Safe Haskell	Safe-Inferred
Language	Haskell2010

Algebra.Classes

Synopsis

type Natural = Integer
timesDefault :: (Additive a1, Additive a2, Integral a1) => a1 -> a2 -> a2
class Additive a where
- (+) :: a -> a -> a
- zero :: a
- times :: Natural -> a -> a
class Show a => TestEqual a where
- (=.=) :: a -> a -> Property
law_refl :: TestEqual a => a -> Property
laws_testEqual :: forall a. Arbitrary a => TestEqual a => Property
nameLaw :: Testable prop => String -> prop -> Property
law_assoc :: forall a. TestEqual a => String -> (a -> a -> a) -> a -> a -> a -> Property
law_left_id :: forall a. TestEqual a => String -> (a -> a -> a) -> a -> a -> Property
law_right_id :: forall a. TestEqual a => String -> (a -> a -> a) -> a -> a -> Property
laws_monoid :: forall a. (Arbitrary a, TestEqual a) => String -> (a -> a -> a) -> a -> Property
law_commutative :: TestEqual a => String -> (a -> a -> a) -> a -> a -> Property
laws_comm_monoid :: forall a. (Arbitrary a, TestEqual a) => String -> (a -> a -> a) -> a -> Property
law_times :: (TestEqual a, Additive a) => Positive Integer -> a -> Property
laws_additive :: forall a. Arbitrary a => (Additive a, TestEqual a) => Property
law_exp_pos :: (TestEqual a, Multiplicative a) => a -> Property
laws_multiplicative :: forall a. Arbitrary a => (Multiplicative a, TestEqual a) => Property
law_fromInteger :: forall a. (TestEqual a, Ring a) => Integer -> Property
laws_ring :: forall a. Arbitrary a => (Ring a, TestEqual a) => Property
sum :: (Foldable t, Additive a) => t a -> a
class Additive r => DecidableZero r where
- isZero :: r -> Bool
law_decidable_zero :: forall a. (DecidableZero a, TestEqual a) => Property
class Additive a => AbelianAdditive a
laws_abelian_additive :: forall a. (Arbitrary a, AbelianAdditive a, TestEqual a) => Property
multDefault :: Group a => Natural -> a -> a
class Additive a => Group a where
- (-) :: a -> a -> a
- subtract :: a -> a -> a
- negate :: a -> a
- mult :: Integer -> a -> a
law_negate_minus :: (TestEqual a, Group a) => a -> a -> Property
law_mult :: (TestEqual a, Group a) => Integer -> a -> Property
laws_group :: forall a. Arbitrary a => (Group a, TestEqual a) => Property
laws_abelian_group :: forall a. Arbitrary a => (Group a, TestEqual a) => Property
(*<) :: (Functor f, Multiplicative a) => a -> f a -> f a
class Scalable s a where
- (*^) :: s -> a -> a
class Scalable' a where
- type Scalar a
- (!*^) :: Scalar a -> a -> a
type SemiModule s a = (AbelianAdditive a, SemiRing s, Scalable s a)
type Module s a = (SemiModule s a, Group s, Group a)
law_module_zero :: forall s a. (Module s a, TestEqual a) => s -> Property
law_module_one :: forall s a. (Module s a, TestEqual a) => a -> Property
law_module_sum :: forall s a. (Module s a, TestEqual a) => s -> a -> a -> Property
law_module_sum_left :: forall s a. (Module s a, TestEqual a) => s -> s -> a -> Property
law_module_mul :: forall s a. (Module s a, TestEqual a) => s -> s -> a -> Property
laws_module :: forall s a. Arbitrary a => (Module s a, TestEqual a, Arbitrary s, Show s) => Property
class Multiplicative a where
- (*) :: a -> a -> a
- one :: a
- (^+) :: a -> Natural -> a
positiveExponentDefault :: Multiplicative a => a -> Natural -> a
product :: (Multiplicative a, Foldable f) => f a -> a
type SemiRing a = (Multiplicative a, AbelianAdditive a)
type PreRing a = (SemiRing a, Group a)
fromIntegerDefault :: PreRing a => Integer -> a
class (Module a a, PreRing a) => Ring a where
- fromInteger :: Integer -> a
class Multiplicative a => Division a where
- recip :: a -> a
- (/) :: a -> a -> a
- (^) :: a -> Integer -> a
class (Ring a, Division a) => Field a where
- fromRational :: Rational -> a
class (Ring a, DecidableZero a) => EuclideanDomain a where
- stdAssociate :: a -> a
- stdUnit :: a -> a
- normalize :: a -> (a, a)
- quot, rem :: a -> a -> a
- quotRem :: a -> a -> (a, a)
gcd :: EuclideanDomain a => a -> a -> a
lcm :: EuclideanDomain a => a -> a -> a
class (Ord a, Ring a, Enum a, EuclideanDomain a) => Integral a where
- div, mod :: a -> a -> a
- divMod :: a -> a -> (a, a)
- toInteger :: a -> Integer
ifThenElse :: Bool -> t -> t -> t
class Division a => Roots a where
- sqrt :: a -> a
- root :: Integer -> a -> a
- (^/) :: a -> Rational -> a
type Algebraic a = (Roots a, Field a)
class Algebraic a => Transcendental a where
- pi :: a
- exp, log :: a -> a
- (**), logBase :: a -> a -> a
- sin, cos, tan :: a -> a
- asin, acos, atan :: a -> a
- sinh, cosh, tanh :: a -> a
- asinh, acosh, atanh :: a -> a
- log1p :: a -> a
- expm1 :: a -> a
- log1pexp :: a -> a
- log1mexp :: a -> a
(^?) :: Transcendental a => a -> a -> a
class Algebraic a => AlgebraicallyClosed a where
- imaginaryUnit :: a
- rootOfUnity :: Integer -> Integer -> a
newtype Sum a = Sum {
- fromSum :: a
}
newtype Product a = Product {
- fromProduct :: a
}
newtype App f x = App (f x)

Documentation

type Natural = Integer Source #

timesDefault :: (Additive a1, Additive a2, Integral a1) => a1 -> a2 -> a2 Source #

