Numeric.Algebra.Complex

Documentation

class Distinguished t where

Minimal complete definition

e

Methods

e :: t

Instances

Distinguished TrigBasis

Methods

e :: TrigBasis

Distinguished QuaternionBasis'

Methods

e :: QuaternionBasis'

Distinguished DualBasis'

Methods

e :: DualBasis'

Distinguished QuaternionBasis

Methods

e :: QuaternionBasis

Distinguished DualBasis

Methods

e :: DualBasis

Distinguished ComplexBasis

Methods

e :: ComplexBasis

Rig r => Distinguished (Trig r)

Methods

e :: Trig r

Rig r => Distinguished (Quaternion' r)

Methods

e :: Quaternion' r

Rig r => Distinguished (Dual' r)

Methods

e :: Dual' r

Rig r => Distinguished (Quaternion r)

Methods

e :: Quaternion r

Rig r => Distinguished (Dual r)

Methods

e :: Dual r

Rig r => Distinguished (Complex r)

Methods

e :: Complex r

Rig r => Distinguished (TrigBasis -> r)

Methods

e :: TrigBasis -> r

Rig r => Distinguished (QuaternionBasis' -> r)

Methods

e :: QuaternionBasis' -> r

Rig r => Distinguished (DualBasis' -> r)

Methods

e :: DualBasis' -> r

Rig r => Distinguished (QuaternionBasis -> r)

Methods

e :: QuaternionBasis -> r

Rig r => Distinguished (DualBasis -> r)

Methods

e :: DualBasis -> r

Rig r => Distinguished (ComplexBasis -> r)

Methods

e :: ComplexBasis -> r

Distinguished a => Distinguished (Covector r a)

Methods

e :: Covector r a

class Distinguished r => Complicated r where

Minimal complete definition

i

Methods

i :: r

Instances

Complicated TrigBasis

Methods

i :: TrigBasis

Complicated QuaternionBasis'

Methods

i :: QuaternionBasis'

Complicated QuaternionBasis

Methods

i :: QuaternionBasis

Complicated ComplexBasis

Methods

i :: ComplexBasis

Rig r => Complicated (Trig r)

Methods

i :: Trig r

Rig r => Complicated (Quaternion' r)

Methods

i :: Quaternion' r

Rig r => Complicated (Quaternion r)

Methods

i :: Quaternion r

Rig r => Complicated (Complex r)

Methods

i :: Complex r

Rig r => Complicated (TrigBasis -> r)

Methods

i :: TrigBasis -> r

Rig r => Complicated (QuaternionBasis' -> r)

Methods

i :: QuaternionBasis' -> r

Rig r => Complicated (QuaternionBasis -> r)

Methods

i :: QuaternionBasis -> r

Rig r => Complicated (ComplexBasis -> r)

Methods

i :: ComplexBasis -> r

Complicated a => Complicated (Covector r a)

Methods

i :: Covector r a

data ComplexBasis

Constructors

E

I

Instances

Bounded ComplexBasis

Methods

minBound :: ComplexBasis

maxBound :: ComplexBasis

Enum ComplexBasis

Methods

succ :: ComplexBasis -> ComplexBasis

pred :: ComplexBasis -> ComplexBasis

toEnum :: Int -> ComplexBasis

fromEnum :: ComplexBasis -> Int

enumFrom :: ComplexBasis -> [ComplexBasis]

enumFromThen :: ComplexBasis -> ComplexBasis -> [ComplexBasis]

enumFromTo :: ComplexBasis -> ComplexBasis -> [ComplexBasis]

enumFromThenTo :: ComplexBasis -> ComplexBasis -> ComplexBasis -> [ComplexBasis]

Eq ComplexBasis

Methods

(==) :: ComplexBasis -> ComplexBasis -> Bool

(/=) :: ComplexBasis -> ComplexBasis -> Bool

Data ComplexBasis

Methods

gfoldl :: (forall d b. Data d => c (d -> b) -> d -> c b) -> (forall g. g -> c g) -> ComplexBasis -> c ComplexBasis

gunfold :: (forall b r. Data b => c (b -> r) -> c r) -> (forall r. r -> c r) -> Constr -> c ComplexBasis

toConstr :: ComplexBasis -> Constr

dataTypeOf :: ComplexBasis -> DataType

dataCast1 :: Typeable (* -> *) t => (forall d. Data d => c (t d)) -> Maybe (c ComplexBasis)

