHC.Types.Int GHC.Integer.Type.Integer
instance NumHask.Data.Integral.ToIntegral GHC.Natural.Natural GHC.Integer.Type.Integer
instance NumHask.Data.Integral.ToIntegral GHC.Int.Int8 GHC.Integer.Type.Integer
instance NumHask.Data.Integral.ToIntegral GHC.Int.Int16 GHC.Integer.Type.Integer
instance NumHask.Data.Integral.ToIntegral GHC.Int.Int32 GHC.Integer.Type.Integer
instance NumHask.Data.Integral.ToIntegral GHC.Int.Int64 GHC.Integer.Type.Integer
instance NumHask.Data.Integral.ToIntegral GHC.Types.Word GHC.Integer.Type.Integer
instance NumHask.Data.Integral.ToIntegral GHC.Word.Word8 GHC.Integer.Type.Integer
instance NumHask.Data.Integral.ToIntegral GHC.Word.Word16 GHC.Integer.Type.Integer
instance NumHask.Data.Integral.ToIntegral GHC.Word.Word32 GHC.Integer.Type.Integer
instance NumHask.Data.Integral.ToIntegral GHC.Word.Word64 GHC.Integer.Type.Integer
instance NumHask.Data.Integral.ToIntegral GHC.Types.Int GHC.Types.Int
instance NumHask.Data.Integral.ToIntegral GHC.Natural.Natural GHC.Natural.Natural
instance NumHask.Data.Integral.ToIntegral GHC.Int.Int8 GHC.Int.Int8
instance NumHask.Data.Integral.ToIntegral GHC.Int.Int16 GHC.Int.Int16
instance NumHask.Data.Integral.ToIntegral GHC.Int.Int32 GHC.Int.Int32
instance NumHask.Data.Integral.ToIntegral GHC.Int.Int64 GHC.Int.Int64
instance NumHask.Data.Integral.ToIntegral GHC.Types.Word GHC.Types.Word
instance NumHask.Data.Integral.ToIntegral GHC.Word.Word8 GHC.Word.Word8
instance NumHask.Data.Integral.ToIntegral GHC.Word.Word16 GHC.Word.Word16
instance NumHask.Data.Integral.ToIntegral GHC.Word.Word32 GHC.Word.Word32
instance NumHask.Data.Integral.ToIntegral GHC.Word.Word64 GHC.Word.Word64
instance NumHask.Data.Integral.Integral GHC.Types.Int
instance NumHask.Data.Integral.Integral GHC.Integer.Type.Integer
instance NumHask.Data.Integral.Integral GHC.Natural.Natural
instance NumHask.Data.Integral.Integral GHC.Int.Int8
instance NumHask.Data.Integral.Integral GHC.Int.Int16
instance NumHask.Data.Integral.Integral GHC.Int.Int32
instance NumHask.Data.Integral.Integral GHC.Int.Int64
instance NumHask.Data.Integral.Integral GHC.Types.Word
instance NumHask.Data.Integral.Integral GHC.Word.Word8
instance NumHask.Data.Integral.Integral GHC.Word.Word16
instance NumHask.Data.Integral.Integral GHC.Word.Word32
instance NumHask.Data.Integral.Integral GHC.Word.Word64
instance NumHask.Data.Integral.Integral b => NumHask.Data.Integral.Integral (a -> b)


-- | Field classes
module NumHask.Algebra.Abstract.Field

-- | A <a>Field</a> is an Integral domain in which every non-zero element
--   has a multiplicative inverse.
--   
--   A summary of the rules inherited from super-classes of Field
--   
--   <pre>
--   zero + a == a
--   a + zero == a
--   (a + b) + c == a + (b + c)
--   a + b == b + a
--   a - a = zero
--   negate a = zero - a
--   negate a + a = zero
--   a + negate a = zero
--   one * a == a
--   a * one == a
--   (a * b) * c == a * (b * c)
--   a * (b + c) == a * b + a * c
--   (a + b) * c == a * c + b * c
--   a * zero == zero
--   zero * a == zero
--   a * b == b * a
--   a / a = one
--   recip a = one / a
--   recip a * a = one
--   a * recip a = one
--   </pre>
class (IntegralDomain a) => Field a

-- | A hyperbolic field class
--   
--   <pre>
--   sqrt . (**2) == identity
--   log . exp == identity
--   for +ive b, a != 0,1: a ** logBase a b ≈ b
--   </pre>
class (Field a) => ExpField a
exp :: ExpField a => a -> a
log :: ExpField a => a -> a
logBase :: ExpField a => a -> a -> a
(**) :: ExpField a => a -> a -> a
sqrt :: ExpField a => a -> a
class (Field a, Subtractive a, Multiplicative b, Additive b) => QuotientField a b
properFraction :: QuotientField a b => a -> (b, a)
round :: QuotientField a b => a -> b
round :: (QuotientField a b, Ord a, Ord b, Subtractive b, Integral b) => a -> b
ceiling :: QuotientField a b => a -> b
ceiling :: (QuotientField a b, Ord a) => a -> b
floor :: QuotientField a b => a -> b
floor :: (QuotientField a b, Ord a, Subtractive b) => a -> b
truncate :: QuotientField a b => a -> b
truncate :: (QuotientField a b, Ord a) => a -> b

-- | A bounded field includes the concepts of infinity and NaN, thus moving
--   away from error throwing.
--   
--   <pre>
--   one / zero + infinity == infinity
--   infinity + a == infinity
--   zero / zero != nan
--   </pre>
--   
--   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) => UpperBoundedField a
infinity :: UpperBoundedField a => a
nan :: UpperBoundedField a => a
class (Subtractive a, Field a) => LowerBoundedField a
negInfinity :: LowerBoundedField a => a

-- | Trigonometric Field
class (Field a) => TrigField a
pi :: TrigField a => a
sin :: TrigField a => a -> a
cos :: TrigField a => a -> a
tan :: TrigField a => a -> a
asin :: TrigField a => a -> a
acos :: TrigField a => a -> a
atan :: TrigField a => a -> a
sinh :: TrigField a => a -> a
cosh :: TrigField a => a -> a
tanh :: TrigField a => a -> a
asinh :: TrigField a => a -> a
acosh :: TrigField a => a -> a
atanh :: TrigField a => a -> a
half :: Field a => a
instance NumHask.Algebra.Abstract.Field.TrigField GHC.Types.Double
instance NumHask.Algebra.Abstract.Field.TrigField GHC.Types.Float
instance NumHask.Algebra.Abstract.Field.TrigField b => NumHask.Algebra.Abstract.Field.TrigField (a -> b)
instance NumHask.Algebra.Abstract.Field.LowerBoundedField GHC.Types.Float
instance NumHask.Algebra.Abstract.Field.LowerBoundedField GHC.Types.Double
instance NumHask.Algebra.Abstract.Field.LowerBoundedField b => NumHask.Algebra.Abstract.Field.LowerBoundedField (a -> b)
instance NumHask.Algebra.Abstract.Field.UpperBoundedField GHC.Types.Float
instance NumHask.Algebra.Abstract.Field.UpperBoundedField GHC.Types.Double
instance NumHask.Algebra.Abstract.Field.UpperBoundedField b => NumHask.Algebra.Abstract.Field.UpperBoundedField (a -> b)
instance NumHask.Algebra.Abstract.Field.QuotientField GHC.Types.Float GHC.Integer.Type.Integer
instance NumHask.Algebra.Abstract.Field.QuotientField GHC.Types.Double GHC.Integer.Type.Integer
instance NumHask.Algebra.Abstract.Field.QuotientField b c => NumHask.Algebra.Abstract.Field.QuotientField (a -> b) (a -> c)
instance NumHask.Algebra.Abstract.Field.ExpField GHC.Types.Double
instance NumHask.Algebra.Abstract.Field.ExpField GHC.Types.Float
instance NumHask.Algebra.Abstract.Field.ExpField b => NumHask.Algebra.Abstract.Field.ExpField (a -> b)
instance NumHask.Algebra.Abstract.Field.Field GHC.Types.Double
instance NumHask.Algebra.Abstract.Field.Field GHC.Types.Float
instance NumHask.Algebra.Abstract.Field.Field b => NumHask.Algebra.Abstract.Field.Field (a -> b)

module NumHask.Algebra.Abstract.Lattice

-- | A algebraic structure with element joins:
--   <a>http://en.wikipedia.org/wiki/Semilattice</a>
--   
--   <pre>
--   Associativity: x \/ (y \/ z) == (x \/ y) \/ z
--   Commutativity: x \/ y == y \/ x
--   Idempotency:   x \/ x == x
--   </pre>
class (Eq a) => JoinSemiLattice a
(\/) :: JoinSemiLattice a => a -> a -> a
infixr 5 \/

-- | The partial ordering induced by the join-semilattice structure
joinLeq :: JoinSemiLattice a => a -> a -> Bool