class Additive a where Source #

Additive monoid

Minimal complete definition

(+), zero

Methods

(+) :: a -> a -> a infixl 6 Source #

zero :: a Source #

times :: Natural -> a -> a Source #

Instances

Instances details

Additive CInt Source #
Instance details Defined in Algebra.Classes Methods (+) :: CInt -> CInt -> CInt Source # zero :: CInt Source # times :: Natural -> CInt -> CInt Source #
Additive Int16 Source #
Instance details Defined in Algebra.Classes Methods (+) :: Int16 -> Int16 -> Int16 Source # zero :: Int16 Source # times :: Natural -> Int16 -> Int16 Source #
Additive Int32 Source #
Instance details Defined in Algebra.Classes Methods (+) :: Int32 -> Int32 -> Int32 Source # zero :: Int32 Source # times :: Natural -> Int32 -> Int32 Source #
Additive Int8 Source #
Instance details Defined in Algebra.Classes Methods (+) :: Int8 -> Int8 -> Int8 Source # zero :: Int8 Source # times :: Natural -> Int8 -> Int8 Source #
Additive Word16 Source #
Instance details Defined in Algebra.Classes Methods (+) :: Word16 -> Word16 -> Word16 Source # zero :: Word16 Source # times :: Natural -> Word16 -> Word16 Source #
Additive Word32 Source #
Instance details Defined in Algebra.Classes Methods (+) :: Word32 -> Word32 -> Word32 Source # zero :: Word32 Source # times :: Natural -> Word32 -> Word32 Source #
Additive Word8 Source #
Instance details Defined in Algebra.Classes Methods (+) :: Word8 -> Word8 -> Word8 Source # zero :: Word8 Source # times :: Natural -> Word8 -> Word8 Source #
Additive Integer Source #
Instance details Defined in Algebra.Classes Methods (+) :: Integer -> Integer -> Integer Source # zero :: Integer Source # times :: Natural -> Integer -> Integer Source #
Additive Bool Source #
Instance details Defined in Algebra.Classes Methods (+) :: Bool -> Bool -> Bool Source # zero :: Bool Source # times :: Natural -> Bool -> Bool Source #
Additive Double Source #
Instance details Defined in Algebra.Classes Methods (+) :: Double -> Double -> Double Source # zero :: Double Source # times :: Natural -> Double -> Double Source #
Additive Float Source #
Instance details Defined in Algebra.Classes Methods (+) :: Float -> Float -> Float Source # zero :: Float Source # times :: Natural -> Float -> Float Source #
Additive Int Source #
Instance details Defined in Algebra.Classes Methods (+) :: Int -> Int -> Int Source # zero :: Int Source # times :: Natural -> Int -> Int Source #
Additive a => Additive (Complex a) Source #
Instance details Defined in Algebra.Classes Methods (+) :: Complex a -> Complex a -> Complex a Source # zero :: Complex a Source # times :: Natural -> Complex a -> Complex a Source #
Integral a => Additive (Ratio a) Source #
Instance details Defined in Algebra.Classes Methods (+) :: Ratio a -> Ratio a -> Ratio a Source # zero :: Ratio a Source # times :: Natural -> Ratio a -> Ratio a Source #
Multiplicative a => Additive (Log a) Source #
Instance details Defined in Algebra.Morphism.Exponential Methods (+) :: Log a -> Log a -> Log a Source # zero :: Log a Source # times :: Natural -> Log a -> Log a Source #
EuclideanDomain a => Additive (Ratio a) Source #	Since: 2.0.1
Instance details Defined in Algebra.Morphism.Ratio Methods (+) :: Ratio a -> Ratio a -> Ratio a Source # zero :: Ratio a Source # times :: Natural -> Ratio a -> Ratio a Source #
(Ord k, AbelianAdditive v) => Additive (Map k v) Source #
Instance details Defined in Algebra.Classes Methods (+) :: Map k v -> Map k v -> Map k v Source # zero :: Map k v Source # times :: Natural -> Map k v -> Map k v Source #
(Applicative f, Additive a) => Additive (App f a) Source #
Instance details Defined in Algebra.Classes Methods (+) :: App f a -> App f a -> App f a Source # zero :: App f a Source # times :: Natural -> App f a -> App f a Source #
(Ord x, AbelianAdditive c, DecidableZero c) => Additive (Affine x c) Source #
Instance details Defined in Algebra.Morphism.Affine Methods (+) :: Affine x c -> Affine x c -> Affine x c Source # zero :: Affine x c Source # times :: Natural -> Affine x c -> Affine x c Source #
(AbelianAdditive c, DecidableZero c, Ord e) => Additive (LinComb e c) Source #
Instance details Defined in Algebra.Morphism.LinComb Methods (+) :: LinComb e c -> LinComb e c -> LinComb e c Source # zero :: LinComb e c Source # times :: Natural -> LinComb e c -> LinComb e c Source #
Additive a => Additive (Pointwise x a) Source #
Instance details Defined in Algebra.Morphism.Pointwise Methods (+) :: Pointwise x a -> Pointwise x a -> Pointwise x a Source # zero :: Pointwise x a Source # times :: Natural -> Pointwise x a -> Pointwise x a Source #
Additive v => Additive (k -> v) Source #
Instance details Defined in Algebra.Classes Methods (+) :: (k -> v) -> (k -> v) -> k -> v Source # zero :: k -> v Source # times :: Natural -> (k -> v) -> k -> v Source #
Additive s => Additive (M s a b) Source #
Instance details Defined in Algebra.Category.BlockMatrix Methods (+) :: M s a b -> M s a b -> M s a b Source # zero :: M s a b Source # times :: Natural -> M s a b -> M s a b Source #
Additive s => Additive (Rel s a b) Source #
Instance details Defined in Algebra.Category.Relation Methods (+) :: Rel s a b -> Rel s a b -> Rel s a b Source # zero :: Rel s a b Source # times :: Natural -> Rel s a b -> Rel s a b Source #
(Applicative f, Additive a) => Additive (Euclid f a) Source #
Instance details Defined in Algebra.Linear Methods (+) :: Euclid f a -> Euclid f a -> Euclid f a Source # zero :: Euclid f a Source # times :: Natural -> Euclid f a -> Euclid f a Source #
Additive (f b a) => Additive (Op f a b) Source #
Instance details Defined in Algebra.Category.Op Methods (+) :: Op f a b -> Op f a b -> Op f a b Source # zero :: Op f a b Source # times :: Natural -> Op f a b -> Op f a b Source #
(Applicative f, Applicative g, Additive a) => Additive (Mat a f g) Source #
Instance details Defined in Algebra.Linear Methods (+) :: Mat a f g -> Mat a f g -> Mat a f g Source # zero :: Mat a f g Source # times :: Natural -> Mat a f g -> Mat a f g Source #

class Show a => TestEqual a where Source #

Methods

(=.=) :: a -> a -> Property infix 0 Source #

Instances

Instances details

TestEqual Double Source #
Instance details Defined in Algebra.Classes Methods (=.=) :: Double -> Double -> Property Source #
TestEqual Int Source #
Instance details Defined in Algebra.Classes Methods (=.=) :: Int -> Int -> Property Source #
(Ord x, Show x, Arbitrary x, TestEqual a, Additive a) => TestEqual (Map x a) Source #
Instance details Defined in Algebra.Classes Methods (=.=) :: Map x a -> Map x a -> Property Source #
(Show s, Additive s, TestEqual s) => TestEqual (M s a b) Source #
Instance details Defined in Algebra.Category.BlockMatrix Methods (=.=) :: M s a b -> M s a b -> Property Source #
TestEqual (f b a) => TestEqual (Op f a b) Source #
Instance details Defined in Algebra.Category.Op Methods (=.=) :: Op f a b -> Op f a b -> Property Source #
(TestEqual s, Arbitrary s, Arbitrary1 a, Arbitrary1 b, Show (a (b s)), VectorR b, VectorR a) => TestEqual (Mat s a b) Source #
Instance details Defined in Algebra.Linear Methods (=.=) :: Mat s a b -> Mat s a b -> Property Source #

law_refl :: TestEqual a => a -> Property Source #

laws_testEqual :: forall a. Arbitrary a => TestEqual a => Property Source #

nameLaw :: Testable prop => String -> prop -> Property Source #

law_assoc :: forall a. TestEqual a => String -> (a -> a -> a) -> a -> a -> a -> Property Source #

law_left_id :: forall a. TestEqual a => String -> (a -> a -> a) -> a -> a -> Property Source #

law_right_id :: forall a. TestEqual a => String -> (a -> a -> a) -> a -> a -> Property Source #

laws_monoid :: forall a. (Arbitrary a, TestEqual a) => String -> (a -> a -> a) -> a -> Property Source #

law_commutative :: TestEqual a => String -> (a -> a -> a) -> a -> a -> Property Source #

laws_comm_monoid :: forall a. (Arbitrary a, TestEqual a) => String -> (a -> a -> a) -> a -> Property Source #

law_times :: (TestEqual a, Additive a) => Positive Integer -> a -> Property Source #

laws_additive :: forall a. Arbitrary a => (Additive a, TestEqual a) => Property Source #

law_exp_pos :: (TestEqual a, Multiplicative a) => a -> Property Source #

laws_multiplicative :: forall a. Arbitrary a => (Multiplicative a, TestEqual a) => Property Source #

law_fromInteger :: forall a. (TestEqual a, Ring a) => Integer -> Property Source #

