Safe Haskell	None
Language	Haskell98

Numeric.Algebra.Complex

Synopsis

Documentation

class Distinguished t where Source #

Minimal complete definition

e

Methods

e :: t Source #

Instances

Distinguished TrigBasis Source #
Methods e :: TrigBasis Source #
Distinguished QuaternionBasis' Source #
Methods e :: QuaternionBasis' Source #
Distinguished DualBasis' Source #
Methods e :: DualBasis' Source #
Distinguished QuaternionBasis Source #
Methods e :: QuaternionBasis Source #
Distinguished DualBasis Source #
Methods e :: DualBasis Source #
Distinguished ComplexBasis Source #
Methods e :: ComplexBasis Source #
Rig r => Distinguished (Trig r) Source #
Methods e :: Trig r Source #
Rig r => Distinguished (Quaternion' r) Source #
Methods e :: Quaternion' r Source #
Rig r => Distinguished (Dual' r) Source #
Methods e :: Dual' r Source #
Rig r => Distinguished (Quaternion r) Source #
Methods e :: Quaternion r Source #
Rig r => Distinguished (Dual r) Source #
Methods e :: Dual r Source #
Rig r => Distinguished (Complex r) Source #
Methods e :: Complex r Source #
Rig r => Distinguished (TrigBasis -> r) Source #
Methods e :: TrigBasis -> r Source #
Rig r => Distinguished (QuaternionBasis' -> r) Source #
Methods e :: QuaternionBasis' -> r Source #
Rig r => Distinguished (DualBasis' -> r) Source #
Methods e :: DualBasis' -> r Source #
Rig r => Distinguished (QuaternionBasis -> r) Source #
Methods e :: QuaternionBasis -> r Source #
Rig r => Distinguished (DualBasis -> r) Source #
Methods e :: DualBasis -> r Source #
Rig r => Distinguished (ComplexBasis -> r) Source #
Methods e :: ComplexBasis -> r Source #
Distinguished a => Distinguished (Covector r a) Source #
Methods e :: Covector r a Source #

class Distinguished r => Complicated r where Source #

Minimal complete definition

i

Methods

i :: r Source #

Instances

Complicated TrigBasis Source #
Methods i :: TrigBasis Source #
Complicated QuaternionBasis' Source #
Methods i :: QuaternionBasis' Source #
Complicated QuaternionBasis Source #
Methods i :: QuaternionBasis Source #
Complicated ComplexBasis Source #
Methods i :: ComplexBasis Source #
Rig r => Complicated (Trig r) Source #
Methods i :: Trig r Source #
Rig r => Complicated (Quaternion' r) Source #
Methods i :: Quaternion' r Source #
Rig r => Complicated (Quaternion r) Source #
Methods i :: Quaternion r Source #
Rig r => Complicated (Complex r) Source #
Methods i :: Complex r Source #
Rig r => Complicated (TrigBasis -> r) Source #
Methods i :: TrigBasis -> r Source #
Rig r => Complicated (QuaternionBasis' -> r) Source #
Methods i :: QuaternionBasis' -> r Source #
Rig r => Complicated (QuaternionBasis -> r) Source #
Methods i :: QuaternionBasis -> r Source #
Rig r => Complicated (ComplexBasis -> r) Source #
Methods i :: ComplexBasis -> r Source #
Complicated a => Complicated (Covector r a) Source #
Methods i :: Covector r a Source #