-- | A algebraic structure with element meets:
--   <a>http://en.wikipedia.org/wiki/Semilattice</a>
--   
--   <pre>
--   Associativity: x /\ (y /\ z) == (x /\ y) /\ z
--   Commutativity: x /\ y == y /\ x
--   Idempotency:   x /\ x == x
--   </pre>
class (Eq a) => MeetSemiLattice a
(/\) :: MeetSemiLattice a => a -> a -> a
infixr 6 /\

-- | The partial ordering induced by the meet-semilattice structure
meetLeq :: MeetSemiLattice a => a -> a -> Bool

-- | The combination of two semi lattices makes a lattice if the absorption
--   law holds: see <a>http://en.wikipedia.org/wiki/Absorption_law</a> and
--   <a>http://en.wikipedia.org/wiki/Lattice_(order)</a>
--   
--   <pre>
--   Absorption: a \/ (a /\ b) == a /\ (a \/ b) == a
--   </pre>
class (JoinSemiLattice a, MeetSemiLattice a) => Lattice a

-- | A join-semilattice with an identity element <a>bottom</a> for
--   <a>\/</a>.
--   
--   <pre>
--   Identity: x \/ bottom == x
--   </pre>
class JoinSemiLattice a => BoundedJoinSemiLattice a
bottom :: BoundedJoinSemiLattice a => a

-- | A meet-semilattice with an identity element <a>top</a> for <a>/\</a>.
--   
--   <pre>
--   Identity: x /\ top == x
--   </pre>
class MeetSemiLattice a => BoundedMeetSemiLattice a
top :: BoundedMeetSemiLattice a => a

-- | Lattices with both bounds
class (JoinSemiLattice a, MeetSemiLattice a, BoundedJoinSemiLattice a, BoundedMeetSemiLattice a) => BoundedLattice a
instance (NumHask.Algebra.Abstract.Lattice.JoinSemiLattice a, NumHask.Algebra.Abstract.Lattice.MeetSemiLattice a, NumHask.Algebra.Abstract.Lattice.BoundedJoinSemiLattice a, NumHask.Algebra.Abstract.Lattice.BoundedMeetSemiLattice a) => NumHask.Algebra.Abstract.Lattice.BoundedLattice a
instance NumHask.Algebra.Abstract.Lattice.BoundedMeetSemiLattice GHC.Types.Float
instance NumHask.Algebra.Abstract.Lattice.BoundedMeetSemiLattice GHC.Types.Double
instance NumHask.Algebra.Abstract.Lattice.BoundedMeetSemiLattice GHC.Types.Int
instance NumHask.Algebra.Abstract.Lattice.BoundedMeetSemiLattice GHC.Types.Bool
instance NumHask.Algebra.Abstract.Lattice.BoundedMeetSemiLattice GHC.Int.Int8
instance NumHask.Algebra.Abstract.Lattice.BoundedMeetSemiLattice GHC.Int.Int16
instance NumHask.Algebra.Abstract.Lattice.BoundedMeetSemiLattice GHC.Int.Int32
instance NumHask.Algebra.Abstract.Lattice.BoundedMeetSemiLattice GHC.Int.Int64
instance NumHask.Algebra.Abstract.Lattice.BoundedMeetSemiLattice GHC.Types.Word
instance NumHask.Algebra.Abstract.Lattice.BoundedMeetSemiLattice GHC.Word.Word8
instance NumHask.Algebra.Abstract.Lattice.BoundedMeetSemiLattice GHC.Word.Word16
instance NumHask.Algebra.Abstract.Lattice.BoundedMeetSemiLattice GHC.Word.Word32
instance NumHask.Algebra.Abstract.Lattice.BoundedMeetSemiLattice GHC.Word.Word64
instance (GHC.Classes.Eq (a -> b), NumHask.Algebra.Abstract.Lattice.BoundedMeetSemiLattice b) => NumHask.Algebra.Abstract.Lattice.BoundedMeetSemiLattice (a -> b)
instance NumHask.Algebra.Abstract.Lattice.BoundedJoinSemiLattice GHC.Types.Float
instance NumHask.Algebra.Abstract.Lattice.BoundedJoinSemiLattice GHC.Types.Double
instance NumHask.Algebra.Abstract.Lattice.BoundedJoinSemiLattice GHC.Types.Int
instance NumHask.Algebra.Abstract.Lattice.BoundedJoinSemiLattice GHC.Types.Bool
instance NumHask.Algebra.Abstract.Lattice.BoundedJoinSemiLattice GHC.Natural.Natural
instance NumHask.Algebra.Abstract.Lattice.BoundedJoinSemiLattice GHC.Int.Int8
instance NumHask.Algebra.Abstract.Lattice.BoundedJoinSemiLattice GHC.Int.Int16
instance NumHask.Algebra.Abstract.Lattice.BoundedJoinSemiLattice GHC.Int.Int32
instance NumHask.Algebra.Abstract.Lattice.BoundedJoinSemiLattice GHC.Int.Int64
instance NumHask.Algebra.Abstract.Lattice.BoundedJoinSemiLattice GHC.Types.Word
instance NumHask.Algebra.Abstract.Lattice.BoundedJoinSemiLattice GHC.Word.Word8
instance NumHask.Algebra.Abstract.Lattice.BoundedJoinSemiLattice GHC.Word.Word16
instance NumHask.Algebra.Abstract.Lattice.BoundedJoinSemiLattice GHC.Word.Word32
instance NumHask.Algebra.Abstract.Lattice.BoundedJoinSemiLattice GHC.Word.Word64
instance (GHC.Classes.Eq (a -> b), NumHask.Algebra.Abstract.Lattice.BoundedJoinSemiLattice b) => NumHask.Algebra.Abstract.Lattice.BoundedJoinSemiLattice (a -> b)
instance (NumHask.Algebra.Abstract.Lattice.JoinSemiLattice a, NumHask.Algebra.Abstract.Lattice.MeetSemiLattice a) => NumHask.Algebra.Abstract.Lattice.Lattice a
instance NumHask.Algebra.Abstract.Lattice.MeetSemiLattice GHC.Types.Float
instance NumHask.Algebra.Abstract.Lattice.MeetSemiLattice GHC.Types.Double
instance NumHask.Algebra.Abstract.Lattice.MeetSemiLattice GHC.Types.Int
instance NumHask.Algebra.Abstract.Lattice.MeetSemiLattice GHC.Integer.Type.Integer
instance NumHask.Algebra.Abstract.Lattice.MeetSemiLattice GHC.Types.Bool
instance NumHask.Algebra.Abstract.Lattice.MeetSemiLattice GHC.Natural.Natural
instance NumHask.Algebra.Abstract.Lattice.MeetSemiLattice GHC.Int.Int8
instance NumHask.Algebra.Abstract.Lattice.MeetSemiLattice GHC.Int.Int16
instance NumHask.Algebra.Abstract.Lattice.MeetSemiLattice GHC.Int.Int32
instance NumHask.Algebra.Abstract.Lattice.MeetSemiLattice GHC.Int.Int64
instance NumHask.Algebra.Abstract.Lattice.MeetSemiLattice GHC.Types.Word
instance NumHask.Algebra.Abstract.Lattice.MeetSemiLattice GHC.Word.Word8
instance NumHask.Algebra.Abstract.Lattice.MeetSemiLattice GHC.Word.Word16
instance NumHask.Algebra.Abstract.Lattice.MeetSemiLattice GHC.Word.Word32
instance NumHask.Algebra.Abstract.Lattice.MeetSemiLattice GHC.Word.Word64
instance (GHC.Classes.Eq (a -> b), NumHask.Algebra.Abstract.Lattice.MeetSemiLattice b) => NumHask.Algebra.Abstract.Lattice.MeetSemiLattice (a -> b)
instance NumHask.Algebra.Abstract.Lattice.JoinSemiLattice GHC.Types.Float
instance NumHask.Algebra.Abstract.Lattice.JoinSemiLattice GHC.Types.Double
instance NumHask.Algebra.Abstract.Lattice.JoinSemiLattice GHC.Types.Int
instance NumHask.Algebra.Abstract.Lattice.JoinSemiLattice GHC.Integer.Type.Integer
instance NumHask.Algebra.Abstract.Lattice.JoinSemiLattice GHC.Types.Bool
instance NumHask.Algebra.Abstract.Lattice.JoinSemiLattice GHC.Natural.Natural
instance NumHask.Algebra.Abstract.Lattice.JoinSemiLattice GHC.Int.Int8
instance NumHask.Algebra.Abstract.Lattice.JoinSemiLattice GHC.Int.Int16
instance NumHask.Algebra.Abstract.Lattice.JoinSemiLattice GHC.Int.Int32
instance NumHask.Algebra.Abstract.Lattice.JoinSemiLattice GHC.Int.Int64
instance NumHask.Algebra.Abstract.Lattice.JoinSemiLattice GHC.Types.Word
instance NumHask.Algebra.Abstract.Lattice.JoinSemiLattice GHC.Word.Word8
instance NumHask.Algebra.Abstract.Lattice.JoinSemiLattice GHC.Word.Word16
instance NumHask.Algebra.Abstract.Lattice.JoinSemiLattice GHC.Word.Word32
instance NumHask.Algebra.Abstract.Lattice.JoinSemiLattice GHC.Word.Word64
instance (GHC.Classes.Eq (a -> b), NumHask.Algebra.Abstract.Lattice.JoinSemiLattice b) => NumHask.Algebra.Abstract.Lattice.JoinSemiLattice (a -> b)