laws_ring :: forall a. Arbitrary a => (Ring a, TestEqual a) => Property Source #

sum :: (Foldable t, Additive a) => t a -> a Source #

class Additive r => DecidableZero r where Source #

Methods

isZero :: r -> Bool Source #

Instances

Instances details

DecidableZero CInt Source #
Instance details Defined in Algebra.Classes Methods isZero :: CInt -> Bool Source #
DecidableZero Word16 Source #
Instance details Defined in Algebra.Classes Methods isZero :: Word16 -> Bool Source #
DecidableZero Word32 Source #
Instance details Defined in Algebra.Classes Methods isZero :: Word32 -> Bool Source #
DecidableZero Word8 Source #
Instance details Defined in Algebra.Classes Methods isZero :: Word8 -> Bool Source #
DecidableZero Integer Source #
Instance details Defined in Algebra.Classes Methods isZero :: Integer -> Bool Source #
DecidableZero Double Source #
Instance details Defined in Algebra.Classes Methods isZero :: Double -> Bool Source #
DecidableZero Float Source #
Instance details Defined in Algebra.Classes Methods isZero :: Float -> Bool Source #
DecidableZero Int Source #
Instance details Defined in Algebra.Classes Methods isZero :: Int -> Bool Source #
DecidableZero x => DecidableZero (Complex x) Source #
Instance details Defined in Algebra.Classes Methods isZero :: Complex x -> Bool Source #
(Integral x, DecidableZero x) => DecidableZero (Ratio x) Source #
Instance details Defined in Algebra.Classes Methods isZero :: Ratio x -> Bool Source #
(Ord k, DecidableZero v, AbelianAdditive v) => DecidableZero (Map k v) Source #
Instance details Defined in Algebra.Classes Methods isZero :: Map k v -> Bool Source #
(AbelianAdditive c, Eq c, DecidableZero c, Ord e) => DecidableZero (LinComb e c) Source #
Instance details Defined in Algebra.Morphism.LinComb Methods isZero :: LinComb e c -> Bool Source #

law_decidable_zero :: forall a. (DecidableZero a, TestEqual a) => Property Source #

class Additive a => AbelianAdditive a Source #

Instances

Instances details

AbelianAdditive CInt Source #
Instance details Defined in Algebra.Classes
AbelianAdditive Int16 Source #
Instance details Defined in Algebra.Classes
AbelianAdditive Int32 Source #
Instance details Defined in Algebra.Classes
AbelianAdditive Int8 Source #
Instance details Defined in Algebra.Classes
AbelianAdditive Word16 Source #
Instance details Defined in Algebra.Classes
AbelianAdditive Word32 Source #
Instance details Defined in Algebra.Classes
AbelianAdditive Word8 Source #
Instance details Defined in Algebra.Classes
AbelianAdditive Integer Source #
Instance details Defined in Algebra.Classes
AbelianAdditive Bool Source #
Instance details Defined in Algebra.Classes
AbelianAdditive Double Source #
Instance details Defined in Algebra.Classes
AbelianAdditive Float Source #
Instance details Defined in Algebra.Classes
AbelianAdditive Int Source #
Instance details Defined in Algebra.Classes
AbelianAdditive a => AbelianAdditive (Complex a) Source #
Instance details Defined in Algebra.Classes
Integral a => AbelianAdditive (Ratio a) Source #
Instance details Defined in Algebra.Classes
EuclideanDomain a => AbelianAdditive (Ratio a) Source #
Instance details Defined in Algebra.Morphism.Ratio
(Ord k, AbelianAdditive v) => AbelianAdditive (Map k v) Source #
Instance details Defined in Algebra.Classes
(Applicative f, AbelianAdditive a) => AbelianAdditive (App f a) Source #
Instance details Defined in Algebra.Classes
(Ord x, AbelianAdditive c, DecidableZero c) => AbelianAdditive (Affine x c) Source #
Instance details Defined in Algebra.Morphism.Affine
(AbelianAdditive c, DecidableZero c, Ord x) => AbelianAdditive (LinComb x c) Source #
Instance details Defined in Algebra.Morphism.LinComb
Additive a => AbelianAdditive (Pointwise x a) Source #
Instance details Defined in Algebra.Morphism.Pointwise
AbelianAdditive v => AbelianAdditive (k -> v) Source #
Instance details Defined in Algebra.Classes
(Applicative f, Applicative g, AbelianAdditive a) => AbelianAdditive (Mat a f g) Source #
Instance details Defined in Algebra.Linear

laws_abelian_additive :: forall a. (Arbitrary a, AbelianAdditive a, TestEqual a) => Property Source #

multDefault :: Group a => Natural -> a -> a Source #

class Additive a => Group a where Source #

Minimal complete definition

(negate | (-) | subtract)

Methods

(-) :: a -> a -> a infixl 6 Source #

subtract :: a -> a -> a Source #

negate :: a -> a Source #

mult :: Integer -> a -> a Source #

Instances

Instances details

Group CInt Source #
Instance details Defined in Algebra.Classes Methods (-) :: CInt -> CInt -> CInt Source # subtract :: CInt -> CInt -> CInt Source # negate :: CInt -> CInt Source # mult :: Integer -> CInt -> CInt Source #
Group Int16 Source #
Instance details Defined in Algebra.Classes Methods (-) :: Int16 -> Int16 -> Int16 Source # subtract :: Int16 -> Int16 -> Int16 Source # negate :: Int16 -> Int16 Source # mult :: Integer -> Int16 -> Int16 Source #
Group Int32 Source #
Instance details Defined in Algebra.Classes Methods (-) :: Int32 -> Int32 -> Int32 Source # subtract :: Int32 -> Int32 -> Int32 Source # negate :: Int32 -> Int32 Source # mult :: Integer -> Int32 -> Int32 Source #
Group Int8 Source #
Instance details Defined in Algebra.Classes Methods (-) :: Int8 -> Int8 -> Int8 Source # subtract :: Int8 -> Int8 -> Int8 Source # negate :: Int8 -> Int8 Source # mult :: Integer -> Int8 -> Int8 Source #
Group Word16 Source #
Instance details Defined in Algebra.Classes Methods (-) :: Word16 -> Word16 -> Word16 Source # subtract :: Word16 -> Word16 -> Word16 Source # negate :: Word16 -> Word16 Source # mult :: Integer -> Word16 -> Word16 Source #
Group Word32 Source #
Instance details Defined in Algebra.Classes Methods (-) :: Word32 -> Word32 -> Word32 Source # subtract :: Word32 -> Word32 -> Word32 Source # negate :: Word32 -> Word32 Source # mult :: Integer -> Word32 -> Word32 Source #
Group Word8 Source #
Instance details Defined in Algebra.Classes Methods (-) :: Word8 -> Word8 -> Word8 Source # subtract :: Word8 -> Word8 -> Word8 Source # negate :: Word8 -> Word8 Source # mult :: Integer -> Word8 -> Word8 Source #
Group Integer Source #
Instance details Defined in Algebra.Classes Methods (-) :: Integer -> Integer -> Integer Source # subtract :: Integer -> Integer -> Integer Source # negate :: Integer -> Integer Source # mult :: Integer -> Integer -> Integer Source #
Group Double Source #
Instance details Defined in Algebra.Classes Methods (-) :: Double -> Double -> Double Source # subtract :: Double -> Double -> Double Source # negate :: Double -> Double Source # mult :: Integer -> Double -> Double Source #
Group Float Source #
Instance details Defined in Algebra.Classes Methods (-) :: Float -> Float -> Float Source # subtract :: Float -> Float -> Float Source # negate :: Float -> Float Source # mult :: Integer -> Float -> Float Source #
Group Int Source #
Instance details Defined in Algebra.Classes Methods (-) :: Int -> Int -> Int Source # subtract :: Int -> Int -> Int Source # negate :: Int -> Int Source # mult :: Integer -> Int -> Int Source #
Group a => Group (Complex a) Source #
Instance details Defined in Algebra.Classes Methods (-) :: Complex a -> Complex a -> Complex a Source # subtract :: Complex a -> Complex a -> Complex a Source # negate :: Complex a -> Complex a Source # mult :: Integer -> Complex a -> Complex a Source #
Integral a => Group (Ratio a) Source #
Instance details Defined in Algebra.Classes Methods (-) :: Ratio a -> Ratio a -> Ratio a Source # subtract :: Ratio a -> Ratio a -> Ratio a Source # negate :: Ratio a -> Ratio a Source # mult :: Integer -> Ratio a -> Ratio a Source #
Division a => Group (Log a) Source #
Instance details Defined in Algebra.Morphism.Exponential Methods (-) :: Log a -> Log a -> Log a Source # subtract :: Log a -> Log a -> Log a Source # negate :: Log a -> Log a Source # mult :: Integer -> Log a -> Log a Source #
EuclideanDomain a => Group (Ratio a) Source #
Instance details Defined in Algebra.Morphism.Ratio Methods (-) :: Ratio a -> Ratio a -> Ratio a Source # subtract :: Ratio a -> Ratio a -> Ratio a Source # negate :: Ratio a -> Ratio a Source # mult :: Integer -> Ratio a -> Ratio a Source #
(Ord k, Group v, AbelianAdditive v) => Group (Map k v) Source #
Instance details Defined in Algebra.Classes Methods (-) :: Map k v -> Map k v -> Map k v Source # subtract :: Map k v -> Map k v -> Map k v Source # negate :: Map k v -> Map k v Source # mult :: Integer -> Map k v -> Map k v Source #
(Applicative f, Group a) => Group (App f a) Source #
Instance details Defined in Algebra.Classes Methods (-) :: App f a -> App f a -> App f a Source # subtract :: App f a -> App f a -> App f a Source # negate :: App f a -> App f a Source # mult :: Integer -> App f a -> App f a Source #
(Ord x, AbelianAdditive c, Group c, DecidableZero c) => Group (Affine x c) Source #
Instance details Defined in Algebra.Morphism.Affine Methods (-) :: Affine x c -> Affine x c -> Affine x c Source # subtract :: Affine x c -> Affine x c -> Affine x c Source # negate :: Affine x c -> Affine x c Source # mult :: Integer -> Affine x c -> Affine x c Source #
(AbelianAdditive c, Group c, DecidableZero c, Ord e) => Group (LinComb e c) Source #
Instance details Defined in Algebra.Morphism.LinComb Methods (-) :: LinComb e c -> LinComb e c -> LinComb e c Source # subtract :: LinComb e c -> LinComb e c -> LinComb e c Source # negate :: LinComb e c -> LinComb e c Source # mult :: Integer -> LinComb e c -> LinComb e c Source #
Group a => Group (Pointwise x a) Source #
Instance details Defined in Algebra.Morphism.Pointwise Methods (-) :: Pointwise x a -> Pointwise x a -> Pointwise x a Source # subtract :: Pointwise x a -> Pointwise x a -> Pointwise x a Source # negate :: Pointwise x a -> Pointwise x a Source # mult :: Integer -> Pointwise x a -> Pointwise x a Source #
Group v => Group (k -> v) Source #
Instance details Defined in Algebra.Classes Methods (-) :: (k -> v) -> (k -> v) -> k -> v Source # subtract :: (k -> v) -> (k -> v) -> k -> v Source # negate :: (k -> v) -> k -> v Source # mult :: Integer -> (k -> v) -> k -> v Source #
Group s => Group (M s a b) Source #
Instance details Defined in Algebra.Category.BlockMatrix Methods (-) :: M s a b -> M s a b -> M s a b Source # subtract :: M s a b -> M s a b -> M s a b Source # negate :: M s a b -> M s a b Source # mult :: Integer -> M s a b -> M s a b Source #
(Applicative f, Group a) => Group (Euclid f a) Source #
Instance details Defined in Algebra.Linear Methods (-) :: Euclid f a -> Euclid f a -> Euclid f a Source # subtract :: Euclid f a -> Euclid f a -> Euclid f a Source # negate :: Euclid f a -> Euclid f a Source # mult :: Integer -> Euclid f a -> Euclid f a Source #
Group (f b a) => Group (Op f a b) Source #
Instance details Defined in Algebra.Category.Op Methods (-) :: Op f a b -> Op f a b -> Op f a b Source # subtract :: Op f a b -> Op f a b -> Op f a b Source # negate :: Op f a b -> Op f a b Source # mult :: Integer -> Op f a b -> Op f a b Source #
(Applicative f, Applicative g, Group a) => Group (Mat a f g) Source #
Instance details Defined in Algebra.Linear Methods (-) :: Mat a f g -> Mat a f g -> Mat a f g Source # subtract :: Mat a f g -> Mat a f g -> Mat a f g Source # negate :: Mat a f g -> Mat a f g Source # mult :: Integer -> Mat a f g -> Mat a f g Source #