dataCast2 :: Typeable (* -> * -> *) t => (forall d e. (Data d, Data e) => c (t d e)) -> Maybe (c ComplexBasis)

gmapT :: (forall b. Data b => b -> b) -> ComplexBasis -> ComplexBasis

gmapQl :: (r -> r' -> r) -> r -> (forall d. Data d => d -> r') -> ComplexBasis -> r

gmapQr :: (r' -> r -> r) -> r -> (forall d. Data d => d -> r') -> ComplexBasis -> r

gmapQ :: (forall d. Data d => d -> u) -> ComplexBasis -> [u]

gmapQi :: Int -> (forall d. Data d => d -> u) -> ComplexBasis -> u

gmapM :: Monad m => (forall d. Data d => d -> m d) -> ComplexBasis -> m ComplexBasis

gmapMp :: MonadPlus m => (forall d. Data d => d -> m d) -> ComplexBasis -> m ComplexBasis

gmapMo :: MonadPlus m => (forall d. Data d => d -> m d) -> ComplexBasis -> m ComplexBasis

Ord ComplexBasis

Methods

compare :: ComplexBasis -> ComplexBasis -> Ordering

(<) :: ComplexBasis -> ComplexBasis -> Bool

(<=) :: ComplexBasis -> ComplexBasis -> Bool

(>) :: ComplexBasis -> ComplexBasis -> Bool

(>=) :: ComplexBasis -> ComplexBasis -> Bool

max :: ComplexBasis -> ComplexBasis -> ComplexBasis

min :: ComplexBasis -> ComplexBasis -> ComplexBasis

Read ComplexBasis

Methods

readsPrec :: Int -> ReadS ComplexBasis

readList :: ReadS [ComplexBasis]

readPrec :: ReadPrec ComplexBasis

readListPrec :: ReadPrec [ComplexBasis]

Show ComplexBasis

Methods

showsPrec :: Int -> ComplexBasis -> ShowS

show :: ComplexBasis -> String

showList :: [ComplexBasis] -> ShowS

Ix ComplexBasis

Methods

range :: (ComplexBasis, ComplexBasis) -> [ComplexBasis]

index :: (ComplexBasis, ComplexBasis) -> ComplexBasis -> Int

unsafeIndex :: (ComplexBasis, ComplexBasis) -> ComplexBasis -> Int

inRange :: (ComplexBasis, ComplexBasis) -> ComplexBasis -> Bool

rangeSize :: (ComplexBasis, ComplexBasis) -> Int

unsafeRangeSize :: (ComplexBasis, ComplexBasis) -> Int

Distinguished ComplexBasis

Methods

e :: ComplexBasis

Complicated ComplexBasis

Methods

i :: ComplexBasis

MonadReader ComplexBasis Complex

Methods

ask :: Complex ComplexBasis

local :: (ComplexBasis -> ComplexBasis) -> Complex a -> Complex a

reader :: (ComplexBasis -> a) -> Complex a

Rng k => Coalgebra k ComplexBasis

Methods

comult :: (ComplexBasis -> k) -> ComplexBasis -> ComplexBasis -> k

Rng k => Algebra k ComplexBasis

Methods

mult :: (ComplexBasis -> ComplexBasis -> k) -> ComplexBasis -> k

Rng k => Bialgebra k ComplexBasis

Rng k => CounitalCoalgebra k ComplexBasis

Methods

counit :: (ComplexBasis -> k) -> k

Rng k => UnitalAlgebra k ComplexBasis

Methods

unit :: k -> ComplexBasis -> k

(InvolutiveSemiring k, Rng k) => HopfAlgebra k ComplexBasis

Methods

antipode :: (ComplexBasis -> k) -> ComplexBasis -> k

(InvolutiveSemiring k, Rng k) => InvolutiveCoalgebra k ComplexBasis

Methods

coinv :: (ComplexBasis -> k) -> ComplexBasis -> k

(InvolutiveSemiring k, Rng k) => InvolutiveAlgebra k ComplexBasis

Methods

inv :: (ComplexBasis -> k) -> ComplexBasis -> k

Rig r => Distinguished (ComplexBasis -> r)

Methods

e :: ComplexBasis -> r

Rig r => Complicated (ComplexBasis -> r)