-- | Metric classes
module NumHask.Analysis.Metric

-- | <a>signum</a> from base is not an operator replicated in numhask,
--   being such a very silly name, and preferred is the much more obvious
--   <a>sign</a>. Compare with <tt>Norm</tt> and <tt>Banach</tt> where
--   there is a change in codomain
--   
--   <pre>
--   abs a * sign a == a
--   </pre>
--   
--   Generalising this class tends towards size and direction (abs is the
--   size on the one-dim number line of a vector with its tail at zero, and
--   sign is the direction, right?).
class (Multiplicative a) => Signed a
sign :: Signed a => a -> a
abs :: Signed a => a -> a

-- | L1 and L2 norms are provided for potential speedups, as well as the
--   generalized p-norm.
--   
--   for p &gt;= 1
--   
--   <pre>
--   normLp p a &gt;= zero
--   normLp p zero == zero
--   </pre>
--   
--   Note that the Normed codomain can be different to the domain.
class (Additive a, Additive b) => Normed a b
normL1 :: Normed a b => a -> b
normL2 :: Normed a b => a -> b

-- | distance between numbers using L1, L2 or Lp-norms
--   
--   <pre>
--   distanceL2 a b &gt;= zero
--   distanceL2 a a == zero
--   \a b c -&gt; distanceL2 a c + distanceL2 b c - distanceL2 a b &gt;= zero &amp;&amp;
--             distanceL2 a b + distanceL2 b c - distanceL2 a c &gt;= zero &amp;&amp;
--             distanceL2 a b + distanceL2 a c - distanceL2 b c &gt;= zero &amp;&amp;
--   </pre>
class Metric a b
distanceL1 :: Metric a b => a -> a -> b
distanceL2 :: Metric a b => a -> a -> b
class (Eq a, Additive a, Subtractive a, MeetSemiLattice a) => Epsilon a
epsilon :: Epsilon a => a
nearZero :: Epsilon a => a -> Bool
aboutEqual :: Epsilon a => a -> a -> Bool
(~=) :: Epsilon a => a -> a -> Bool
infixl 4 ~=
instance NumHask.Analysis.Metric.Epsilon GHC.Types.Double
instance NumHask.Analysis.Metric.Epsilon GHC.Types.Float
instance NumHask.Analysis.Metric.Epsilon GHC.Types.Int
instance NumHask.Analysis.Metric.Epsilon GHC.Integer.Type.Integer
instance NumHask.Analysis.Metric.Epsilon GHC.Int.Int8
instance NumHask.Analysis.Metric.Epsilon GHC.Int.Int16
instance NumHask.Analysis.Metric.Epsilon GHC.Int.Int32
instance NumHask.Analysis.Metric.Epsilon GHC.Int.Int64
instance NumHask.Analysis.Metric.Epsilon GHC.Types.Word
instance NumHask.Analysis.Metric.Epsilon GHC.Word.Word8
instance NumHask.Analysis.Metric.Epsilon GHC.Word.Word16
instance NumHask.Analysis.Metric.Epsilon GHC.Word.Word32
instance NumHask.Analysis.Metric.Epsilon GHC.Word.Word64
instance NumHask.Analysis.Metric.Metric GHC.Types.Double GHC.Types.Double
instance NumHask.Analysis.Metric.Metric GHC.Types.Float GHC.Types.Float
instance NumHask.Analysis.Metric.Metric GHC.Types.Int GHC.Types.Int
instance NumHask.Analysis.Metric.Metric GHC.Integer.Type.Integer GHC.Integer.Type.Integer
instance NumHask.Analysis.Metric.Metric GHC.Natural.Natural GHC.Natural.Natural
instance NumHask.Analysis.Metric.Metric GHC.Int.Int8 GHC.Int.Int8
instance NumHask.Analysis.Metric.Metric GHC.Int.Int16 GHC.Int.Int16
instance NumHask.Analysis.Metric.Metric GHC.Int.Int32 GHC.Int.Int32
instance NumHask.Analysis.Metric.Metric GHC.Int.Int64 GHC.Int.Int64
instance NumHask.Analysis.Metric.Metric GHC.Types.Word GHC.Types.Word
instance NumHask.Analysis.Metric.Metric GHC.Word.Word8 GHC.Word.Word8
instance NumHask.Analysis.Metric.Metric GHC.Word.Word16 GHC.Word.Word16
instance NumHask.Analysis.Metric.Metric GHC.Word.Word32 GHC.Word.Word32
instance NumHask.Analysis.Metric.Metric GHC.Word.Word64 GHC.Word.Word64
instance NumHask.Analysis.Metric.Normed GHC.Types.Double GHC.Types.Double
instance NumHask.Analysis.Metric.Normed GHC.Types.Float GHC.Types.Float
instance NumHask.Analysis.Metric.Normed GHC.Types.Int GHC.Types.Int
instance NumHask.Analysis.Metric.Normed GHC.Integer.Type.Integer GHC.Integer.Type.Integer
instance NumHask.Analysis.Metric.Normed GHC.Natural.Natural GHC.Natural.Natural
instance NumHask.Analysis.Metric.Normed GHC.Int.Int8 GHC.Int.Int8
instance NumHask.Analysis.Metric.Normed GHC.Int.Int16 GHC.Int.Int16
instance NumHask.Analysis.Metric.Normed GHC.Int.Int32 GHC.Int.Int32
instance NumHask.Analysis.Metric.Normed GHC.Int.Int64 GHC.Int.Int64
instance NumHask.Analysis.Metric.Normed GHC.Types.Word GHC.Types.Word
instance NumHask.Analysis.Metric.Normed GHC.Word.Word8 GHC.Word.Word8
instance NumHask.Analysis.Metric.Normed GHC.Word.Word16 GHC.Word.Word16
instance NumHask.Analysis.Metric.Normed GHC.Word.Word32 GHC.Word.Word32
instance NumHask.Analysis.Metric.Normed GHC.Word.Word64 GHC.Word.Word64
instance NumHask.Analysis.Metric.Signed GHC.Types.Double
instance NumHask.Analysis.Metric.Signed GHC.Types.Float
instance NumHask.Analysis.Metric.Signed GHC.Types.Int
instance NumHask.Analysis.Metric.Signed GHC.Integer.Type.Integer
instance NumHask.Analysis.Metric.Signed GHC.Natural.Natural
instance NumHask.Analysis.Metric.Signed GHC.Int.Int8
instance NumHask.Analysis.Metric.Signed GHC.Int.Int16
instance NumHask.Analysis.Metric.Signed GHC.Int.Int32
instance NumHask.Analysis.Metric.Signed GHC.Int.Int64
instance NumHask.Analysis.Metric.Signed GHC.Types.Word
instance NumHask.Analysis.Metric.Signed GHC.Word.Word8
instance NumHask.Analysis.Metric.Signed GHC.Word.Word16
instance NumHask.Analysis.Metric.Signed GHC.Word.Word32
instance NumHask.Analysis.Metric.Signed GHC.Word.Word64

module NumHask.Data.Complex

-- | Complex numbers are an algebraic type.
--   
--   For a complex number <tt>z</tt>, <tt><a>abs</a> z</tt> is a number
--   with the magnitude of <tt>z</tt>, but oriented in the positive real
--   direction, whereas <tt><a>sign</a> z</tt> has the phase of <tt>z</tt>,
--   but unit magnitude.
--   
--   The <a>Foldable</a> and <a>Traversable</a> instances traverse the real
--   part first.
data Complex a

-- | forms a complex number from its real and imaginary rectangular
--   components.
(:+) :: !a -> !a -> Complex a
infix 6 :+

-- | Extracts the real part of a complex number.
realPart :: Complex a -> a

-- | Extracts the imaginary part of a complex number.
imagPart :: Complex a -> a
mkPolar :: TrigField a => a -> a -> Complex a

-- | <tt><a>cis</a> t</tt> is a complex value with magnitude <tt>1</tt> and
--   phase <tt>t</tt> (modulo <tt>2*<a>pi</a></tt>).
cis :: TrigField a => a -> Complex a

-- | The function <a>polar</a> takes a complex number and returns a
--   (magnitude, phase) pair in canonical form: the magnitude is
--   nonnegative, and the phase in the range <tt>(-<a>pi</a>,
--   <a>pi</a>]</tt>; if the magnitude is zero, then so is the phase.
polar :: (RealFloat a, ExpField a) => Complex a -> (a, a)

-- | The nonnegative magnitude of a complex number.
magnitude :: (ExpField a, RealFloat a) => Complex a -> a