data ComplexBasis Source #

Constructors

E
I

Instances

Bounded ComplexBasis Source #
Methods minBound :: ComplexBasis # maxBound :: ComplexBasis #
Enum ComplexBasis Source #
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 Source #
Methods (==) :: ComplexBasis -> ComplexBasis -> Bool # (/=) :: ComplexBasis -> ComplexBasis -> Bool #
Data ComplexBasis Source #
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 Source #
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 Source #
Methods readsPrec :: Int -> ReadS ComplexBasis # readList :: ReadS [ComplexBasis] # readPrec :: ReadPrec ComplexBasis # readListPrec :: ReadPrec [ComplexBasis] #
Show ComplexBasis Source #
Methods showsPrec :: Int -> ComplexBasis -> ShowS # show :: ComplexBasis -> String # showList :: [ComplexBasis] -> ShowS #
Ix ComplexBasis Source #
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 Source #
Methods e :: ComplexBasis Source #
Complicated ComplexBasis Source #
Methods i :: ComplexBasis Source #
MonadReader ComplexBasis Complex Source #
Methods ask :: Complex ComplexBasis # local :: (ComplexBasis -> ComplexBasis) -> Complex a -> Complex a # reader :: (ComplexBasis -> a) -> Complex a #
Rng k => Coalgebra k ComplexBasis Source #
Methods comult :: (ComplexBasis -> k) -> ComplexBasis -> ComplexBasis -> k Source #
Rng k => Algebra k ComplexBasis Source #
Methods mult :: (ComplexBasis -> ComplexBasis -> k) -> ComplexBasis -> k Source #
Rng k => Bialgebra k ComplexBasis Source #

Rng k => CounitalCoalgebra k ComplexBasis Source #
Methods counit :: (ComplexBasis -> k) -> k Source #
Rng k => UnitalAlgebra k ComplexBasis Source #
Methods unit :: k -> ComplexBasis -> k Source #
(InvolutiveSemiring k, Rng k) => HopfAlgebra k ComplexBasis Source #
Methods antipode :: (ComplexBasis -> k) -> ComplexBasis -> k Source #
(InvolutiveSemiring k, Rng k) => InvolutiveCoalgebra k ComplexBasis Source #
Methods coinv :: (ComplexBasis -> k) -> ComplexBasis -> k Source #
(InvolutiveSemiring k, Rng k) => InvolutiveAlgebra k ComplexBasis Source #
Methods inv :: (ComplexBasis -> k) -> ComplexBasis -> k Source #
Rig r => Distinguished (ComplexBasis -> r) Source #
Methods e :: ComplexBasis -> r Source #
Rig r => Complicated (ComplexBasis -> r) Source #
Methods i :: ComplexBasis -> r Source #

data Complex a Source #

Constructors

Complex a a

Instances

Monad Complex Source #
Methods (>>=) :: Complex a -> (a -> Complex b) -> Complex b # (>>) :: Complex a -> Complex b -> Complex b # return :: a -> Complex a # fail :: String -> Complex a #
Functor Complex Source #
Methods fmap :: (a -> b) -> Complex a -> Complex b # (<$) :: a -> Complex b -> Complex a #
Applicative Complex Source #
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 Source #
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 Source #
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 Source #
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 Source #
Associated Types type Rep (Complex :: * -> ) :: # Methods tabulate :: (Rep Complex -> a) -> Complex a # index :: Complex a -> Rep Complex -> a #
Traversable1 Complex Source #
Methods traverse1 :: Apply f => (a -> f b) -> Complex a -> f (Complex b) # sequence1 :: Apply f => Complex (f b) -> f (Complex b) #
Foldable1 Complex Source #
Methods fold1 :: Semigroup m => Complex m -> m # foldMap1 :: Semigroup m => (a -> m) -> Complex a -> m # toNonEmpty :: Complex a -> NonEmpty a #
Apply Complex Source #
Methods (<.>) :: Complex (a -> b) -> Complex a -> Complex b # (.>) :: Complex a -> Complex b -> Complex b # (<.) :: Complex a -> Complex b -> Complex a #
Bind Complex Source #
Methods (>>-) :: Complex a -> (a -> Complex b) -> Complex b # join :: Complex (Complex a) -> Complex a #
MonadReader ComplexBasis Complex Source #
Methods ask :: Complex ComplexBasis # local :: (ComplexBasis -> ComplexBasis) -> Complex a -> Complex a # reader :: (ComplexBasis -> a) -> Complex a #
RightModule r s => RightModule r (Complex s) Source #
Methods (*.) :: Complex s -> r -> Complex s Source #
LeftModule r s => LeftModule r (Complex s) Source #
Methods (.*) :: r -> Complex s -> Complex s Source #
(Commutative r, Rng r, InvolutiveSemiring r) => Quadrance r (Complex r) Source #
Methods quadrance :: Complex r -> r Source #
Eq a => Eq (Complex a) Source #
Methods (==) :: Complex a -> Complex a -> Bool # (/=) :: Complex a -> Complex a -> Bool #
Data a => Data (Complex a) Source #
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) Source #
Methods readsPrec :: Int -> ReadS (Complex a) # readList :: ReadS [Complex a] # readPrec :: ReadPrec (Complex a) # readListPrec :: ReadPrec [Complex a] #
Show a => Show (Complex a) Source #
Methods showsPrec :: Int -> Complex a -> ShowS # show :: Complex a -> String # showList :: [Complex a] -> ShowS #
Idempotent r => Idempotent (Complex r) Source #