Methods

i :: ComplexBasis -> r

data Complex a

Constructors

Complex a a

Instances

Monad Complex

Methods

(>>=) :: Complex a -> (a -> Complex b) -> Complex b

(>>) :: Complex a -> Complex b -> Complex b

return :: a -> Complex a

fail :: String -> Complex a

Functor Complex

Methods

fmap :: (a -> b) -> Complex a -> Complex b

(<$) :: a -> Complex b -> Complex a

Applicative Complex

Methods

pure :: a -> Complex a

(<*>) :: Complex (a -> b) -> Complex a -> Complex b

liftA2 :: (a -> b -> c) -> Complex a -> Complex b -> Complex c

(*>) :: Complex a -> Complex b -> Complex b

(<*) :: Complex a -> Complex b -> Complex a

Foldable Complex

Methods

fold :: Monoid m => Complex m -> m

foldMap :: Monoid m => (a -> m) -> Complex a -> m

foldr :: (a -> b -> b) -> b -> Complex a -> b

foldr' :: (a -> b -> b) -> b -> Complex a -> b

foldl :: (b -> a -> b) -> b -> Complex a -> b

foldl' :: (b -> a -> b) -> b -> Complex a -> b

foldr1 :: (a -> a -> a) -> Complex a -> a

foldl1 :: (a -> a -> a) -> Complex a -> a

toList :: Complex a -> [a]

null :: Complex a -> Bool

length :: Complex a -> Int

elem :: Eq a => a -> Complex a -> Bool

maximum :: Ord a => Complex a -> a

minimum :: Ord a => Complex a -> a

sum :: Num a => Complex a -> a

product :: Num a => Complex a -> a

Traversable Complex

Methods

traverse :: Applicative f => (a -> f b) -> Complex a -> f (Complex b)

sequenceA :: Applicative f => Complex (f a) -> f (Complex a)

mapM :: Monad m => (a -> m b) -> Complex a -> m (Complex b)

sequence :: Monad m => Complex (m a) -> m (Complex a)

Distributive Complex

Methods

distribute :: Functor f => f (Complex a) -> Complex (f a)

collect :: Functor f => (a -> Complex b) -> f a -> Complex (f b)

distributeM :: Monad m => m (Complex a) -> Complex (m a)

collectM :: Monad m => (a -> Complex b) -> m a -> Complex (m b)

Representable Complex

Associated Types

type Rep (Complex :: * -> *) :: *

Methods

tabulate :: (Rep Complex -> a) -> Complex a

index :: Complex a -> Rep Complex -> a

Traversable1 Complex

Methods

traverse1 :: Apply f => (a -> f b) -> Complex a -> f (Complex b)

sequence1 :: Apply f => Complex (f b) -> f (Complex b)

Foldable1 Complex

Methods

fold1 :: Semigroup m => Complex m -> m

foldMap1 :: Semigroup m => (a -> m) -> Complex a -> m

toNonEmpty :: Complex a -> NonEmpty a

Apply Complex

Methods

(<.>) :: Complex (a -> b) -> Complex a -> Complex b

(.>) :: Complex a -> Complex b -> Complex b

(<.) :: Complex a -> Complex b -> Complex a

Bind Complex

Methods

(>>-) :: Complex a -> (a -> Complex b) -> Complex b

join :: Complex (Complex a) -> Complex a

MonadReader ComplexBasis Complex

Methods

ask :: Complex ComplexBasis

local :: (ComplexBasis -> ComplexBasis) -> Complex a -> Complex a

reader :: (ComplexBasis -> a) -> Complex a

RightModule r s => RightModule r (Complex s)

Methods

(*.) :: Complex s -> r -> Complex s

LeftModule r s => LeftModule r (Complex s)

Methods

(.*) :: r -> Complex s -> Complex s

(Commutative r, Rng r, InvolutiveSemiring r) => Quadrance r (Complex r)

Methods

quadrance :: Complex r -> r

Eq a => Eq (Complex a)

Methods

(==) :: Complex a -> Complex a -> Bool

(/=) :: Complex a -> Complex a -> Bool

Data a => Data (Complex a)