-- | The phase of a complex number, in the range <tt>(-<a>pi</a>,
--   <a>pi</a>]</tt>. If the magnitude is zero, then so is the phase.
phase :: RealFloat a => Complex a -> a
instance Data.Traversable.Traversable NumHask.Data.Complex.Complex
instance Data.Foldable.Foldable NumHask.Data.Complex.Complex
instance GHC.Base.Functor NumHask.Data.Complex.Complex
instance GHC.Generics.Generic1 NumHask.Data.Complex.Complex
instance GHC.Generics.Generic (NumHask.Data.Complex.Complex a)
instance Data.Data.Data a => Data.Data.Data (NumHask.Data.Complex.Complex a)
instance GHC.Read.Read a => GHC.Read.Read (NumHask.Data.Complex.Complex a)
instance GHC.Show.Show a => GHC.Show.Show (NumHask.Data.Complex.Complex a)
instance GHC.Classes.Eq a => GHC.Classes.Eq (NumHask.Data.Complex.Complex a)
instance NumHask.Algebra.Abstract.Additive.Additive a => NumHask.Algebra.Abstract.Additive.Additive (NumHask.Data.Complex.Complex a)
instance NumHask.Algebra.Abstract.Additive.Subtractive a => NumHask.Algebra.Abstract.Additive.Subtractive (NumHask.Data.Complex.Complex a)
instance (NumHask.Algebra.Abstract.Ring.Distributive a, NumHask.Algebra.Abstract.Additive.Subtractive a) => NumHask.Algebra.Abstract.Ring.Distributive (NumHask.Data.Complex.Complex a)
instance (NumHask.Algebra.Abstract.Additive.Subtractive a, NumHask.Algebra.Abstract.Multiplicative.Multiplicative a) => NumHask.Algebra.Abstract.Multiplicative.Multiplicative (NumHask.Data.Complex.Complex a)
instance (NumHask.Algebra.Abstract.Additive.Subtractive a, NumHask.Algebra.Abstract.Multiplicative.Divisive a) => NumHask.Algebra.Abstract.Multiplicative.Divisive (NumHask.Data.Complex.Complex a)
instance (NumHask.Algebra.Abstract.Additive.Additive a, NumHask.Data.Integral.FromIntegral a b) => NumHask.Data.Integral.FromIntegral (NumHask.Data.Complex.Complex a) b
instance (NumHask.Algebra.Abstract.Multiplicative.Multiplicative a, NumHask.Algebra.Abstract.Field.ExpField a, NumHask.Analysis.Metric.Normed a a) => NumHask.Analysis.Metric.Normed (NumHask.Data.Complex.Complex a) a
instance (NumHask.Algebra.Abstract.Multiplicative.Multiplicative a, NumHask.Algebra.Abstract.Additive.Subtractive a, NumHask.Algebra.Abstract.Field.ExpField a, NumHask.Analysis.Metric.Normed a a) => NumHask.Analysis.Metric.Metric (NumHask.Data.Complex.Complex a) a
instance (GHC.Classes.Ord a, NumHask.Analysis.Metric.Signed a, NumHask.Algebra.Abstract.Additive.Subtractive a, NumHask.Analysis.Metric.Epsilon a) => NumHask.Analysis.Metric.Epsilon (NumHask.Data.Complex.Complex a)
instance (NumHask.Algebra.Abstract.Ring.IntegralDomain a, NumHask.Algebra.Abstract.Additive.Subtractive a) => NumHask.Algebra.Abstract.Ring.IntegralDomain (NumHask.Data.Complex.Complex a)
instance (NumHask.Algebra.Abstract.Field.Field a, NumHask.Algebra.Abstract.Additive.Subtractive a) => NumHask.Algebra.Abstract.Field.Field (NumHask.Data.Complex.Complex a)
instance (GHC.Classes.Ord a, NumHask.Algebra.Abstract.Field.TrigField a, NumHask.Algebra.Abstract.Field.ExpField a, NumHask.Algebra.Abstract.Additive.Subtractive a) => NumHask.Algebra.Abstract.Field.ExpField (NumHask.Data.Complex.Complex a)
instance (NumHask.Algebra.Abstract.Ring.Distributive a, NumHask.Algebra.Abstract.Additive.Subtractive a) => NumHask.Algebra.Abstract.Ring.InvolutiveRing (NumHask.Data.Complex.Complex a)
instance (NumHask.Algebra.Abstract.Field.UpperBoundedField a, NumHask.Algebra.Abstract.Ring.IntegralDomain a, NumHask.Algebra.Abstract.Additive.Subtractive a) => NumHask.Algebra.Abstract.Field.UpperBoundedField (NumHask.Data.Complex.Complex a)
instance NumHask.Algebra.Abstract.Field.LowerBoundedField a => NumHask.Algebra.Abstract.Field.LowerBoundedField (NumHask.Data.Complex.Complex a)
instance NumHask.Algebra.Abstract.Lattice.JoinSemiLattice a => NumHask.Algebra.Abstract.Lattice.JoinSemiLattice (NumHask.Data.Complex.Complex a)
instance NumHask.Algebra.Abstract.Lattice.MeetSemiLattice a => NumHask.Algebra.Abstract.Lattice.MeetSemiLattice (NumHask.Data.Complex.Complex a)
instance NumHask.Algebra.Abstract.Lattice.BoundedJoinSemiLattice a => NumHask.Algebra.Abstract.Lattice.BoundedJoinSemiLattice (NumHask.Data.Complex.Complex a)
instance NumHask.Algebra.Abstract.Lattice.BoundedMeetSemiLattice a => NumHask.Algebra.Abstract.Lattice.BoundedMeetSemiLattice (NumHask.Data.Complex.Complex a)


-- | Metric classes
module NumHask.Analysis.Banach

-- | Banach (with Norm) laws form rules around size and direction of a
--   number, with a potential crossing into another codomain.
--   
--   <pre>
--   a == singleton zero || normalizeL2 a *. normL2 a == a
--   </pre>
class (ExpField (Actor h), Normed h (Actor h), DivisiveAction h) => Banach h
normalizeL1 :: Banach h => h -> h
normalizeL2 :: Banach h => h -> h

-- | the inner product: a distributive fold
--   
--   <pre>
--   a &lt;.&gt; b == b &lt;.&gt; a
--   a &lt;.&gt; (b +c) == a &lt;.&gt; b + a &lt;.&gt; c
--   a &lt;.&gt; (s *. b + c) == s * (a &lt;.&gt; b) + a &lt;.&gt; c
--   </pre>
--   
--   (s0 *. a) <a>.</a> (s1 *. b) == s0 * s1 * (a <a>.</a> b)
class (Distributive (Actor h)) => Hilbert h
(<.>) :: Hilbert h => h -> h -> Actor h
infix 8 <.>


-- | The abstract algebraic class structure of a number.
module NumHask.Algebra.Abstract


-- | Integral classes
module NumHask.Data.Rational
data Ratio a
(:%) :: !a -> !a -> Ratio a

-- | Arbitrary-precision rational numbers, represented as a ratio of two
--   <a>Integer</a> values. A rational number may be constructed using the
--   <a>%</a> operator.
type Rational = Ratio Integer

-- | toRatio is equivalent to <tt>Real</tt> in base, but is polymorphic in
--   the Integral type.
class ToRatio a b
toRatio :: ToRatio a b => a -> Ratio b
toRatio :: (ToRatio a b, Ratio c ~ a, ToIntegral c Integer, ToRatio (Ratio b) b, FromInteger b) => a -> Ratio b
type ToRational a = ToRatio a Integer
toRational :: ToRatio a Integer => a -> Ratio Integer

-- | <tt>Fractional</tt> in base splits into fromRatio and Field FIXME:
--   work out why the default type isn't firing so that an explicit
--   instance is needed for `FromRatio (Ratio Integer) Integer`
class FromRatio a b
fromRatio :: FromRatio a b => Ratio b -> a
fromRatio :: (FromRatio a b, Ratio b ~ a) => Ratio b -> a

-- | with RebindableSyntax the literal '1.0' mean exactly `fromRational
--   (1.0::GHC.Real.Rational)`.
class FromRational a
fromRational :: FromRational a => Rational -> a

-- | Given that fromRational is reserved, fromRational' provides general
--   conversion between numhask rationals.
fromRational' :: (FromRatio b Integer, ToRatio a Integer) => a -> b
fromBaseRational :: Rational -> Ratio Integer

-- | <a>reduce</a> is a subsidiary function used only in this module. It
--   normalises a ratio by dividing both numerator and denominator by their
--   greatest common divisor.
reduce :: (Eq a, Subtractive a, Signed a, Integral a) => a -> a -> Ratio a