law_negate_minus :: (TestEqual a, Group a) => a -> a -> Property Source #

law_mult :: (TestEqual a, Group a) => Integer -> a -> Property Source #

laws_group :: forall a. Arbitrary a => (Group a, TestEqual a) => Property Source #

laws_abelian_group :: forall a. Arbitrary a => (Group a, TestEqual a) => Property Source #

(*<) :: (Functor f, Multiplicative a) => a -> f a -> f a infixr 7 Source #

Functorial scaling. Compared to (*^) this operator disambiguates the scalar type, by using the functor structure and using the multiplicative instance for scalars.

class Scalable s a where Source #

Any instance must preserve the following invariants: 1. if Multiplicative a and Scalable a a, then (*) = (*^) for a. 2. Scalable must define a partial order relation, in particular, instances of the form (Scalable s a) => Scalable s (T ... a ...) are acceptable, and should be declared overlappable.

Methods

(*^) :: s -> a -> a infixr 7 Source #

Instances

Instances details

Scalable CInt CInt Source #
Instance details Defined in Algebra.Classes Methods (*^) :: CInt -> CInt -> CInt Source #
Scalable Int16 Int16 Source #
Instance details Defined in Algebra.Classes Methods (*^) :: Int16 -> Int16 -> Int16 Source #
Scalable Int32 Int32 Source #
Instance details Defined in Algebra.Classes Methods (*^) :: Int32 -> Int32 -> Int32 Source #
Scalable Int8 Int8 Source #
Instance details Defined in Algebra.Classes Methods (*^) :: Int8 -> Int8 -> Int8 Source #
Scalable Rational Double Source #
Instance details Defined in Algebra.Classes Methods (*^) :: Rational -> Double -> Double Source #
Scalable Word16 Word16 Source #
Instance details Defined in Algebra.Classes Methods (*^) :: Word16 -> Word16 -> Word16 Source #
Scalable Word32 Word32 Source #
Instance details Defined in Algebra.Classes Methods (*^) :: Word32 -> Word32 -> Word32 Source #
Scalable Word8 Word8 Source #
Instance details Defined in Algebra.Classes Methods (*^) :: Word8 -> Word8 -> Word8 Source #
Scalable Integer Integer Source #
Instance details Defined in Algebra.Classes Methods (*^) :: Integer -> Integer -> Integer Source #
Scalable Double Double Source #
Instance details Defined in Algebra.Classes Methods (*^) :: Double -> Double -> Double Source #
Scalable Float Float Source #
Instance details Defined in Algebra.Classes Methods (*^) :: Float -> Float -> Float Source #
Scalable Int Int Source #
Instance details Defined in Algebra.Classes Methods (*^) :: Int -> Int -> Int Source #
Multiplicative a => Scalable Integer (Log a) Source #
Instance details Defined in Algebra.Morphism.Exponential Methods (*^) :: Integer -> Log a -> Log a Source #
Scalable s a => Scalable s (Complex a) Source #
Instance details Defined in Algebra.Classes Methods (*^) :: s -> Complex a -> Complex a Source #
Multiplicative c => Scalable c (Affine x c) Source #
Instance details Defined in Algebra.Morphism.Affine Methods (*^) :: c -> Affine x c -> Affine x c Source #
Scalable s a => Scalable s (Map k a) Source #
Instance details Defined in Algebra.Classes Methods (*^) :: s -> Map k a -> Map k a Source #
Scalable s a => Scalable s (LinComb k a) Source #
Instance details Defined in Algebra.Morphism.LinComb Methods (*^) :: s -> LinComb k a -> LinComb k a Source #
Scalable s a => Scalable s (k -> a) Source #
Instance details Defined in Algebra.Classes Methods (*^) :: s -> (k -> a) -> k -> a Source #
Ring s => Scalable s (M s a b) Source #
Instance details Defined in Algebra.Category.BlockMatrix Methods (*^) :: s -> M s a b -> M s a b Source #
(Functor f, Scalable s a) => Scalable s (Euclid f a) Source #
Instance details Defined in Algebra.Linear Methods (*^) :: s -> Euclid f a -> Euclid f a Source #
(Functor f, Functor g, Scalable s a) => Scalable s (Mat a f g) Source #
Instance details Defined in Algebra.Linear Methods (*^) :: s -> Mat a f g -> Mat a f g Source #
Ring a => Scalable (Complex a) (Complex a) Source #
Instance details Defined in Algebra.Classes Methods (*^) :: Complex a -> Complex a -> Complex a Source #
Integral a => Scalable (Ratio a) (Ratio a) Source #
Instance details Defined in Algebra.Classes Methods (*^) :: Ratio a -> Ratio a -> Ratio a Source #
EuclideanDomain a => Scalable (Ratio a) (Ratio a) Source #
Instance details Defined in Algebra.Morphism.Ratio Methods (*^) :: Ratio a -> Ratio a -> Ratio a Source #
(Applicative f, Scalable s a) => Scalable (App f s) (App f a) Source #
Instance details Defined in Algebra.Classes Methods (*^) :: App f s -> App f a -> App f a Source #
Multiplicative a => Scalable (Pointwise x a) (Pointwise x a) Source #
Instance details Defined in Algebra.Morphism.Pointwise Methods (*^) :: Pointwise x a -> Pointwise x a -> Pointwise x a Source #

class Scalable' a where Source #

"Most natural" scaling. Also disambiguates the scalar type, but using a fundep.