Methods

gfoldl :: (forall d b. Data d => c (d -> b) -> d -> c b) -> (forall g. g -> c g) -> Complex a -> c (Complex a)

gunfold :: (forall b r. Data b => c (b -> r) -> c r) -> (forall r. r -> c r) -> Constr -> c (Complex a)

toConstr :: Complex a -> Constr

dataTypeOf :: Complex a -> DataType

dataCast1 :: Typeable (* -> *) t => (forall d. Data d => c (t d)) -> Maybe (c (Complex a))

dataCast2 :: Typeable (* -> * -> *) t => (forall d e. (Data d, Data e) => c (t d e)) -> Maybe (c (Complex a))

gmapT :: (forall b. Data b => b -> b) -> Complex a -> Complex a

gmapQl :: (r -> r' -> r) -> r -> (forall d. Data d => d -> r') -> Complex a -> r

gmapQr :: (r' -> r -> r) -> r -> (forall d. Data d => d -> r') -> Complex a -> r

gmapQ :: (forall d. Data d => d -> u) -> Complex a -> [u]

gmapQi :: Int -> (forall d. Data d => d -> u) -> Complex a -> u

gmapM :: Monad m => (forall d. Data d => d -> m d) -> Complex a -> m (Complex a)

gmapMp :: MonadPlus m => (forall d. Data d => d -> m d) -> Complex a -> m (Complex a)

gmapMo :: MonadPlus m => (forall d. Data d => d -> m d) -> Complex a -> m (Complex a)

Read a => Read (Complex a)

Methods

readsPrec :: Int -> ReadS (Complex a)

readList :: ReadS [Complex a]

readPrec :: ReadPrec (Complex a)

readListPrec :: ReadPrec [Complex a]

Show a => Show (Complex a)

Methods

showsPrec :: Int -> Complex a -> ShowS

show :: Complex a -> String

showList :: [Complex a] -> ShowS

Idempotent r => Idempotent (Complex r)

Abelian r => Abelian (Complex r)

Partitionable r => Partitionable (Complex r)

Methods

partitionWith :: (Complex r -> Complex r -> r) -> Complex r -> NonEmpty r

Additive r => Additive (Complex r)

Methods

(+) :: Complex r -> Complex r -> Complex r

sinnum1p :: Natural -> Complex r -> Complex r

sumWith1 :: Foldable1 f => (a -> Complex r) -> f a -> Complex r

Monoidal r => Monoidal (Complex r)

Methods

zero :: Complex r

sinnum :: Natural -> Complex r -> Complex r

sumWith :: Foldable f => (a -> Complex r) -> f a -> Complex r

(Commutative r, Rng r) => Semiring (Complex r)

(Commutative r, Rng r) => Multiplicative (Complex r)

Methods

(*) :: Complex r -> Complex r -> Complex r

pow1p :: Complex r -> Natural -> Complex r

productWith1 :: Foldable1 f => (a -> Complex r) -> f a -> Complex r

Group r => Group (Complex r)

Methods

(-) :: Complex r -> Complex r -> Complex r

negate :: Complex r -> Complex r

subtract :: Complex r -> Complex r -> Complex r

times :: Integral n => n -> Complex r -> Complex r

(Commutative r, Ring r) => Unital (Complex r)

Methods

one :: Complex r

pow :: Complex r -> Natural -> Complex r

productWith :: Foldable f => (a -> Complex r) -> f a -> Complex r

(Commutative r, InvolutiveSemiring r, DivisionRing r) => Division (Complex r)

Methods

recip :: Complex r -> Complex r

(/) :: Complex r -> Complex r -> Complex r

(\\) :: Complex r -> Complex r -> Complex r

(^) :: Integral n => Complex r -> n -> Complex r

(TriviallyInvolutive r, Rng r) => Commutative (Complex r)

(Commutative r, Rng r, InvolutiveSemiring r) => InvolutiveSemiring (Complex r)

(Commutative r, Rng r, InvolutiveMultiplication r) => InvolutiveMultiplication (Complex r)

Methods

adjoint :: Complex r -> Complex r

(Commutative r, Ring r) => Rig (Complex r)

Methods

fromNatural :: Natural -> Complex r

(Commutative r, Ring r) => Ring (Complex r)

Methods

fromInteger :: Integer -> Complex r

Rig r => Distinguished (Complex r)

Methods

e :: Complex r

Rig r => Complicated (Complex r)

Methods

i :: Complex r

(Commutative r, Rng r) => RightModule (Complex r) (Complex r)

Methods

(*.) :: Complex r -> Complex r -> Complex r

(Commutative r, Rng r) => LeftModule (Complex r) (Complex r)

Methods

(.*) :: Complex r -> Complex r -> Complex r

type Rep Complex

type Rep Complex = ComplexBasis

realPart :: (Representable f, Rep f ~ ComplexBasis) => f a -> a

imagPart :: (Representable f, Rep f ~ ComplexBasis) => f a -> a

uncomplicate :: Hamiltonian q => ComplexBasis -> ComplexBasis -> q

half of the Cayley-Dickson quaternion isomorphism