-- | <tt><a>gcd</a> x y</tt> is the non-negative factor of both <tt>x</tt>
--   and <tt>y</tt> of which every common factor of <tt>x</tt> and
--   <tt>y</tt> is also a factor; for example <tt><a>gcd</a> 4 2 = 2</tt>,
--   <tt><a>gcd</a> (-4) 6 = 2</tt>, <tt><a>gcd</a> 0 4</tt> = <tt>4</tt>.
--   <tt><a>gcd</a> 0 0</tt> = <tt>0</tt>. (That is, the common divisor
--   that is "greatest" in the divisibility preordering.)
--   
--   Note: Since for signed fixed-width integer types, <tt><a>abs</a>
--   <tt>minBound</tt> &lt; 0</tt>, the result may be negative if one of
--   the arguments is <tt><tt>minBound</tt></tt> (and necessarily is if the
--   other is <tt>0</tt> or <tt><tt>minBound</tt></tt>) for such types.
gcd :: (Eq a, Signed a, Integral a) => a -> a -> a
instance GHC.Show.Show a => GHC.Show.Show (NumHask.Data.Rational.Ratio a)
instance NumHask.Data.Rational.FromRational GHC.Types.Double
instance NumHask.Data.Rational.FromRational GHC.Types.Float
instance NumHask.Data.Rational.FromRational GHC.Real.Rational
instance NumHask.Data.Rational.FromRatio GHC.Types.Double GHC.Integer.Type.Integer
instance NumHask.Data.Rational.FromRatio GHC.Types.Float GHC.Integer.Type.Integer
instance NumHask.Data.Rational.FromRatio GHC.Real.Rational GHC.Integer.Type.Integer
instance NumHask.Data.Rational.FromRatio (NumHask.Data.Rational.Ratio GHC.Integer.Type.Integer) GHC.Integer.Type.Integer
instance NumHask.Data.Rational.ToRatio GHC.Types.Double GHC.Integer.Type.Integer
instance NumHask.Data.Rational.ToRatio GHC.Types.Float GHC.Integer.Type.Integer
instance NumHask.Data.Rational.ToRatio GHC.Real.Rational GHC.Integer.Type.Integer
instance NumHask.Data.Rational.ToRatio (NumHask.Data.Rational.Ratio GHC.Integer.Type.Integer) GHC.Integer.Type.Integer
instance NumHask.Data.Rational.ToRatio GHC.Types.Int GHC.Integer.Type.Integer
instance NumHask.Data.Rational.ToRatio GHC.Integer.Type.Integer GHC.Integer.Type.Integer
instance NumHask.Data.Rational.ToRatio GHC.Natural.Natural GHC.Integer.Type.Integer
instance NumHask.Data.Rational.ToRatio GHC.Int.Int8 GHC.Integer.Type.Integer
instance NumHask.Data.Rational.ToRatio GHC.Int.Int16 GHC.Integer.Type.Integer
instance NumHask.Data.Rational.ToRatio GHC.Int.Int32 GHC.Integer.Type.Integer
instance NumHask.Data.Rational.ToRatio GHC.Int.Int64 GHC.Integer.Type.Integer
instance NumHask.Data.Rational.ToRatio GHC.Types.Word GHC.Integer.Type.Integer
instance NumHask.Data.Rational.ToRatio GHC.Word.Word8 GHC.Integer.Type.Integer
instance NumHask.Data.Rational.ToRatio GHC.Word.Word16 GHC.Integer.Type.Integer
instance NumHask.Data.Rational.ToRatio GHC.Word.Word32 GHC.Integer.Type.Integer
instance NumHask.Data.Rational.ToRatio GHC.Word.Word64 GHC.Integer.Type.Integer
instance NumHask.Data.Rational.GCDConstraints a => NumHask.Algebra.Abstract.Additive.Additive (NumHask.Data.Rational.Ratio a)
instance NumHask.Data.Rational.GCDConstraints a => NumHask.Algebra.Abstract.Additive.Subtractive (NumHask.Data.Rational.Ratio a)
instance NumHask.Data.Rational.GCDConstraints a => NumHask.Algebra.Abstract.Multiplicative.Multiplicative (NumHask.Data.Rational.Ratio a)
instance NumHask.Data.Rational.GCDConstraints a => NumHask.Algebra.Abstract.Multiplicative.Divisive (NumHask.Data.Rational.Ratio a)
instance NumHask.Data.Rational.GCDConstraints a => NumHask.Algebra.Abstract.Ring.Distributive (NumHask.Data.Rational.Ratio a)
instance NumHask.Data.Rational.GCDConstraints a => NumHask.Algebra.Abstract.Ring.IntegralDomain (NumHask.Data.Rational.Ratio a)
instance NumHask.Data.Rational.GCDConstraints a => NumHask.Algebra.Abstract.Field.Field (NumHask.Data.Rational.Ratio a)
instance (NumHask.Data.Rational.GCDConstraints a, NumHask.Data.Rational.GCDConstraints b, NumHask.Data.Integral.ToInteger a, NumHask.Algebra.Abstract.Field.Field a, NumHask.Data.Integral.FromIntegral b a) => NumHask.Algebra.Abstract.Field.QuotientField (NumHask.Data.Rational.Ratio a) b
instance (NumHask.Data.Rational.GCDConstraints a, NumHask.Algebra.Abstract.Ring.Distributive a, NumHask.Algebra.Abstract.Ring.IntegralDomain a) => NumHask.Algebra.Abstract.Field.UpperBoundedField (NumHask.Data.Rational.Ratio a)
instance (NumHask.Data.Rational.GCDConstraints a, NumHask.Algebra.Abstract.Field.Field a) => NumHask.Algebra.Abstract.Field.LowerBoundedField (NumHask.Data.Rational.Ratio a)
instance NumHask.Data.Rational.GCDConstraints a => NumHask.Analysis.Metric.Signed (NumHask.Data.Rational.Ratio a)
instance NumHask.Data.Rational.GCDConstraints a => NumHask.Analysis.Metric.Normed (NumHask.Data.Rational.Ratio a) (NumHask.Data.Rational.Ratio a)
instance NumHask.Data.Rational.GCDConstraints a => NumHask.Analysis.Metric.Metric (NumHask.Data.Rational.Ratio a) (NumHask.Data.Rational.Ratio a)
instance (NumHask.Data.Rational.GCDConstraints a, NumHask.Algebra.Abstract.Lattice.MeetSemiLattice a) => NumHask.Analysis.Metric.Epsilon (NumHask.Data.Rational.Ratio a)
instance NumHask.Data.Rational.GCDConstraints a => NumHask.Algebra.Abstract.Lattice.JoinSemiLattice (NumHask.Data.Rational.Ratio a)
instance NumHask.Data.Rational.GCDConstraints a => NumHask.Algebra.Abstract.Lattice.MeetSemiLattice (NumHask.Data.Rational.Ratio a)
instance (GHC.Classes.Eq a, NumHask.Algebra.Abstract.Additive.Additive a) => GHC.Classes.Eq (NumHask.Data.Rational.Ratio a)
instance (GHC.Classes.Ord a, NumHask.Algebra.Abstract.Multiplicative.Multiplicative a, NumHask.Algebra.Abstract.Additive.Additive a) => GHC.Classes.Ord (NumHask.Data.Rational.Ratio a)
instance (NumHask.Data.Integral.FromIntegral a b, NumHask.Algebra.Abstract.Multiplicative.Multiplicative a) => NumHask.Data.Integral.FromIntegral (NumHask.Data.Rational.Ratio a) b


-- | A Pair is *the* classical higher-kinded number but there is no canon.
module NumHask.Data.Pair

-- | A pair of a's, implemented as a tuple, but api represented as a Pair
--   of a's.
--   
--   <pre>
--   &gt;&gt;&gt; fmap (+1) (Pair 1 2)
--   Pair 2 3
--   
--   &gt;&gt;&gt; pure one :: Pair Int
--   Pair 1 1
--   
--   &gt;&gt;&gt; (*) &lt;$&gt; Pair 1 2 &lt;*&gt; pure 2
--   Pair 2 4
--   
--   &gt;&gt;&gt; foldr (++) [] (Pair [1,2] [3])
--   [1,2,3]
--   
--   &gt;&gt;&gt; Pair "a" "pair" `mappend` pure " " `mappend` Pair "string" "mappended"
--   Pair "a string" "pair mappended"
--   </pre>
--   
--   As a Ring and Field class
--   
--   <pre>
--   &gt;&gt;&gt; Pair 0 1 + zero
--   Pair 0 1
--   
--   &gt;&gt;&gt; Pair 0 1 + Pair 2 3
--   Pair 2 4
--   
--   &gt;&gt;&gt; Pair 1 1 - one
--   Pair 0 0
--   
--   &gt;&gt;&gt; Pair 0 1 * one
--   Pair 0 1
--   
--   &gt;&gt;&gt; Pair 0.0 1.0 / one
--   Pair 0.0 1.0
--   
--   &gt;&gt;&gt; Pair 11 12 `mod` (pure 6)
--   Pair 5 0
--   </pre>
--   
--   As an action
--   
--   <pre>
--   &gt;&gt;&gt; Pair 1 2 .+ 3
--   Pair 4 5
--   </pre>
newtype Pair a
Pair' :: (a, a) -> Pair a