Associated Types

type Scalar a Source #

Methods

(!*^) :: Scalar a -> a -> a Source #

type SemiModule s a = (AbelianAdditive a, SemiRing s, Scalable s a) Source #

A prefix variant of (*^), useful when using type applications. scale :: forall s a. Scalable s a => s -> a -> a scale = (*^)

type Module s a = (SemiModule s a, Group s, Group a) Source #

law_module_zero :: forall s a. (Module s a, TestEqual a) => s -> Property Source #

law_module_one :: forall s a. (Module s a, TestEqual a) => a -> Property Source #

law_module_sum :: forall s a. (Module s a, TestEqual a) => s -> a -> a -> Property Source #

law_module_sum_left :: forall s a. (Module s a, TestEqual a) => s -> s -> a -> Property Source #

law_module_mul :: forall s a. (Module s a, TestEqual a) => s -> s -> a -> Property Source #

laws_module :: forall s a. Arbitrary a => (Module s a, TestEqual a, Arbitrary s, Show s) => Property Source #

class Multiplicative a where Source #

Multiplicative monoid

Minimal complete definition

(*), one

Methods

(*) :: a -> a -> a infixl 7 Source #

one :: a Source #

(^+) :: a -> Natural -> a infixr 8 Source #

Instances

Instances details

Multiplicative Property Source #
Instance details Defined in Algebra.Classes Methods (*) :: Property -> Property -> Property Source # one :: Property Source # (^+) :: Property -> Natural -> Property Source #
Multiplicative CInt Source #
Instance details Defined in Algebra.Classes Methods (*) :: CInt -> CInt -> CInt Source # one :: CInt Source # (^+) :: CInt -> Natural -> CInt Source #
Multiplicative Int16 Source #
Instance details Defined in Algebra.Classes Methods (*) :: Int16 -> Int16 -> Int16 Source # one :: Int16 Source # (^+) :: Int16 -> Natural -> Int16 Source #
Multiplicative Int32 Source #
Instance details Defined in Algebra.Classes Methods (*) :: Int32 -> Int32 -> Int32 Source # one :: Int32 Source # (^+) :: Int32 -> Natural -> Int32 Source #
Multiplicative Int8 Source #
Instance details Defined in Algebra.Classes Methods (*) :: Int8 -> Int8 -> Int8 Source # one :: Int8 Source # (^+) :: Int8 -> Natural -> Int8 Source #
Multiplicative Word16 Source #
Instance details Defined in Algebra.Classes Methods (*) :: Word16 -> Word16 -> Word16 Source # one :: Word16 Source # (^+) :: Word16 -> Natural -> Word16 Source #
Multiplicative Word32 Source #
Instance details Defined in Algebra.Classes Methods (*) :: Word32 -> Word32 -> Word32 Source # one :: Word32 Source # (^+) :: Word32 -> Natural -> Word32 Source #
Multiplicative Word8 Source #
Instance details Defined in Algebra.Classes Methods (*) :: Word8 -> Word8 -> Word8 Source # one :: Word8 Source # (^+) :: Word8 -> Natural -> Word8 Source #
Multiplicative Integer Source #
Instance details Defined in Algebra.Classes Methods (*) :: Integer -> Integer -> Integer Source # one :: Integer Source # (^+) :: Integer -> Natural -> Integer Source #
Multiplicative Bool Source #
Instance details Defined in Algebra.Classes Methods (*) :: Bool -> Bool -> Bool Source # one :: Bool Source # (^+) :: Bool -> Natural -> Bool Source #
Multiplicative Double Source #
Instance details Defined in Algebra.Classes Methods (*) :: Double -> Double -> Double Source # one :: Double Source # (^+) :: Double -> Natural -> Double Source #
Multiplicative Float Source #
Instance details Defined in Algebra.Classes Methods (*) :: Float -> Float -> Float Source # one :: Float Source # (^+) :: Float -> Natural -> Float Source #
Multiplicative Int Source #
Instance details Defined in Algebra.Classes Methods (*) :: Int -> Int -> Int Source # one :: Int Source # (^+) :: Int -> Natural -> Int Source #
Ring a => Multiplicative (Complex a) Source #
Instance details Defined in Algebra.Classes Methods (*) :: Complex a -> Complex a -> Complex a Source # one :: Complex a Source # (^+) :: Complex a -> Natural -> Complex a Source #
Integral a => Multiplicative (Ratio a) Source #
Instance details Defined in Algebra.Classes Methods (*) :: Ratio a -> Ratio a -> Ratio a Source # one :: Ratio a Source # (^+) :: Ratio a -> Natural -> Ratio a Source #
Additive a => Multiplicative (Exp a) Source #
Instance details Defined in Algebra.Morphism.Exponential Methods (*) :: Exp a -> Exp a -> Exp a Source # one :: Exp a Source # (^+) :: Exp a -> Natural -> Exp a Source #
EuclideanDomain a => Multiplicative (Ratio a) Source #
Instance details Defined in Algebra.Morphism.Ratio Methods (*) :: Ratio a -> Ratio a -> Ratio a Source # one :: Ratio a Source # (^+) :: Ratio a -> Natural -> Ratio a Source #
(Category cat, Obj cat a) => Multiplicative (Endo cat a) Source #
Instance details Defined in Algebra.Category.Endo Methods (*) :: Endo cat a -> Endo cat a -> Endo cat a Source # one :: Endo cat a Source # (^+) :: Endo cat a -> Natural -> Endo cat a Source #
(Applicative f, Multiplicative a) => Multiplicative (App f a) Source #
Instance details Defined in Algebra.Classes Methods (*) :: App f a -> App f a -> App f a Source # one :: App f a Source # (^+) :: App f a -> Natural -> App f a Source #
Multiplicative a => Multiplicative (Pointwise x a) Source #
Instance details Defined in Algebra.Morphism.Pointwise Methods (*) :: Pointwise x a -> Pointwise x a -> Pointwise x a Source # one :: Pointwise x a Source # (^+) :: Pointwise x a -> Natural -> Pointwise x a Source #

positiveExponentDefault :: Multiplicative a => a -> Natural -> a Source #

product :: (Multiplicative a, Foldable f) => f a -> a Source #

type SemiRing a = (Multiplicative a, AbelianAdditive a) Source #

type PreRing a = (SemiRing a, Group a) Source #

fromIntegerDefault :: PreRing a => Integer -> a Source #

class (Module a a, PreRing a) => Ring a where Source #

Minimal complete definition

Nothing

Methods

fromInteger :: Integer -> a Source #

Instances

Instances details

Ring CInt Source #
Instance details Defined in Algebra.Classes Methods fromInteger :: Integer -> CInt Source #
Ring Int16 Source #
Instance details Defined in Algebra.Classes Methods fromInteger :: Integer -> Int16 Source #
Ring Int32 Source #
Instance details Defined in Algebra.Classes Methods fromInteger :: Integer -> Int32 Source #
Ring Int8 Source #
Instance details Defined in Algebra.Classes Methods fromInteger :: Integer -> Int8 Source #
Ring Word16 Source #
Instance details Defined in Algebra.Classes Methods fromInteger :: Integer -> Word16 Source #
Ring Word32 Source #
Instance details Defined in Algebra.Classes Methods fromInteger :: Integer -> Word32 Source #
Ring Word8 Source #
Instance details Defined in Algebra.Classes Methods fromInteger :: Integer -> Word8 Source #
Ring Integer Source #
Instance details Defined in Algebra.Classes Methods fromInteger :: Integer -> Integer Source #
Ring Double Source #
Instance details Defined in Algebra.Classes Methods fromInteger :: Integer -> Double Source #
Ring Float Source #
Instance details Defined in Algebra.Classes Methods fromInteger :: Integer -> Float Source #
Ring Int Source #
Instance details Defined in Algebra.Classes Methods fromInteger :: Integer -> Int Source #
Ring a => Ring (Complex a) Source #
Instance details Defined in Algebra.Classes Methods fromInteger :: Integer -> Complex a Source #
Integral a => Ring (Ratio a) Source #
Instance details Defined in Algebra.Classes Methods fromInteger :: Integer -> Ratio a Source #
EuclideanDomain a => Ring (Ratio a) Source #
Instance details Defined in Algebra.Morphism.Ratio Methods fromInteger :: Integer -> Ratio a Source #
(Applicative f, Ring a) => Ring (App f a) Source #
Instance details Defined in Algebra.Classes Methods fromInteger :: Integer -> App f a Source #
Ring a => Ring (Pointwise x a) Source #
Instance details Defined in Algebra.Morphism.Pointwise Methods fromInteger :: Integer -> Pointwise x a Source #

class Multiplicative a => Division a where Source #

Minimal complete definition

(recip | (/))