Abelian r => Abelian (Complex r) Source #

Partitionable r => Partitionable (Complex r) Source #
Methods partitionWith :: (Complex r -> Complex r -> r) -> Complex r -> NonEmpty r Source #
Additive r => Additive (Complex r) Source #
Methods (+) :: Complex r -> Complex r -> Complex r Source # sinnum1p :: Natural -> Complex r -> Complex r Source # sumWith1 :: Foldable1 f => (a -> Complex r) -> f a -> Complex r Source #
Monoidal r => Monoidal (Complex r) Source #
Methods zero :: Complex r Source # sinnum :: Natural -> Complex r -> Complex r Source # sumWith :: Foldable f => (a -> Complex r) -> f a -> Complex r Source #
(Commutative r, Rng r) => Semiring (Complex r) Source #

(Commutative r, Rng r) => Multiplicative (Complex r) Source #
Methods (*) :: Complex r -> Complex r -> Complex r Source # pow1p :: Complex r -> Natural -> Complex r Source # productWith1 :: Foldable1 f => (a -> Complex r) -> f a -> Complex r Source #
Group r => Group (Complex r) Source #
Methods (-) :: Complex r -> Complex r -> Complex r Source # negate :: Complex r -> Complex r Source # subtract :: Complex r -> Complex r -> Complex r Source # times :: Integral n => n -> Complex r -> Complex r Source #
(Commutative r, Ring r) => Unital (Complex r) Source #
Methods one :: Complex r Source # pow :: Complex r -> Natural -> Complex r Source # productWith :: Foldable f => (a -> Complex r) -> f a -> Complex r Source #
(Commutative r, InvolutiveSemiring r, DivisionRing r) => Division (Complex r) Source #
Methods recip :: Complex r -> Complex r Source # (/) :: Complex r -> Complex r -> Complex r Source # (\\) :: Complex r -> Complex r -> Complex r Source # (^) :: Integral n => Complex r -> n -> Complex r Source #
(TriviallyInvolutive r, Rng r) => Commutative (Complex r) Source #

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

(Commutative r, Rng r, InvolutiveMultiplication r) => InvolutiveMultiplication (Complex r) Source #
Methods adjoint :: Complex r -> Complex r Source #
(Commutative r, Ring r) => Rig (Complex r) Source #
Methods fromNatural :: Natural -> Complex r Source #
(Commutative r, Ring r) => Ring (Complex r) Source #
Methods fromInteger :: Integer -> Complex r Source #
Rig r => Distinguished (Complex r) Source #
Methods e :: Complex r Source #
Rig r => Complicated (Complex r) Source #
Methods i :: Complex r Source #
(Commutative r, Rng r) => RightModule (Complex r) (Complex r) Source #
Methods (*.) :: Complex r -> Complex r -> Complex r Source #
(Commutative r, Rng r) => LeftModule (Complex r) (Complex r) Source #
Methods (.*) :: Complex r -> Complex r -> Complex r Source #
type Rep Complex Source #
type Rep Complex = ComplexBasis

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

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

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

half of the Cayley-Dickson quaternion isomorphism