-- | the preferred pattern
pattern Pair :: a -> a -> Pair a
instance GHC.Generics.Generic (NumHask.Data.Pair.Pair a)
instance GHC.Classes.Eq a => GHC.Classes.Eq (NumHask.Data.Pair.Pair a)
instance GHC.Show.Show a => GHC.Show.Show (NumHask.Data.Pair.Pair a)
instance GHC.Base.Functor NumHask.Data.Pair.Pair
instance Data.Functor.Classes.Eq1 NumHask.Data.Pair.Pair
instance Data.Functor.Classes.Show1 NumHask.Data.Pair.Pair
instance GHC.Base.Applicative NumHask.Data.Pair.Pair
instance GHC.Base.Monad NumHask.Data.Pair.Pair
instance Data.Foldable.Foldable NumHask.Data.Pair.Pair
instance Data.Traversable.Traversable NumHask.Data.Pair.Pair
instance GHC.Base.Semigroup a => GHC.Base.Semigroup (NumHask.Data.Pair.Pair a)
instance (GHC.Base.Semigroup a, GHC.Base.Monoid a) => GHC.Base.Monoid (NumHask.Data.Pair.Pair a)
instance GHC.Enum.Bounded a => GHC.Enum.Bounded (NumHask.Data.Pair.Pair a)
instance NumHask.Algebra.Abstract.Additive.Additive a => NumHask.Algebra.Abstract.Additive.Additive (NumHask.Data.Pair.Pair a)
instance NumHask.Algebra.Abstract.Additive.Subtractive a => NumHask.Algebra.Abstract.Additive.Subtractive (NumHask.Data.Pair.Pair a)
instance NumHask.Algebra.Abstract.Multiplicative.Multiplicative a => NumHask.Algebra.Abstract.Multiplicative.Multiplicative (NumHask.Data.Pair.Pair a)
instance NumHask.Algebra.Abstract.Multiplicative.Divisive a => NumHask.Algebra.Abstract.Multiplicative.Divisive (NumHask.Data.Pair.Pair a)
instance NumHask.Data.Integral.Integral a => NumHask.Data.Integral.Integral (NumHask.Data.Pair.Pair a)
instance NumHask.Analysis.Metric.Signed a => NumHask.Analysis.Metric.Signed (NumHask.Data.Pair.Pair a)
instance (NumHask.Algebra.Abstract.Field.ExpField a, NumHask.Analysis.Metric.Normed a a) => NumHask.Analysis.Metric.Normed (NumHask.Data.Pair.Pair a) a
instance (NumHask.Algebra.Abstract.Additive.Subtractive a, NumHask.Analysis.Metric.Epsilon a) => NumHask.Analysis.Metric.Epsilon (NumHask.Data.Pair.Pair a)
instance (NumHask.Algebra.Abstract.Field.ExpField a, NumHask.Algebra.Abstract.Additive.Subtractive a, NumHask.Analysis.Metric.Normed a a) => NumHask.Analysis.Metric.Metric (NumHask.Data.Pair.Pair a) a
instance NumHask.Algebra.Abstract.Ring.Distributive a => NumHask.Algebra.Abstract.Ring.Distributive (NumHask.Data.Pair.Pair a)
instance NumHask.Algebra.Abstract.Field.Field a => NumHask.Algebra.Abstract.Field.Field (NumHask.Data.Pair.Pair a)
instance NumHask.Algebra.Abstract.Ring.IntegralDomain a => NumHask.Algebra.Abstract.Ring.IntegralDomain (NumHask.Data.Pair.Pair a)
instance NumHask.Algebra.Abstract.Field.ExpField a => NumHask.Algebra.Abstract.Field.ExpField (NumHask.Data.Pair.Pair a)
instance NumHask.Algebra.Abstract.Field.UpperBoundedField a => NumHask.Algebra.Abstract.Field.UpperBoundedField (NumHask.Data.Pair.Pair a)
instance NumHask.Algebra.Abstract.Field.LowerBoundedField a => NumHask.Algebra.Abstract.Field.LowerBoundedField (NumHask.Data.Pair.Pair a)
instance NumHask.Algebra.Abstract.Additive.Additive a => NumHask.Algebra.Abstract.Action.AdditiveAction (NumHask.Data.Pair.Pair a)
instance NumHask.Algebra.Abstract.Additive.Subtractive a => NumHask.Algebra.Abstract.Action.SubtractiveAction (NumHask.Data.Pair.Pair a)
instance NumHask.Algebra.Abstract.Multiplicative.Multiplicative a => NumHask.Algebra.Abstract.Action.MultiplicativeAction (NumHask.Data.Pair.Pair a)
instance NumHask.Algebra.Abstract.Multiplicative.Divisive a => NumHask.Algebra.Abstract.Action.DivisiveAction (NumHask.Data.Pair.Pair a)
instance NumHask.Algebra.Abstract.Lattice.JoinSemiLattice a => NumHask.Algebra.Abstract.Lattice.JoinSemiLattice (NumHask.Data.Pair.Pair a)
instance NumHask.Algebra.Abstract.Lattice.MeetSemiLattice a => NumHask.Algebra.Abstract.Lattice.MeetSemiLattice (NumHask.Data.Pair.Pair a)
instance NumHask.Algebra.Abstract.Lattice.BoundedJoinSemiLattice a => NumHask.Algebra.Abstract.Lattice.BoundedJoinSemiLattice (NumHask.Data.Pair.Pair a)
instance NumHask.Algebra.Abstract.Lattice.BoundedMeetSemiLattice a => NumHask.Algebra.Abstract.Lattice.BoundedMeetSemiLattice (NumHask.Data.Pair.Pair a)
instance NumHask.Data.Integral.FromIntegral a b => NumHask.Data.Integral.FromIntegral (NumHask.Data.Pair.Pair a) b
instance NumHask.Data.Rational.FromRatio a b => NumHask.Data.Rational.FromRatio (NumHask.Data.Pair.Pair a) b
instance NumHask.Analysis.Metric.Normed a a => NumHask.Analysis.Metric.Normed (NumHask.Data.Pair.Pair a) (NumHask.Data.Pair.Pair a)
instance (NumHask.Algebra.Abstract.Additive.Subtractive a, NumHask.Analysis.Metric.Normed a a) => NumHask.Analysis.Metric.Metric (NumHask.Data.Pair.Pair a) (NumHask.Data.Pair.Pair a)

module NumHask.Data.LogField

-- | Module : Data.Number.LogFloat Copyright : Copyright (c) 2007--2015
--   wren gayle romano License : BSD3 Maintainer :
--   wren@community.haskell.org Stability : stable Portability : portable
--   (with CPP, FFI) Link :
--   <a>https://hackage.haskell.org/package/logfloat</a>
--   
--   A <tt>LogField</tt> is just a <a>Field</a> with a special
--   interpretation. The <a>LogField</a> function is presented instead of
--   the constructor, in order to ensure semantic conversion. At present
--   the <a>Show</a> instance will convert back to the normal-domain, and
--   hence will underflow at that point. This behavior may change in the
--   future.
--   
--   Because <a>logField</a> performs the semantic conversion, we can use
--   operators which say what we *mean* rather than saying what we're
--   actually doing to the underlying representation. That is, equivalences
--   like the following are true[1] thanks to type-class overloading:
--   
--   <pre>
--   logField (p + q) == logField p + logField q
--   logField (p * q) == logField p * logField q
--   </pre>
--   
--   Performing operations in the log-domain is cheap, prevents underflow,
--   and is otherwise very nice for dealing with miniscule probabilities.
--   However, crossing into and out of the log-domain is expensive and
--   should be avoided as much as possible. In particular, if you're doing
--   a series of multiplications as in <tt>lp * LogField q * LogField
--   r</tt> it's faster to do <tt>lp * LogField (q * r)</tt> if you're
--   reasonably sure the normal-domain multiplication won't underflow;
--   because that way you enter the log-domain only once, instead of twice.
--   Also note that, for precision, if you're doing more than a few
--   multiplications in the log-domain, you should use <a>product</a>
--   rather than using '(*)' repeatedly.
--   
--   Even more particularly, you should <i>avoid addition</i> whenever
--   possible. Addition is provided because sometimes we need it, and the
--   proper implementation is not immediately apparent. However, between
--   two <tt>LogField</tt>s addition requires crossing the exp/log boundary
--   twice; with a <tt>LogField</tt> and a <a>Double</a> it's three times,
--   since the regular number needs to enter the log-domain first. This
--   makes addition incredibly slow. Again, if you can parenthesize to do
--   normal-domain operations first, do it!
--   
--   <ul>
--   <li><i>1</i> That is, true up-to underflow and floating point
--   fuzziness. Which is, of course, the whole point of this module.</li>
--   </ul>
data LogField a

-- | Constructor which does semantic conversion from normal-domain to
--   log-domain. Throws errors on negative and NaN inputs. If <tt>p</tt> is
--   non-negative, then following equivalence holds:
--   
--   <pre>
--   logField p == logToLogField (log p)
--   </pre>
logField :: ExpField a => a -> LogField a

-- | Semantically convert our log-domain value back into the normal-domain.
--   Beware of overflow/underflow. The following equivalence holds (without
--   qualification):
--   
--   <pre>
--   fromLogField == exp . logFromLogField
--   </pre>
fromLogField :: ExpField a => LogField a -> a

-- | Constructor which assumes the argument is already in the log-domain.
logToLogField :: a -> LogField a