Methods

recip :: a -> a Source #

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

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

Instances

Instances details

Division Double Source #
Instance details Defined in Algebra.Classes Methods recip :: Double -> Double Source # (/) :: Double -> Double -> Double Source # (^) :: Double -> Integer -> Double Source #
Division Float Source #
Instance details Defined in Algebra.Classes Methods recip :: Float -> Float Source # (/) :: Float -> Float -> Float Source # (^) :: Float -> Integer -> Float Source #
Field a => Division (Complex a) Source #
Instance details Defined in Algebra.Classes Methods recip :: Complex a -> Complex a Source # (/) :: Complex a -> Complex a -> Complex a Source # (^) :: Complex a -> Integer -> Complex a Source #
Integral a => Division (Ratio a) Source #
Instance details Defined in Algebra.Classes Methods recip :: Ratio a -> Ratio a Source # (/) :: Ratio a -> Ratio a -> Ratio a Source # (^) :: Ratio a -> Integer -> Ratio a Source #
Group a => Division (Exp a) Source #
Instance details Defined in Algebra.Morphism.Exponential Methods recip :: Exp a -> Exp a Source # (/) :: Exp a -> Exp a -> Exp a Source # (^) :: Exp a -> Integer -> Exp a Source #
EuclideanDomain a => Division (Ratio a) Source #	Since: 2.0.1
Instance details Defined in Algebra.Morphism.Ratio Methods recip :: Ratio a -> Ratio a Source # (/) :: Ratio a -> Ratio a -> Ratio a Source # (^) :: Ratio a -> Integer -> Ratio a Source #
(Dagger cat, Obj cat a) => Division (Endo cat a) Source #
Instance details Defined in Algebra.Category.Endo Methods recip :: Endo cat a -> Endo cat a Source # (/) :: Endo cat a -> Endo cat a -> Endo cat a Source # (^) :: Endo cat a -> Integer -> Endo cat a Source #
(Applicative f, Division s) => Division (App f s) Source #
Instance details Defined in Algebra.Classes Methods recip :: App f s -> App f s Source # (/) :: App f s -> App f s -> App f s Source # (^) :: App f s -> Integer -> App f s Source #
Division a => Division (Pointwise x a) Source #
Instance details Defined in Algebra.Morphism.Pointwise Methods recip :: Pointwise x a -> Pointwise x a Source # (/) :: Pointwise x a -> Pointwise x a -> Pointwise x a Source # (^) :: Pointwise x a -> Integer -> Pointwise x a Source #

class (Ring a, Division a) => Field a where Source #

Minimal complete definition

Nothing

Methods

fromRational :: Rational -> a Source #

Instances

Instances details

Field Double Source #
Instance details Defined in Algebra.Classes Methods fromRational :: Rational -> Double Source #
Field Float Source #
Instance details Defined in Algebra.Classes Methods fromRational :: Rational -> Float Source #
Field a => Field (Complex a) Source #
Instance details Defined in Algebra.Classes Methods fromRational :: Rational -> Complex a Source #
Integral a => Field (Ratio a) Source #
Instance details Defined in Algebra.Classes Methods fromRational :: Rational -> Ratio a Source #
EuclideanDomain a => Field (Ratio a) Source #
Instance details Defined in Algebra.Morphism.Ratio Methods fromRational :: Rational -> Ratio a Source #
(Applicative f, Field s) => Field (App f s) Source #
Instance details Defined in Algebra.Classes Methods fromRational :: Rational -> App f s Source #
Field a => Field (Pointwise x a) Source #
Instance details Defined in Algebra.Morphism.Pointwise Methods fromRational :: Rational -> Pointwise x a Source #

class (Ring a, DecidableZero a) => EuclideanDomain a where Source #

Minimal complete definition

(stdUnit | normalize), (quotRem | quot, rem)

Methods

stdAssociate :: a -> a Source #

stdUnit :: a -> a Source #

normalize :: a -> (a, a) Source #

quot :: a -> a -> a infixl 7 Source #

rem :: a -> a -> a infixl 7 Source #

quotRem :: a -> a -> (a, a) Source #

Instances

Instances details

EuclideanDomain CInt Source #
Instance details Defined in Algebra.Classes Methods stdAssociate :: CInt -> CInt Source # stdUnit :: CInt -> CInt Source # normalize :: CInt -> (CInt, CInt) Source # quot :: CInt -> CInt -> CInt Source # rem :: CInt -> CInt -> CInt Source # quotRem :: CInt -> CInt -> (CInt, CInt) Source #
EuclideanDomain Integer Source #
Instance details Defined in Algebra.Classes Methods stdAssociate :: Integer -> Integer Source # stdUnit :: Integer -> Integer Source # normalize :: Integer -> (Integer, Integer) Source # quot :: Integer -> Integer -> Integer Source # rem :: Integer -> Integer -> Integer Source # quotRem :: Integer -> Integer -> (Integer, Integer) Source #
EuclideanDomain Int Source #
Instance details Defined in Algebra.Classes Methods stdAssociate :: Int -> Int Source # stdUnit :: Int -> Int Source # normalize :: Int -> (Int, Int) Source # quot :: Int -> Int -> Int Source # rem :: Int -> Int -> Int Source # quotRem :: Int -> Int -> (Int, Int) Source #

gcd :: EuclideanDomain a => a -> a -> a Source #

lcm :: EuclideanDomain a => a -> a -> a Source #

lcm x y is the smallest positive integer that both x and y divide.

class (Ord a, Ring a, Enum a, EuclideanDomain a) => Integral a where Source #

Minimal complete definition

toInteger

Methods

div :: a -> a -> a infixl 7 Source #

mod :: a -> a -> a infixl 7 Source #

divMod :: a -> a -> (a, a) Source #

toInteger :: a -> Integer Source #

Instances

Instances details

Integral Integer Source #
Instance details Defined in Algebra.Classes Methods div :: Integer -> Integer -> Integer Source # mod :: Integer -> Integer -> Integer Source # divMod :: Integer -> Integer -> (Integer, Integer) Source # toInteger :: Integer -> Integer Source #
Integral Int Source #
Instance details Defined in Algebra.Classes Methods div :: Int -> Int -> Int Source # mod :: Int -> Int -> Int Source # divMod :: Int -> Int -> (Int, Int) Source # toInteger :: Int -> Integer Source #

ifThenElse :: Bool -> t -> t -> t Source #

class Division a => Roots a where Source #

Minimal complete definition

root | (^/)

Methods

sqrt :: a -> a Source #

root :: Integer -> a -> a Source #

(^/) :: a -> Rational -> a infixr 8 Source #

Instances

Instances details

Roots Double Source #
Instance details Defined in Algebra.Classes Methods sqrt :: Double -> Double Source # root :: Integer -> Double -> Double Source # (^/) :: Double -> Rational -> Double Source #
Roots Float Source #
Instance details Defined in Algebra.Classes Methods sqrt :: Float -> Float Source # root :: Integer -> Float -> Float Source # (^/) :: Float -> Rational -> Float Source #
(RealFloat a, Ord a, Algebraic a) => Roots (Complex a) Source #
Instance details Defined in Algebra.Classes Methods sqrt :: Complex a -> Complex a Source # root :: Integer -> Complex a -> Complex a Source # (^/) :: Complex a -> Rational -> Complex a Source #
Field a => Roots (Exp a) Source #
Instance details Defined in Algebra.Morphism.Exponential Methods sqrt :: Exp a -> Exp a Source # root :: Integer -> Exp a -> Exp a Source # (^/) :: Exp a -> Rational -> Exp a Source #
(Applicative f, Roots s) => Roots (App f s) Source #
Instance details Defined in Algebra.Classes Methods sqrt :: App f s -> App f s Source # root :: Integer -> App f s -> App f s Source # (^/) :: App f s -> Rational -> App f s Source #
Roots a => Roots (Pointwise x a) Source #
Instance details Defined in Algebra.Morphism.Pointwise Methods sqrt :: Pointwise x a -> Pointwise x a Source # root :: Integer -> Pointwise x a -> Pointwise x a Source # (^/) :: Pointwise x a -> Rational -> Pointwise x a Source #

type Algebraic a = (Roots a, Field a) Source #

class Algebraic a => Transcendental a where Source #

Class providing transcendental functions

Minimal complete definition

pi, exp, log, sin, cos, asin, acos, atan, sinh, cosh, asinh, acosh, atanh

Methods

pi :: a Source #

exp :: a -> a Source #

log :: a -> a Source #

(**) :: a -> a -> a infixr 8 Source #

logBase :: a -> a -> a Source #

sin :: a -> a Source #

cos :: a -> a Source #

tan :: a -> a Source #

asin :: a -> a Source #

acos :: a -> a Source #

atan :: a -> a Source #

sinh :: a -> a Source #

cosh :: a -> a Source #

tanh :: a -> a Source #

asinh :: a -> a Source #

acosh :: a -> a Source #

atanh :: a -> a Source #

log1p :: a -> a Source #

log1p x computes log (1 + x), but provides more precise results for small (absolute) values of x if possible.

Since: 4.9.0.0

expm1 :: a -> a Source #

expm1 x computes exp x - 1, but provides more precise results for small (absolute) values of x if possible.

Since: 4.9.0.0

log1pexp :: a -> a Source #

log1pexp x computes log (1 + exp x), but provides more precise results if possible.

Examples:

if x is a large negative number, log (1 + exp x) will be imprecise for the reasons given in log1p.
if exp x is close to -1, log (1 + exp x) will be imprecise for the reasons given in expm1.

Since: 4.9.0.0

log1mexp :: a -> a Source #

log1mexp x computes log (1 - exp x), but provides more precise results if possible.

Examples:

if x is a large negative number, log (1 - exp x) will be imprecise for the reasons given in log1p.
if exp x is close to 1, log (1 - exp x) will be imprecise for the reasons given in expm1.

Since: 4.9.0.0

Instances

Instances details

Transcendental Double Source #
Instance details Defined in Algebra.Classes Methods pi :: Double Source # exp :: Double -> Double Source # log :: Double -> Double Source # (**) :: Double -> Double -> Double Source # logBase :: Double -> Double -> Double Source # sin :: Double -> Double Source # cos :: Double -> Double Source # tan :: Double -> Double Source # asin :: Double -> Double Source # acos :: Double -> Double Source # atan :: Double -> Double Source # sinh :: Double -> Double Source # cosh :: Double -> Double Source # tanh :: Double -> Double Source # asinh :: Double -> Double Source # acosh :: Double -> Double Source # atanh :: Double -> Double Source # log1p :: Double -> Double Source # expm1 :: Double -> Double Source # log1pexp :: Double -> Double Source # log1mexp :: Double -> Double Source #
Transcendental Float Source #
Instance details Defined in Algebra.Classes Methods pi :: Float Source # exp :: Float -> Float Source # log :: Float -> Float Source # (**) :: Float -> Float -> Float Source # logBase :: Float -> Float -> Float Source # sin :: Float -> Float Source # cos :: Float -> Float Source # tan :: Float -> Float Source # asin :: Float -> Float Source # acos :: Float -> Float Source # atan :: Float -> Float Source # sinh :: Float -> Float Source # cosh :: Float -> Float Source # tanh :: Float -> Float Source # asinh :: Float -> Float Source # acosh :: Float -> Float Source # atanh :: Float -> Float Source # log1p :: Float -> Float Source # expm1 :: Float -> Float Source # log1pexp :: Float -> Float Source # log1mexp :: Float -> Float Source #
(RealFloat a, Transcendental a) => Transcendental (Complex a) Source #
Instance details Defined in Algebra.Classes Methods pi :: Complex a Source # exp :: Complex a -> Complex a Source # log :: Complex a -> Complex a Source # (**) :: Complex a -> Complex a -> Complex a Source # logBase :: Complex a -> Complex a -> Complex a Source # sin :: Complex a -> Complex a Source # cos :: Complex a -> Complex a Source # tan :: Complex a -> Complex a Source # asin :: Complex a -> Complex a Source # acos :: Complex a -> Complex a Source # atan :: Complex a -> Complex a Source # sinh :: Complex a -> Complex a Source # cosh :: Complex a -> Complex a Source # tanh :: Complex a -> Complex a Source # asinh :: Complex a -> Complex a Source # acosh :: Complex a -> Complex a Source # atanh :: Complex a -> Complex a Source # log1p :: Complex a -> Complex a Source # expm1 :: Complex a -> Complex a Source # log1pexp :: Complex a -> Complex a Source # log1mexp :: Complex a -> Complex a Source #
(Applicative f, Transcendental s) => Transcendental (App f s) Source #
Instance details Defined in Algebra.Classes Methods pi :: App f s Source # exp :: App f s -> App f s Source # log :: App f s -> App f s Source # (**) :: App f s -> App f s -> App f s Source # logBase :: App f s -> App f s -> App f s Source # sin :: App f s -> App f s Source # cos :: App f s -> App f s Source # tan :: App f s -> App f s Source # asin :: App f s -> App f s Source # acos :: App f s -> App f s Source # atan :: App f s -> App f s Source # sinh :: App f s -> App f s Source # cosh :: App f s -> App f s Source # tanh :: App f s -> App f s Source # asinh :: App f s -> App f s Source # acosh :: App f s -> App f s Source # atanh :: App f s -> App f s Source # log1p :: App f s -> App f s Source # expm1 :: App f s -> App f s Source # log1pexp :: App f s -> App f s Source # log1mexp :: App f s -> App f s Source #
Transcendental a => Transcendental (Pointwise x a) Source #
Instance details Defined in Algebra.Morphism.Pointwise Methods pi :: Pointwise x a Source # exp :: Pointwise x a -> Pointwise x a Source # log :: Pointwise x a -> Pointwise x a Source # (**) :: Pointwise x a -> Pointwise x a -> Pointwise x a Source # logBase :: Pointwise x a -> Pointwise x a -> Pointwise x a Source # sin :: Pointwise x a -> Pointwise x a Source # cos :: Pointwise x a -> Pointwise x a Source # tan :: Pointwise x a -> Pointwise x a Source # asin :: Pointwise x a -> Pointwise x a Source # acos :: Pointwise x a -> Pointwise x a Source # atan :: Pointwise x a -> Pointwise x a Source # sinh :: Pointwise x a -> Pointwise x a Source # cosh :: Pointwise x a -> Pointwise x a Source # tanh :: Pointwise x a -> Pointwise x a Source # asinh :: Pointwise x a -> Pointwise x a Source # acosh :: Pointwise x a -> Pointwise x a Source # atanh :: Pointwise x a -> Pointwise x a Source # log1p :: Pointwise x a -> Pointwise x a Source # expm1 :: Pointwise x a -> Pointwise x a Source # log1pexp :: Pointwise x a -> Pointwise x a Source # log1mexp :: Pointwise x a -> Pointwise x a Source #

(^?) :: Transcendental a => a -> a -> a infixr 8 Source #

class Algebraic a => AlgebraicallyClosed a where Source #

Minimal complete definition

rootOfUnity

Methods

imaginaryUnit :: a Source #

rootOfUnity :: Integer -> Integer -> a Source #

rootOfUnity n give the nth roots of unity. The 2nd argument specifies which one is demanded