-- | Return the log-domain value itself without conversion.
logFromLogField :: LogField a -> a

-- | <i>O(n)</i>. Compute the sum of a finite list of <a>LogField</a>s,
--   being careful to avoid underflow issues. That is, the following
--   equivalence holds (modulo underflow and all that):
--   
--   <pre>
--   LogField . accurateSum == accurateSum . map LogField
--   </pre>
--   
--   <i>N.B.</i>, this function requires two passes over the input. Thus,
--   it is not amenable to list fusion, and hence will use a lot of memory
--   when summing long lists.
accurateSum :: (ExpField a, Subtractive a, Foldable f, Ord a) => f (LogField a) -> LogField a

-- | <i>O(n)</i>. Compute the product of a finite list of <a>LogField</a>s,
--   being careful to avoid numerical error due to loss of precision. That
--   is, the following equivalence holds (modulo underflow and all that):
--   
--   <pre>
--   LogField . accurateProduct == accurateProduct . map LogField
--   </pre>
accurateProduct :: (ExpField a, Subtractive a, Foldable f) => f (LogField a) -> LogField a

-- | <i>O(1)</i>. Compute powers in the log-domain; that is, the following
--   equivalence holds (modulo underflow and all that):
--   
--   <pre>
--   LogField (p ** m) == LogField p `pow` m
--   </pre>
--   
--   <i>Since: 0.13</i>
pow :: (ExpField a, LowerBoundedField a, Ord a) => LogField a -> a -> LogField a
infixr 8 `pow`
instance Data.Traversable.Traversable NumHask.Data.LogField.LogField
instance Data.Foldable.Foldable NumHask.Data.LogField.LogField
instance GHC.Base.Functor NumHask.Data.LogField.LogField
instance GHC.Generics.Generic1 NumHask.Data.LogField.LogField
instance GHC.Generics.Generic (NumHask.Data.LogField.LogField a)
instance Data.Data.Data a => Data.Data.Data (NumHask.Data.LogField.LogField a)
instance GHC.Read.Read a => GHC.Read.Read (NumHask.Data.LogField.LogField a)
instance GHC.Classes.Ord a => GHC.Classes.Ord (NumHask.Data.LogField.LogField a)
instance GHC.Classes.Eq a => GHC.Classes.Eq (NumHask.Data.LogField.LogField a)
instance (NumHask.Algebra.Abstract.Field.ExpField a, GHC.Show.Show a) => GHC.Show.Show (NumHask.Data.LogField.LogField a)
instance (NumHask.Algebra.Abstract.Field.ExpField a, NumHask.Algebra.Abstract.Field.LowerBoundedField a, GHC.Classes.Ord a) => NumHask.Algebra.Abstract.Additive.Additive (NumHask.Data.LogField.LogField a)
instance (NumHask.Algebra.Abstract.Field.ExpField a, GHC.Classes.Ord a, NumHask.Algebra.Abstract.Field.LowerBoundedField a, NumHask.Algebra.Abstract.Field.UpperBoundedField a) => NumHask.Algebra.Abstract.Additive.Subtractive (NumHask.Data.LogField.LogField a)
instance (NumHask.Algebra.Abstract.Field.LowerBoundedField a, GHC.Classes.Eq a) => NumHask.Algebra.Abstract.Multiplicative.Multiplicative (NumHask.Data.LogField.LogField a)
instance (NumHask.Algebra.Abstract.Field.LowerBoundedField a, GHC.Classes.Eq a) => NumHask.Algebra.Abstract.Multiplicative.Divisive (NumHask.Data.LogField.LogField a)
instance (GHC.Classes.Ord a, NumHask.Algebra.Abstract.Field.LowerBoundedField a, NumHask.Algebra.Abstract.Field.ExpField a) => NumHask.Algebra.Abstract.Ring.Distributive (NumHask.Data.LogField.LogField a)
instance (NumHask.Algebra.Abstract.Field.Field (NumHask.Data.LogField.LogField a), NumHask.Algebra.Abstract.Field.ExpField a, NumHask.Algebra.Abstract.Field.LowerBoundedField a, GHC.Classes.Ord a) => NumHask.Algebra.Abstract.Field.ExpField (NumHask.Data.LogField.LogField a)
instance (NumHask.Data.Integral.FromIntegral a b, NumHask.Algebra.Abstract.Field.ExpField a) => NumHask.Data.Integral.FromIntegral (NumHask.Data.LogField.LogField a) b
instance (NumHask.Data.Integral.ToIntegral a b, NumHask.Algebra.Abstract.Field.ExpField a) => NumHask.Data.Integral.ToIntegral (NumHask.Data.LogField.LogField a) b
instance (NumHask.Data.Rational.FromRatio a b, NumHask.Algebra.Abstract.Field.ExpField a) => NumHask.Data.Rational.FromRatio (NumHask.Data.LogField.LogField a) b
instance (NumHask.Data.Rational.ToRatio a b, NumHask.Algebra.Abstract.Field.ExpField a) => NumHask.Data.Rational.ToRatio (NumHask.Data.LogField.LogField a) b
instance GHC.Classes.Ord a => NumHask.Algebra.Abstract.Lattice.JoinSemiLattice (NumHask.Data.LogField.LogField a)
instance GHC.Classes.Ord a => NumHask.Algebra.Abstract.Lattice.MeetSemiLattice (NumHask.Data.LogField.LogField a)
instance (NumHask.Analysis.Metric.Epsilon a, NumHask.Algebra.Abstract.Field.ExpField a, NumHask.Algebra.Abstract.Field.LowerBoundedField a, NumHask.Algebra.Abstract.Field.UpperBoundedField a, GHC.Classes.Ord a) => NumHask.Analysis.Metric.Epsilon (NumHask.Data.LogField.LogField a)
instance (GHC.Classes.Ord a, NumHask.Algebra.Abstract.Field.ExpField a, NumHask.Algebra.Abstract.Field.LowerBoundedField a) => NumHask.Algebra.Abstract.Field.Field (NumHask.Data.LogField.LogField a)
instance (GHC.Classes.Ord a, NumHask.Algebra.Abstract.Field.ExpField a, NumHask.Algebra.Abstract.Field.LowerBoundedField a, NumHask.Algebra.Abstract.Field.UpperBoundedField a) => NumHask.Algebra.Abstract.Field.LowerBoundedField (NumHask.Data.LogField.LogField a)
instance (GHC.Classes.Ord a, NumHask.Algebra.Abstract.Field.ExpField a, NumHask.Algebra.Abstract.Field.LowerBoundedField a) => NumHask.Algebra.Abstract.Ring.IntegralDomain (NumHask.Data.LogField.LogField a)
instance (GHC.Classes.Ord a, NumHask.Algebra.Abstract.Field.ExpField a, NumHask.Algebra.Abstract.Field.LowerBoundedField a, NumHask.Algebra.Abstract.Field.UpperBoundedField a) => NumHask.Algebra.Abstract.Field.UpperBoundedField (NumHask.Data.LogField.LogField a)
instance (GHC.Classes.Ord a, NumHask.Algebra.Abstract.Field.LowerBoundedField a, NumHask.Algebra.Abstract.Field.UpperBoundedField a, NumHask.Algebra.Abstract.Field.ExpField a) => NumHask.Analysis.Metric.Signed (NumHask.Data.LogField.LogField a)