Instances

Instances details

(RealFloat a, Transcendental a) => AlgebraicallyClosed (Complex a) Source #
Instance details Defined in Algebra.Classes Methods imaginaryUnit :: Complex a Source # rootOfUnity :: Integer -> Integer -> Complex a Source #

newtype Sum a Source #

Constructors

Sum
Fields fromSum :: a

Instances

Instances details

Additive a => Monoid (Sum a) Source #
Instance details Defined in Algebra.Classes Methods mempty :: Sum a # mappend :: Sum a -> Sum a -> Sum a # mconcat :: [Sum a] -> Sum a #
Additive a => Semigroup (Sum a) Source #
Instance details Defined in Algebra.Classes Methods (<>) :: Sum a -> Sum a -> Sum a # sconcat :: NonEmpty (Sum a) -> Sum a # stimes :: Integral b => b -> Sum a -> Sum a #
Generic (Sum a) Source #
Instance details Defined in Algebra.Classes Associated Types type Rep (Sum a) :: Type -> Type # Methods from :: Sum a -> Rep (Sum a) x # to :: Rep (Sum a) x -> Sum a #
Show a => Show (Sum a) Source #
Instance details Defined in Algebra.Classes Methods showsPrec :: Int -> Sum a -> ShowS # show :: Sum a -> String # showList :: [Sum a] -> ShowS #
Binary a => Binary (Sum a) Source #
Instance details Defined in Algebra.Classes Methods put :: Sum a -> Put # get :: Get (Sum a) # putList :: [Sum a] -> Put #
Eq a => Eq (Sum a) Source #
Instance details Defined in Algebra.Classes Methods (==) :: Sum a -> Sum a -> Bool # (/=) :: Sum a -> Sum a -> Bool #
Ord a => Ord (Sum a) Source #
Instance details Defined in Algebra.Classes Methods compare :: Sum a -> Sum a -> Ordering # (<) :: Sum a -> Sum a -> Bool # (<=) :: Sum a -> Sum a -> Bool # (>) :: Sum a -> Sum a -> Bool # (>=) :: Sum a -> Sum a -> Bool # max :: Sum a -> Sum a -> Sum a # min :: Sum a -> Sum a -> Sum a #
type Rep (Sum a) Source #
Instance details Defined in Algebra.Classes type Rep (Sum a) = D1 ('MetaData "Sum" "Algebra.Classes" "gasp-1.4.0.0-4Z31dYmTpJbCzuuYkBBWRm" 'True) (C1 ('MetaCons "Sum" 'PrefixI 'True) (S1 ('MetaSel ('Just "fromSum") 'NoSourceUnpackedness 'NoSourceStrictness 'DecidedLazy) (Rec0 a)))

newtype Product a Source #

Constructors

Product
Fields fromProduct :: a

Instances

Instances details

Multiplicative a => Monoid (Product a) Source #
Instance details Defined in Algebra.Classes Methods mempty :: Product a # mappend :: Product a -> Product a -> Product a # mconcat :: [Product a] -> Product a #
Multiplicative a => Semigroup (Product a) Source #
Instance details Defined in Algebra.Classes Methods (<>) :: Product a -> Product a -> Product a # sconcat :: NonEmpty (Product a) -> Product a # stimes :: Integral b => b -> Product a -> Product a #
Generic (Product a) Source #
Instance details Defined in Algebra.Classes Associated Types type Rep (Product a) :: Type -> Type # Methods from :: Product a -> Rep (Product a) x # to :: Rep (Product a) x -> Product a #
Show a => Show (Product a) Source #
Instance details Defined in Algebra.Classes Methods showsPrec :: Int -> Product a -> ShowS # show :: Product a -> String # showList :: [Product a] -> ShowS #
Eq a => Eq (Product a) Source #
Instance details Defined in Algebra.Classes Methods (==) :: Product a -> Product a -> Bool # (/=) :: Product a -> Product a -> Bool #
Ord a => Ord (Product a) Source #
Instance details Defined in Algebra.Classes Methods compare :: Product a -> Product a -> Ordering # (<) :: Product a -> Product a -> Bool # (<=) :: Product a -> Product a -> Bool # (>) :: Product a -> Product a -> Bool # (>=) :: Product a -> Product a -> Bool # max :: Product a -> Product a -> Product a # min :: Product a -> Product a -> Product a #
type Rep (Product a) Source #
Instance details Defined in Algebra.Classes type Rep (Product a) = D1 ('MetaData "Product" "Algebra.Classes" "gasp-1.4.0.0-4Z31dYmTpJbCzuuYkBBWRm" 'True) (C1 ('MetaCons "Product" 'PrefixI 'True) (S1 ('MetaSel ('Just "fromProduct") 'NoSourceUnpackedness 'NoSourceStrictness 'DecidedLazy) (Rec0 a)))

newtype App f x Source #

Constructors

App (f x)

Instances

Instances details

Applicative f => Applicative (App f) Source #
Instance details Defined in Algebra.Classes Methods pure :: a -> App f a # (<>) :: App f (a -> b) -> App f a -> App f b # liftA2 :: (a -> b -> c) -> App f a -> App f b -> App f c # (>) :: App f a -> App f b -> App f b # (<*) :: App f a -> App f b -> App f a #
Functor f => Functor (App f) Source #
Instance details Defined in Algebra.Classes Methods fmap :: (a -> b) -> App f a -> App f b # (<$) :: a -> App f b -> App f a #
(Applicative f, AbelianAdditive a) => AbelianAdditive (App f a) Source #
Instance details Defined in Algebra.Classes
(Applicative f, Additive a) => Additive (App f a) Source #
Instance details Defined in Algebra.Classes Methods (+) :: App f a -> App f a -> App f a Source # zero :: App f a Source # times :: Natural -> App f a -> App f a Source #
(Applicative f, Division s) => Division (App f s) Source #
Instance details Defined in Algebra.Classes Methods recip :: App f s -> App f s Source # (/) :: App f s -> App f s -> App f s Source # (^) :: App f s -> Integer -> App f s Source #
(Applicative f, Field s) => Field (App f s) Source #
Instance details Defined in Algebra.Classes Methods fromRational :: Rational -> App f s Source #
(Applicative f, Group a) => Group (App f a) Source #
Instance details Defined in Algebra.Classes Methods (-) :: App f a -> App f a -> App f a Source # subtract :: App f a -> App f a -> App f a Source # negate :: App f a -> App f a Source # mult :: Integer -> App f a -> App f a Source #
(Applicative f, Multiplicative a) => Multiplicative (App f a) Source #
Instance details Defined in Algebra.Classes Methods (*) :: App f a -> App f a -> App f a Source # one :: App f a Source # (^+) :: App f a -> Natural -> App f a Source #
(Applicative f, Ring a) => Ring (App f a) Source #
Instance details Defined in Algebra.Classes Methods fromInteger :: Integer -> App f a Source #
(Applicative f, Roots s) => Roots (App f s) Source #
Instance details Defined in Algebra.Classes Methods sqrt :: App f s -> App f s Source # root :: Integer -> App f s -> App f s Source # (^/) :: App f s -> Rational -> App f s Source #
(Applicative f, Transcendental s) => Transcendental (App f s) Source #
Instance details Defined in Algebra.Classes Methods pi :: App f s Source # exp :: App f s -> App f s Source # log :: App f s -> App f s Source # (**) :: App f s -> App f s -> App f s Source # logBase :: App f s -> App f s -> App f s Source # sin :: App f s -> App f s Source # cos :: App f s -> App f s Source # tan :: App f s -> App f s Source # asin :: App f s -> App f s Source # acos :: App f s -> App f s Source # atan :: App f s -> App f s Source # sinh :: App f s -> App f s Source # cosh :: App f s -> App f s Source # tanh :: App f s -> App f s Source # asinh :: App f s -> App f s Source # acosh :: App f s -> App f s Source # atanh :: App f s -> App f s Source # log1p :: App f s -> App f s Source # expm1 :: App f s -> App f s Source # log1pexp :: App f s -> App f s Source # log1mexp :: App f s -> App f s Source #
(Applicative f, Scalable s a) => Scalable (App f s) (App f a) Source #
Instance details Defined in Algebra.Classes Methods (*^) :: App f s -> App f a -> App f a Source #