module NumHask.Data.Wrapped
newtype Wrapped a
Wrapped :: a -> Wrapped a
[unWrapped] :: Wrapped a -> a
instance NumHask.Algebra.Abstract.Field.LowerBoundedField a => NumHask.Algebra.Abstract.Field.LowerBoundedField (NumHask.Data.Wrapped.Wrapped a)
instance NumHask.Algebra.Abstract.Field.UpperBoundedField a => NumHask.Algebra.Abstract.Field.UpperBoundedField (NumHask.Data.Wrapped.Wrapped a)
instance NumHask.Analysis.Metric.Epsilon a => NumHask.Analysis.Metric.Epsilon (NumHask.Data.Wrapped.Wrapped a)
instance NumHask.Algebra.Abstract.Lattice.JoinSemiLattice a => NumHask.Algebra.Abstract.Lattice.JoinSemiLattice (NumHask.Data.Wrapped.Wrapped a)
instance NumHask.Algebra.Abstract.Lattice.MeetSemiLattice a => NumHask.Algebra.Abstract.Lattice.MeetSemiLattice (NumHask.Data.Wrapped.Wrapped a)
instance NumHask.Analysis.Metric.Signed a => NumHask.Analysis.Metric.Signed (NumHask.Data.Wrapped.Wrapped a)
instance NumHask.Data.Integral.Integral a => NumHask.Data.Integral.Integral (NumHask.Data.Wrapped.Wrapped a)
instance NumHask.Algebra.Abstract.Field.TrigField a => NumHask.Algebra.Abstract.Field.TrigField (NumHask.Data.Wrapped.Wrapped a)
instance NumHask.Algebra.Abstract.Field.ExpField a => NumHask.Algebra.Abstract.Field.ExpField (NumHask.Data.Wrapped.Wrapped a)
instance (NumHask.Algebra.Abstract.Additive.Additive a, NumHask.Algebra.Abstract.Multiplicative.Divisive a) => NumHask.Algebra.Abstract.Field.Field (NumHask.Data.Wrapped.Wrapped a)
instance (NumHask.Algebra.Abstract.Ring.StarSemiring a, NumHask.Algebra.Abstract.Group.Magma a) => NumHask.Algebra.Abstract.Ring.KleeneAlgebra (NumHask.Data.Wrapped.Wrapped a)
instance NumHask.Algebra.Abstract.Ring.StarSemiring a => NumHask.Algebra.Abstract.Ring.StarSemiring (NumHask.Data.Wrapped.Wrapped a)
instance NumHask.Algebra.Abstract.Ring.InvolutiveRing a => NumHask.Algebra.Abstract.Ring.InvolutiveRing (NumHask.Data.Wrapped.Wrapped a)
instance (NumHask.Algebra.Abstract.Additive.Additive a, NumHask.Algebra.Abstract.Multiplicative.Divisive a) => NumHask.Algebra.Abstract.Ring.IntegralDomain (NumHask.Data.Wrapped.Wrapped a)
instance (NumHask.Algebra.Abstract.Additive.Additive a, NumHask.Algebra.Abstract.Multiplicative.Multiplicative a) => NumHask.Algebra.Abstract.Ring.Distributive (NumHask.Data.Wrapped.Wrapped a)
instance NumHask.Algebra.Abstract.Multiplicative.Divisive a => NumHask.Algebra.Abstract.Multiplicative.Divisive (NumHask.Data.Wrapped.Wrapped a)
instance NumHask.Algebra.Abstract.Multiplicative.Multiplicative a => NumHask.Algebra.Abstract.Multiplicative.Multiplicative (NumHask.Data.Wrapped.Wrapped a)
instance NumHask.Algebra.Abstract.Additive.Subtractive a => NumHask.Algebra.Abstract.Additive.Subtractive (NumHask.Data.Wrapped.Wrapped a)
instance NumHask.Algebra.Abstract.Additive.Additive a => NumHask.Algebra.Abstract.Additive.Additive (NumHask.Data.Wrapped.Wrapped a)
instance NumHask.Algebra.Abstract.Group.Magma a => NumHask.Algebra.Abstract.Group.Idempotent (NumHask.Data.Wrapped.Wrapped a)
instance NumHask.Algebra.Abstract.Group.Magma a => NumHask.Algebra.Abstract.Group.Magma (NumHask.Data.Wrapped.Wrapped a)
instance GHC.Classes.Ord a => GHC.Classes.Ord (NumHask.Data.Wrapped.Wrapped a)
instance GHC.Classes.Eq a => GHC.Classes.Eq (NumHask.Data.Wrapped.Wrapped a)
instance GHC.Show.Show a => GHC.Show.Show (NumHask.Data.Wrapped.Wrapped a)
instance (GHC.Classes.Ord a, NumHask.Algebra.Abstract.Field.QuotientField a GHC.Integer.Type.Integer) => NumHask.Algebra.Abstract.Field.QuotientField (NumHask.Data.Wrapped.Wrapped a) (NumHask.Data.Wrapped.Wrapped GHC.Integer.Type.Integer)
instance NumHask.Data.Integral.FromIntegral a b => NumHask.Data.Integral.FromIntegral (NumHask.Data.Wrapped.Wrapped a) b
instance NumHask.Data.Integral.ToIntegral a b => NumHask.Data.Integral.ToIntegral (NumHask.Data.Wrapped.Wrapped a) b
instance NumHask.Data.Rational.FromRatio a b => NumHask.Data.Rational.FromRatio (NumHask.Data.Wrapped.Wrapped a) b
instance NumHask.Data.Rational.ToRatio a b => NumHask.Data.Rational.ToRatio (NumHask.Data.Wrapped.Wrapped a) b
instance NumHask.Analysis.Metric.Normed a b => NumHask.Analysis.Metric.Normed (NumHask.Data.Wrapped.Wrapped a) (NumHask.Data.Wrapped.Wrapped b)
instance NumHask.Analysis.Metric.Metric a b => NumHask.Analysis.Metric.Metric (NumHask.Data.Wrapped.Wrapped a) (NumHask.Data.Wrapped.Wrapped b)

module NumHask.Exception
newtype NumHaskException
NumHaskException :: String -> NumHaskException
[errorMessage] :: NumHaskException -> String

-- | Throw an exception. Exceptions may be thrown from purely functional
--   code, but may only be caught within the <a>IO</a> monad.
throw :: Exception e => e -> a
instance GHC.Show.Show NumHask.Exception.NumHaskException
instance GHC.Exception.Type.Exception NumHask.Exception.NumHaskException

module NumHask.Data.Positive
newtype Positive a
Positive :: a -> Positive a
[unPositive] :: Positive a -> a
positive :: (Ord a, Additive a) => a -> Maybe (Positive a)
positive_ :: (Ord a, Additive a) => a -> Positive a
instance (NumHask.Analysis.Metric.Epsilon a, GHC.Classes.Ord a) => NumHask.Analysis.Metric.Epsilon (NumHask.Data.Positive.Positive a)
instance NumHask.Algebra.Abstract.Lattice.MeetSemiLattice a => NumHask.Algebra.Abstract.Lattice.MeetSemiLattice (NumHask.Data.Positive.Positive a)
instance NumHask.Algebra.Abstract.Lattice.JoinSemiLattice a => NumHask.Algebra.Abstract.Lattice.JoinSemiLattice (NumHask.Data.Positive.Positive a)
instance NumHask.Analysis.Metric.Signed a => NumHask.Analysis.Metric.Signed (NumHask.Data.Positive.Positive a)
instance NumHask.Data.Integral.Integral a => NumHask.Data.Integral.Integral (NumHask.Data.Positive.Positive a)
instance NumHask.Algebra.Abstract.Field.TrigField a => NumHask.Algebra.Abstract.Field.TrigField (NumHask.Data.Positive.Positive a)
instance NumHask.Algebra.Abstract.Field.ExpField a => NumHask.Algebra.Abstract.Field.ExpField (NumHask.Data.Positive.Positive a)
instance (NumHask.Algebra.Abstract.Additive.Additive a, NumHask.Algebra.Abstract.Multiplicative.Divisive a) => NumHask.Algebra.Abstract.Field.Field (NumHask.Data.Positive.Positive a)
instance (NumHask.Algebra.Abstract.Additive.Additive a, NumHask.Algebra.Abstract.Multiplicative.Divisive a) => NumHask.Algebra.Abstract.Ring.IntegralDomain (NumHask.Data.Positive.Positive a)
instance (NumHask.Algebra.Abstract.Additive.Additive a, NumHask.Algebra.Abstract.Multiplicative.Multiplicative a) => NumHask.Algebra.Abstract.Ring.Distributive (NumHask.Data.Positive.Positive a)
instance NumHask.Algebra.Abstract.Multiplicative.Divisive a => NumHask.Algebra.Abstract.Multiplicative.Divisive (NumHask.Data.Positive.Positive a)
instance NumHask.Algebra.Abstract.Multiplicative.Multiplicative a => NumHask.Algebra.Abstract.Multiplicative.Multiplicative (NumHask.Data.Positive.Positive a)
instance NumHask.Algebra.Abstract.Additive.Additive a => NumHask.Algebra.Abstract.Additive.Additive (NumHask.Data.Positive.Positive a)
instance GHC.Classes.Ord a => GHC.Classes.Ord (NumHask.Data.Positive.Positive a)
instance GHC.Classes.Eq a => GHC.Classes.Eq (NumHask.Data.Positive.Positive a)
instance GHC.Show.Show a => GHC.Show.Show (NumHask.Data.Positive.Positive a)
instance (GHC.Classes.Ord a, NumHask.Algebra.Abstract.Additive.Subtractive a) => NumHask.Algebra.Abstract.Additive.Subtractive (NumHask.Data.Positive.Positive a)
instance (GHC.Classes.Ord a, NumHask.Algebra.Abstract.Field.QuotientField a GHC.Integer.Type.Integer) => NumHask.Algebra.Abstract.Field.QuotientField (NumHask.Data.Positive.Positive a) (NumHask.Data.Positive.Positive GHC.Integer.Type.Integer)
instance NumHask.Algebra.Abstract.Field.UpperBoundedField a => NumHask.Algebra.Abstract.Field.UpperBoundedField (NumHask.Data.Positive.Positive a)
instance NumHask.Algebra.Abstract.Field.UpperBoundedField a => GHC.Enum.Bounded (NumHask.Data.Positive.Positive a)
instance NumHask.Analysis.Metric.Normed a a => NumHask.Analysis.Metric.Normed a (NumHask.Data.Positive.Positive a)
instance (NumHask.Algebra.Abstract.Additive.Subtractive a, NumHask.Analysis.Metric.Normed a a) => NumHask.Analysis.Metric.Metric a (NumHask.Data.Positive.Positive a)